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Portfolio Optimization
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Portfolio Optimization

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AACIMP 2010 Summer School lecture by Gerhard Wilhelm Weber. "Applied Mathematics" stream. "Modern Operational Research and Its Mathematical Methods with a Focus on Financial Mathematics" course. Part …

AACIMP 2010 Summer School lecture by Gerhard Wilhelm Weber. "Applied Mathematics" stream. "Modern Operational Research and Its Mathematical Methods with a Focus on Financial Mathematics" course. Part 1.
More info at http://summerschool.ssa.org.ua

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  • 1. Portfolio Optimization Gerhard-Wilhelm Weber1 Erik Kropat2 Zafer-Korcan Görgülü3 1 Institute of Applied Mathematics Middle East Technical University Ankara, Turkey 2 Department of Mathematics University of Erlangen-Nuremberg Erlangen, Germany 3 University of the Federal Armed Forces Munich, Germany 2008 1 / 477
  • 2. Outline I 1 The Mean-Variance Approach in a One-Period Model Introduction 2 The Continuous-Time Market Model Modeling the Security Prices Excursion 1: Brownian Motion and Martingales Continuation: Modeling the Security prices ˆ Excursion 2: The Ito Integral ˆ Excursion 3: The Ito Formula Trading Strategy and Wealth Process Properties of the Continuous-Time Market Model Excursion 4: The Martingale Representation Theorem 2 / 477
  • 3. Outline II 3 Option Pricing Introduction Examples The Replication Principle Arbitrage Opportunity Continuation Partial Differential Approach (PDA) Arbitrage & Option Pricing 3 / 477
  • 4. Outline III 4 Pricing of Exotic Options and Numerical Algorithms Introduction Examples Examples Equivalent Martingale Measure Exotic Options with Explicit Pricing Formulae Weak Convergence of Stochastic Processes Monte-Carlo Simulation Approximation via Binomial Trees The Pathwise Binomial Approach of Rogers and Stapleton 4 / 477
  • 5. Outline IV 5 Optimal Portfolios Introduction and Formulation of the Problem The martingale method Optimal Option Portfolios Excursion 8: Stochastic Control Maximize expected value in presence of quadratic control costs Introduction Portfolio Optimization via Stochastic Control Method 5 / 477
  • 6. Outline 1 The Mean-Variance Approach in a One-Period Model 6 / 477
  • 7. Outline 1 The Mean-Variance Approach in a One-Period Model Introduction 7 / 477
  • 8. Introduction MVA Based on H. M ARKOWITZ OPM • Decisions on investment strategies only at the beginning of the period • Consequences of these decisions will be observed at the end of the period (−→ no action in between: static model) 8 / 477
  • 9. The one-period model Market with d traded securities d different securities with positive prices p1 , . . . , pd at time t = 0 Security prices P1 (T ), . . . , Pd (T ) at final time t = T not foreseeable −→ modeled as non-negative random variables on probability space (Ω, F , P) 9 / 477
  • 10. Securities in a OPM Returns of Securities Pi (T ) Ri (T ) := pi (1 ≤ i ≤ d ) Estimated Means, Variances and Covariances E (Ri (T )) = µi , Cov Ri (T ), Rj (T ) = σij (1 ≤ i ≤ d ) Remark The matrix σ := σij i,j∈{1,...,d} is positive semi-definite as it is a variance-covariance matrix. 10 / 477
  • 11. Securities in a OPM Each security perfectly divisable Hold ϕi ∈ R shares of security i (1 ≤ i ≤ d ) Negative position (ϕi < 0 for some i) corresponds to a selling −→ Not allowed in OPM −→ No negative positions: pi ≥ 0 (1 ≤ i ≤ d) −→ No transaction costs 11 / 477
  • 12. Budget equation and portfolio return The Budget Equation Investor with initial wealth x > 0 holds ϕi ≥ 0 shares of security i with ϕi · pi = x 1≤i≤d The Portfolio Vector π := (π1 , . . . , πd )T ϕi · pi πi := (1 ≤ i ≤ d ) x Portfolio Return R π := πi · Ri (T ) = π T R 1≤i≤d 12 / 477
  • 13. Budget equation and portfolio return Remark πi . . . fraction of total wealth invested in security i ϕi · pi 1≤i≤d x πi = = =1 x x 1≤i≤d X π (T ) . . . final wealth corresponding to x and π X π (T ) = ϕi · Pi (T ) 1≤i≤d 13 / 477
  • 14. Budget equation and portfolio return Remark (continued) Portfolio Return ϕi · pi Pi (T ) X π (T ) Rπ = πi · Ri (T ) = · = x pi x 1≤i≤d 1≤i≤d Portfolio Mean and Portfolio Variance E (R π ) = πi · µi , Var (R π ) = πi · σij · πj 1≤i≤d 1≤i,j≤d 14 / 477
  • 15. Selection of a portfolio–criterion (i) Maximize mean return (choose security of highest mean return) −→ risky, big fluctuations of return (ii) Minimize risk of fluction 15 / 477
  • 16. Selection of a portfolio–approach by Markowitz (MVA) Balance Risk (Portfolio Variance) and Return (Portfolio Mean) (i) Maximize E (R π ) under given upper bound c1 for Var (R π )   πi ≥ 0 (1 ≤ i ≤ d )    π πi = 1 max E (R ) subject to π∈Rd  1≤i≤d    Var (R π ) ≤ c1 (ii) Minimize Var (R π ) under given lower bound c2 for E (R π )   πi ≥ 0  (1 ≤ i ≤ d )  min Var (R π ) subject to πi = 1 π∈Rd   1≤i≤d  E (R π ) ≥ c2 16 / 477
  • 17. Solution methods (i) Linear Optimization Problem with quadratic constraint −→ No standard algorithms, numerical inefficient (ii) Quadratic Optimization Problem with positive semidefinite objective matrix σ −→ efficient algorithms (i.e., G OLDFARB/I DNANI, G ILL/M URRAY) Feasible region non-empty if c2 ≤ max µi 1≤i≤d σ positive definite and feasible region non-empty −→ unique solution (even if one security riskless) 17 / 477
  • 18. Relations between the formulations (i) and (ii) Theorem Assume: σ positive definite min µi ≤ c2 ≤ max µi c2 ∈ R+ 0 1≤i≤d 1≤i≤d min σ 2 (π) ≤ c1 ≤ max σ 2 (π) c1 ∈ R+ 0 πi ≥0, 1≤i≤d πi =1 πi ≥0, 1≤i≤d πi =1 Then ∗ (1) π ∗ solves (i) with Var R π = c1 =⇒ π ∗ solves (ii) with ∗ c2 := E R π (2) π solves (ii) with E R π = c2 =⇒ π solves (i) with c1 := Var R π 18 / 477
  • 19. The diversification effect–example Holding different Securities reduces Variance Both security prices fluctuate randomly σ11 , σ22 > 0 independent σ12 = σ21 = 0 0.5 Then for the Portfolio π = we get 0.5 σ11 σ22 Var (R π ) = Var (0.5 · R1 + 0.5 · R2 ) = + 4 4 19 / 477
  • 20. The diversification effect–example Holding different Securities reduces Variance 0.5 −→ If σ11 = σ22 then the Variance of Portfolio is half as big 0.5 1 0 as that of or 0 1 −→ Reduction of Variance . . . Diversification Effect depends on number of traded securities 20 / 477
  • 21. Example Mean-Variance Criterion Investing into seemingly bad security can be optimal. Let be 1 0.1 −0.1 µ= , σ= 0.9 −0.1 0.15 Formulation (ii) becomes (II) min Var (R π ) = min 2 2 0.1 · π1 + 0.15 · π2 − 0.2 · π1 π2 π π   π1 , π2 ≥ 0 subject to π1 + π2 = 1  E (R π ) = π1 + 0.9 · π2 ≥ 0.96 21 / 477
  • 22. Example 1 0.5 Consider Portfolios and (does not satify 0 0.5 expectation constraint) T T Var R (1,0) = 0.1 , E R (1,0) =1 T T Var R (0.5,0.5) = 0.125 , E R (0.5,0.5) = 0.95 22 / 477
  • 23. Example Ignore expectation constraint and remember π1 , π2 ≥ 0 π1 + π2 = 1. Hence min 0.1 · π1 + 0.15 · (1 − π1 )2 − 0.2 · π1 · (1 − π1 ) 2 π 2 = min 0.45 · π1 − 0.5 · π1 + 0.15 π 0.5 −→ Minimizing Portfolio (No solution of (II) but better than ) 0.5 1 5 π= · 9 4 T −→ Portfolio Return Variance Var R ( 9 , 9 ) 5 4 ¯ = 0.001 T −→ Portfolio Return Mean E R ( 9 , 9 ) 5 4 ¯ = 0.95 23 / 477
  • 24. Example 1.0 π2 0.5 0.4 0.0 0.0 0.5 0.6 1.0 π1 Pairs (π1 , π2 ) satisfying expectation constraint are above the dotted line Intersect with line π1 + π2 = 1 −→ Feasible region of MeanVariance Problem (bold line) 24 / 477
  • 25. Example 0.15 0.1 Var 0.05 0 0 0.5 0.6 1.0 1.5 π1 Portfolio Return Variance (as function of π1 ) of all pairs satisfying π1 + π2 = 1 Minimum in feasible region π ∈ [0.6, 1] is attained at π = 0.6 Optimal Portfolio in (II) →∗ = − 0.6 ∗ ∗ π with Var R π = 0.012 , E Rπ = 0.96 0.4 25 / 477
  • 26. Stock price model OPM No assumption on distribution of security returns Solving MV Problem just needed expectations and covariances 26 / 477
  • 27. Stock price model OPM with just one security (price p1 at time t = 0 ) At time T security may have price d · p1 or u · p1 q: probability of decreasing by factor d 1−q : probability of increasing by factor u (u > d ) Mean and Variance of Return P1 (T ) E (R1 (T )) =E = q · u + (1 − q) · d p1 P1 (T ) Var (R1 (T )) = Var = q · u 2 + (1 − q) · d 2 p1 − (q · u + (1 − q) · d )2 27 / 477
  • 28. Stock price model OPM with just one security (price p1 at time t = 0 ) After n periods the security has price P1 (n · T ) = p1 · u Xn · d n−Xn with Xn ∼ B(n, p) number of up-movements of price in n periods 28 / 477
  • 29. Comments on MVA Only trading at initial time t = 0 No reaction to current price changes possible ( −→ static model) Risk of investment only modeled by variance of return Need of Continuous-Time Market Models Discrete-time multi-period models (many periods −→ no fast algorithms) 29 / 477
  • 30. Outline 2 The Continuous-Time Market Model 30 / 477
  • 31. Outline 2 The Continuous-Time Market Model Modeling the Security Prices Excursion 1: Brownian Motion and Martingales Continuation: Modeling the Security prices ˆ Excursion 2: The Ito Integral ˆ Excursion 3: The Ito Formula Trading Strategy and Wealth Process Properties of the Continuous-Time Market Model Excursion 4: The Martingale Representation Theorem 31 / 477
  • 32. Modeling the security prices Market with d+1 securities d risky stocks with prices p1 , p2 , . . . , pd at time t = 0 and random prices P1 (t), P2 (t), . . . , Pd (t) at times t > 0 1 bond with price p0 at time t = 0 and deterministic price P0 (t) at times t > 0. Assume: Perfectly devisible securities, no transaction costs. ⇒ Modeling of the price development on the time interval [0, T ]. 32 / 477
  • 33. The bond price Assume: Continuous compounding of interest at a constant rate r : Bond price: P0 (t) = p0 · er ·t for t ∈ [0, T ] a non-constant, time-dependent and integrable rate r (t): t r (s) ds Bond price: P0 (t) = p0 · e 0 for t ∈ [0, T ] 33 / 477
  • 34. The stock price Stock price = random fluctuations around an intrinsic bond part 2 1.8 1.6 1.4 1.2 1 0.8 0 0.2 0.4 0.6 0.8 1 log-linear model for a stock price ln(Pi (t)) = ln(pi ) + bi · t + ”randomness” 34 / 477
  • 35. The stock price Randomness is assumed to have no tendency, i.e., E("randomness") = 0, to be time-dependent, to represent the sum of all deviations of ln(Pi (t)) from ln(pi ) + bi · t on [0, T ], ∼ N (0, σ 2 t) for some σ > 0. 35 / 477
  • 36. The stock price Deviation at time t Y (t) := ln(Pi (t)) − ln(pi ) − bi · t with Y (t) ∼ N (0, σ 2 t) Properties: E (Y (t)) = 0, Y (t) is time-dependent. 36 / 477
  • 37. The stock price Y (t) = Y (δ) + (Y (t) − Y (δ)), δ ∈ (0, t) Distribution of the increments of the deviation Y (t) − Y (δ) depends only on the time span t − δ is independent of Y (s), s ≤ δ =⇒ Y (t) − Y (δ) ∼ N 0, σ 2 (t − δ) 37 / 477
  • 38. The stock price Existence and properties of the stochastic process {Y (t)}t∈[0,∞) will be studied in the excursion on the Brownian motion. 38 / 477
  • 39. Outline 2 The Continuous-Time Market Model Modeling the Security Prices Excursion 1: Brownian Motion and Martingales Continuation: Modeling the Security prices ˆ Excursion 2: The Ito Integral ˆ Excursion 3: The Ito Formula Trading Strategy and Wealth Process Properties of the Continuous-Time Market Model Excursion 4: The Martingale Representation Theorem 39 / 477
  • 40. General assumptions General assumptions Let (Ω, F, P) be a complete probability space with sample space Ω, σ-field F and probability measure P. 40 / 477
  • 41. Filtration Definition Let {Ft }t∈I be a family of sub-σ-fields of F, I be an ordered index set with Fs ⊂ Ft for s < t, s, t ∈ I. The family {Ft }t∈I is called a filtration. A filtration describes flow of information over time. Ft models events observable up to time t. If a random variable Xt is Ft -measurable, we are able to determine its value from the information given at time t. 41 / 477
  • 42. Stochastic process Definition A set {(Xt , Ft )}t∈I consisting of a filtration {Ft }t∈I and a family of Rn -valued random variables {Xt }t∈I with Xt being Ft -measurable is called a stochastic process with filtration {Ft }t∈I . 42 / 477
  • 43. Remark Remark I = [0, ∞) or I = [0, T ]. Canoncial filtration (natural filtration) of {Xt }t∈I : Ft := FtX := σ{Xs | s ≤ t, s ∈ I}. Notation: {Xt }t∈I = {X (t)}t∈I = X . 43 / 477
  • 44. Sample path Sample path For fixed ω ∈ Ω the set X .(ω) := {Xt (ω)}t∈I = {X (t, ω)}t∈I is called a sample path or a realization of the stochastic process. 44 / 477
  • 45. Identification of stochastic processes Can two stochastic processes be identified with each other? Definition Let {(Xt , Ft )}t∈[0,∞) and {(Yt , Gt )}t∈[0,∞) be two stochastic processes. Y is a modification of X , if P{ω | Xt (ω) = Yt (ω)} = 1 for all t ≥ 0. Definition Let {(Xt , Ft )}t∈[0,∞) and {(Yt , Gt )}t∈[0,∞) be two stochastic processes. X and Y are indistinguishable, if P{ω | Xt (ω) = Yt (ω) for all t ∈ [0, ∞)} = 1. 45 / 477
  • 46. Identification of stochastic processes Remark X , Y indistinguishable ⇒ Y modification of X . Theorem Let the stochastic process Y be a modification of X . If both processes have continuous sample paths P-almost surely, then X and Y are indistinguishable. 46 / 477
  • 47. Brownian motion Definition The real-valued process {Wt }t≥0 with continuous sample paths and i) W0 = 0 P-a.s. ii) Wt − Ws ∼ N (0, t − s) for 0 ≤ s < t "stationary increments" iii) Wt − Ws independent of Wu − Wr for 0 ≤ r ≤ u ≤ s < t "independent increments" is called a one-dimensional Brownian motion. 47 / 477
  • 48. Brownian motion Remark By an n-dimensional Brownian motion we mean the Rn -valued process W (t) = (W1 (t), . . . , Wn (t)), with components Wi being independent one-dimensional Brownian motions. 48 / 477
  • 49. Brownian motion and filtration Brownian motion can be associated with natural filtration FtW := σ{Ws | 0 ≤ s ≤ t}, t ∈ [0, ∞) P-augmentation of the natural filtration (Brownian filtration) Ft := σ{FtW ∪ N | N ∈ F, P(N) = 0}, t ∈ [0, ∞) 49 / 477
  • 50. Brownian motion and filtration Requirement iii) of a Brownian motion with respect to a filtration {Ft }t≥0 is often stated as iii)∗ Wt − Ws independent of Fs , 0 ≤ s < t. {Ft }t≥0 natural filtration (Brownian filtration) ⇒ iii) and iii)∗ are equivalent. {Ft }t≥0 arbitrary filtration ⇒ iii) and iii)∗ are usually not equivalent. Convention If we consider a Brownian motion {(Wt , Ft )}t≥0 with an arbitrary filtration we implicitly assume iii)∗ to be fulfilled. 50 / 477
  • 51. Existence of the Brownian motion How can we show the existence of a stochastic process satisfying the requirements of a Brownian motion? Construction and existence proofs are long and technical. Construction based on weak convergence and approximation by random walks [Billingsley 1968]. Wiener measure, Wiener process. 51 / 477
  • 52. Brownian motion and filtration Theorem The Brownian filtration {Ft }t≥0 is right-continuous as well as left-continuous, i.e., we have Ft = Ft+ := Ft+ε and Ft = Ft− := σ Fs . ε>0 s<t Definition A filtration {Gt }t≥0 satifies the usual conditions, if it is right-continuous and G0 contains all P-null sets of F. General assumption for this section Let {Ft }t≥0 be a filtration which satisfies the usual conditions. 52 / 477
  • 53. Martingales Definition The real-valued process {(Xt , Ft )}t∈I with E |Xt | < ∞ for all t ∈ I (where I is an ordered index set), is called a super-martingale, if for all s, t ∈ I with s ≤ t we have E (Xt |Fs ) ≤ Xs P-a.s. , a sub-martingale, if for all s, t ∈ I with s ≤ t we have E (Xt |Fs ) ≥ Xs P-a.s. , a martingale, if for all s, t ∈ I with s ≤ t we have E (Xt |Fs ) = Xs P-a.s. . 53 / 477
  • 54. Interpretation of the martingale concept Example: Modeling games of chance Xn : Wealth of a gambler after n-th participation in a fair game Martingale condition: E (Xn+1 |Fn ) = Xn P-a.s. ⇒ "After the game the player is as rich as he was before" favorable game = sub-martingale non-favorable game = super-martingale 54 / 477
  • 55. Interpretation of the martingale concept Example: Tossing a fair coin "Head": Gambler receives one dollar "Tail": Gambler loses one dollar ⇒ Martingale 55 / 477
  • 56. Interpretation of the martingale concept Theorem A one-dimensional Brownian motion Wt is a martingale. Remark Each stochastic process with independent, centered increments is a martingale with respect to its natural filtration. The Brownian motion with drift µ and volatility σ Xt := µt + σWt , µ ∈ R, σ ∈ R is a martingale if µ = 0, a super-martingale if µ ≤ 0 and a sub-martingale if µ ≥ 0. 56 / 477
  • 57. Interpretation of the martingale concept Theorem (1) Let {(Xt , Ft )}t∈I be a martingale and ϕ : R → R be a convex function with E |ϕ(Xt )| < ∞ for all t ∈ I. Then {(ϕ(Xt ), Ft )}t∈I is a sub-martingale. (2) Let {(Xt , Ft )}t∈I be a sub-martingale and ϕ : R → R a convex, non-decreasing function with E |ϕ(Xt )| < ∞ for all t ∈ I. Then {(ϕ(Xt ), Ft )}t∈I is a sub-martingale. 57 / 477
  • 58. Interpretation of the martingale concept Remark (1) Typical applications are given by ϕ(x) = x 2 , ϕ(x) = |x|. (2) The theorem is also valid for d -dimensional vectors X (t) = (X1 (t), . . . , Xd (t)) of martingales and convex functions ϕ : Rd → R. 58 / 477
  • 59. Stopping time Definition A stopping time with respect to a filtration {Ft }t∈[0,∞) (or {Fn }n∈N ) is an F-measurable random variable τ : Ω → [0, ∞] (or τ : Ω → N ∪ {∞}) with {ω ∈ Ω | τ (ω) ≤ t} ∈ Ft for all t ∈ [0, ∞) (or {ω ∈ Ω | τ (ω) ≤ n} ∈ Fn for all n ∈ N). Theorem If τ1 , τ2 are both stopping times then τ1 ∧ τ2 := min{τ1 , τ2 } is also a stopping time. 59 / 477
  • 60. The stopped process The stopped process Let {(Xt , Ft )}t∈I be a stochastic process, let I be either N or [0, ∞), and τ a stopping time. The stopped process {Xt∧τ }t∈I is defined by Xt (ω) if t ≤ τ (ω), Xt∧τ (ω) := Xτ (ω) (ω) if t > τ (ω). Example: Wealth of a gambler who participates in a sequence of games until he is either bankrupt or has reached a given level of wealth. 60 / 477
  • 61. The stopped filtration The stopped filtration Let τ be a stopping time with respect to a filtration {Ft }t∈[0,∞) . σ-field of events determined prior to the stopping time τ Fτ := {A ∈ F | A ∩ {τ ≤ t} ∈ Ft for all t ∈ [0, ∞)} Stopped filtration {Fτ ∧t }t∈[0,∞) . 61 / 477
  • 62. The stopped filtration What will happen if we stop a martingale or a sub-martingale? Theorem: Optional sampling Let {(Xt , Ft )}t∈[0,∞) be a right-continuous sub-martingale (or martingale). Let τ1 , τ2 be stopping times with τ1 ≤ τ2 . Then for all t ∈ [0, ∞) we have E (Xt∧τ2 | Ft∧τ1 ) ≥ Xt∧τ1 P-a.s. (or E (Xt∧τ2 | Ft∧τ1 ) = Xt∧τ1 P-a.s.). 62 / 477
  • 63. The stopped filtration Corollary Let τ be a stopping time and {(Xt , Ft )}t∈[0,∞) a right-continuous sub-martingale (or martingale). Then the stopped process {(Xt∧τ , Ft )}t∈[0,∞) is also a sub-martingale (or martingale). 63 / 477
  • 64. The stopped filtration Theorem Let {(Xt , Ft )}t∈[0,∞) be a right-continuous process. Then Xt is a martingale if and only if for all bounded stopping times τ we have EXτ = EX0 . → Characterization of a martingale 64 / 477
  • 65. The stopped filtration Definition Let {(Xt , Ft )}t∈[0,∞) be a stochastic process with X0 = 0. If there is a non-decreasing sequence {τn }n∈N of stopping times with P lim τn = ∞ = 1, n→∞ such that (n) Xt := (Xt∧τn , Ft ) t∈[0,∞) is a martingale for all n ∈ N, then X is a local martingale. The sequence {τn }n∈N is called a localizing sequence corresponding to X . 65 / 477
  • 66. The stopped filtration Remark (1) Each martingale is a local martingale. (2) A local martingale with continuous paths is called continuous local martingale. (3) There exist local martingales which are not martingales. E (Xt ) need not exist for a local martingale. However, the expectation has to exist along the localizing sequence t ∧ τn . The local martingale is a martingale on the random time intervals [0, τn ]. Theorem A non-negative local martingale is a super-martingale. 66 / 477
  • 67. The stopped filtration Theorem: Doob’s inequality Let {Mt }t≥0 be a martingale with right-continuous paths and 2 E (MT ) < ∞ or all T > 0. Then, we have 2 2 E sup |Mt | ≤ 4 · E (MT ). 0≤t≤T Theorem Let {(Xt , Ft )}t∈[0,∞) be a non-negative super-martingale with right-continuous paths. Then, for λ > 0 we obtain λ·P ω sup Xs (ω) ≥ λ ≤ E (X0 ). 0≤s≤t 67 / 477
  • 68. Outline 2 The Continuous-Time Market Model Modeling the Security Prices Excursion 1: Brownian Motion and Martingales Continuation: Modeling the Security prices ˆ Excursion 2: The Ito Integral ˆ Excursion 3: The Ito Formula Trading Strategy and Wealth Process Properties of the Continuous-Time Market Model Excursion 4: The Martingale Representation Theorem 68 / 477
  • 69. Continuation: The stock price log-linear model for a stock price ln(Pi (t)) = ln(pi ) + bi · t + ”randomness” Brownian motion {(Wt , Ft )}t≥0 is the appropriate stochastic process to model the "randomness" 69 / 477
  • 70. Continuation: The stock price Market with one stock and one bond (d=1) ln(P1 (t)) = ln(p1 ) + b1 · t + σ11 Wt P1 (t) = p1 · exp b1 · t + σ11 Wt Market with d stocks and one bond (d>1) m ln(Pi (t)) = ln(pi ) + bi · t + σij Wj (t), i = 1, . . . , d j=1 m Pi (t) = pi · exp bi · t + σij Wj (t) , i = 1, . . . , d j=1 70 / 477
  • 71. Continuation: The stock price Distribution of the logarithm of the stock prices m 2 ln(Pi (t)) ∼ N ln(pi ) + bi · t, σij · t j=1 ⇒ Pi (t) is log-normally distributed. 71 / 477
  • 72. Continuation: The stock price Lemma m 1 2 Let bi := bi + 2 σij for i = 1, . . . , d . j=1 (1) E (Pi (t)) = pi · ebi t . m (2) Var (Pi (t)) = pi2 · exp(2bi t) · exp 2 σij t −1 . j=1 m 1 (3) Xt := a · exp cj Wj (t) − cj2 t with a, cj ∈ R, j = 1, . . . , m 2 j=1 is a martingale. 72 / 477
  • 73. Interpretation of the stock price model The stock price model m 1 2 Pi (t) = pi · exp(bi t) · exp σij Wj (t) − σij t , 2 j=1 Pi (0) = pi , i = 1, . . . , d . The stock price is the product of the mean stock price pi · exp(bi t) and a martingale with expectation 1, namely m 1 2 exp σij Wj (t) − σij t 2 j=1 which models the stock price around its mean value. 73 / 477
  • 74. Interpretation of the stock price model Vector of mean rates of stock returns b = (b1 , . . . , bd )T Volatility matrix   σ11 . . . σ1m  . .  σ= . . .. . .  . σd1 . . . σdm Pi (t) is a geometric Brownian motion with drift bi and volatility σi. = (σi1 , . . . , σim )T . 74 / 477
  • 75. Summary: Stock prices Bond price and stock prices P0 (t) = p0 · ert Bond price P0 (0)= p0 m 1 2 Pi (t) = pi · exp(bi t) · exp σij Wj (t) − σ t 2 ij Stock prices j=1 Pi (0) = pi , i = 1, . . . , d . 75 / 477
  • 76. Extension Extension: Model with non-constant, time-dependent, and integrable rates of return bi (t) and volatilities σ(t). Stock prices: t m 1 2 Pi (t) = pi · exp bi (s) − σij (s) ds 2 0 j=1 m t · exp σij (s) dWj (s) j=1 0 t Problem: σij (s) dWj (s) 0 ˆ ⇒ Stochastic integral (Ito integral) 76 / 477
  • 77. Outline 2 The Continuous-Time Market Model Modeling the Security Prices Excursion 1: Brownian Motion and Martingales Continuation: Modeling the Security prices ˆ Excursion 2: The Ito Integral ˆ Excursion 3: The Ito Formula Trading Strategy and Wealth Process Properties of the Continuous-Time Market Model Excursion 4: The Martingale Representation Theorem 77 / 477
  • 78. ˆ The Ito integral Is it possible to define the stochastic integral t Xs (ω) dWs (ω) 0 ω-wise in a reasonable way? 78 / 477
  • 79. ˆ The Ito integral Theorem P-almost all paths of the Brownian motion {Wt }t∈[0,∞) are nowhere differentiable. ⇒ A definition of the form t t dWs (ω) Xs (ω) dWs (ω) = Xs (ω) ds ds 0 0 is impossible. 79 / 477
  • 80. ˆ The Ito integral Theorem With the definition 2n Zn (ω) := W i (ω) − W i−1 (ω) , n ∈ N, ω ∈ Ω 2n 2n i=1 we have n→∞ Zn (ω) − − ∞ −→ P-a.s. , i.e., the paths Wt (ω) of the Brownian motion admit infinite variation on the interval [0, 1] P-almost surely. The paths Wt (ω) of the Brownian motion have infinite variation on each non-empty finite interval [s1 , s2 ] ⊂ [0, ∞) P-almost surely. 80 / 477
  • 81. General assumptions General assumptions for this section Let (Ω, F, P) be a complete probability space equipped with a filtration {Ft }t satisfying the usual conditions. Further assume that on this space a Brownian motion {(Wt , Ft )}t∈[0,∞) with respect to this filtration is defined. 81 / 477
  • 82. Simple process Definition A stochastic process {Xt }t∈[0,T ] is called a simple process if there exist real numbers 0 = t0 < t1 < . . . < tp = T , p ∈ N, and bounded random variables Φi : Ω → R, i = 0, 1, . . . , p, with Φ0 F0 -measurable, Φi Fti−1 -measurable, i = 1, . . . , p such that Xt (ω) has the representation p Xt (ω) = X (t, ω) = Φ0 (ω) · 1{0} (t) + Φi (ω) · 1(ti−1 ,ti ] (t) i=1 for each ω ∈ Ω. 82 / 477
  • 83. Simple process Remark Xt is Fti−1 -measurable for all t ∈ (ti−1 , ti ]. The paths X (., ω) of the simple process Xt are left-continuous step functions with height Φi (ω) · 1(ti−1 ,ti ] (t). 1 0.9 0.8 0.7 0.6 X(.,ω) 0.5 0.4 0.3 0.2 0.1 0 0 t t2 t3 T 1 t 83 / 477
  • 84. Stochastic integral Definition For a simple process {Xt }t∈[0,T ] the stochastic integral I.(X ) for t ∈ (tk , tk +1 ] is defined according to t It (X ) := Xs dWs := Φi (Wti − Wti−1 ) + Φk +1 (Wt − Wtk ), 0 1≤i≤k or more generally for t ∈ [0, T ]: t It (X ) := Xs dWs := Φi (Wti ∧t − Wti−1 ∧t ). 0 1≤i≤p 84 / 477
  • 85. Stochastic integral Theorem: Elementary properties of the stochastic integral Let X := {Xt }t∈[0,T ] be a simple process. Then we have (1) {It (X )}t∈[0,T ] is a continuous martingale with respect to {Ft }t∈[0,T ] . In particular, we have E (It (X )) = 0 for all t ∈ [0, T ]. t 2 t (2) E Xs dWs =E 2 Xs ds for t ∈ [0, T ]. 0 0 t 2 T (3) E sup Xs dWs ≤4·E 2 Xs ds . 0≤t≤T 0 0 85 / 477
  • 86. Stochastic integral Remark (1) By (2) the stochastic integral is a square-integrable stochastic process. (2) For the simple process X ≡ 1 we obtain t 1 dWs = Wt 0 and t t 2 E dWs = E (Wt2 ) =t= ds. 0 0 86 / 477
  • 87. Stochastic integral Remark (1) Integrals with general boundaries: T T t Xs dWs := Xs dWs − Xs dWs for t ≤ T . t 0 0 For t ≤ T , A ∈ Ft we have T T 1A (ω) · Xs (ω) · 1[t,T ] (s) dWs = 1A (ω) · Xs (ω) dWs . 0 t (2) Let X , Y be simple processes, a, b ∈ R. Then we have It (aX + bY ) = a · It (X ) + b · It (Y ) (linearity) 87 / 477
  • 88. Measurability Definition A stochastic process {(Xt , Gt )}t∈[0,∞) is called measurable if the mapping [0, ∞) × Ω → Rn (s, ω) → Xs (ω) is B([0, ∞)) ⊗ F-B(Rn )-measurable. Remark Measurability of the process X implies that X (., ω) is B([0, ∞))-B(Rn )-measurable for a fixed ω ∈ Ω. Thus, for all t ∈ [0, ∞), t i = 1, . . . , n, the integral Xi2 (s) ds is defined. 0 88 / 477
  • 89. Measurability Definition A stochastic process {(Xt , Gt )}t∈[0,∞) is called progressively measurable if for all t ≥ 0 the mapping [0, t] × Ω → Rn (s, ω) → Xs (ω) is B([0, t]) ⊗ Gt -B(Rn )-measurable. 89 / 477
  • 90. Measurability Remark (1) If the real-valued process {(Xt , Gt )}t∈[0,∞) is progressively measurable and bounded, then for all t ∈ [0, ∞) the integral t Xs ds is Gt -measurable. 0 (2) Every progressively measurable process is measurable. (3) Each measurable process possesses a progressively measurable modification. 90 / 477
  • 91. Measurability Theorem If all paths of the stochastic process {(Xt , Gt )}t∈[0,∞) are right-continuous (or left-continuous), then the process is progressively measurable. Theorem Let τ be a stopping time with respect to the filtration {Gt }t∈[0,∞) . If the stochastic process {(Xt , Gt )}t∈[0,∞) is progressively measurable, then so is the stopped process {(Xt∧τ , Gt )}t∈[0,∞) . In particular, Xt∧τ is Gt and Gt∧τ -measurable. 91 / 477
  • 92. Extension of the stochastic integral to L2 [0, T ]-processes Definition L2 [0, T ] := L2 [0, T ], Ω, F, {Ft }t∈[0,T ] , P := {(Xt , Ft )}t∈[0,T ] real-valued stochastic process T {Xt }t progressively measurable, E Xt2 dt < ∞ 0 T Norm on L2 [0, T ]: X 2 T := E Xt2 dt . 0 92 / 477
  • 93. Extension of the stochastic integral to L2 [0, T ]-processes · 2 L2 -norm on the probability space T [0, T ] × Ω, B([0, T ]) ⊗ F, λ ⊗ P . · 2 semi-norm ( X −Y 2 = 0 ⇒ X = Y ). T T X equivalent to Y :⇔ X = Y a.s. λ ⊗ P. 93 / 477
  • 94. Extension of the stochastic integral to L2 [0, T ]-processes ˆ Ito isometry Let X be a simple process. The mapping X → I.(X ) induces by T T 2 2 2 2 I.(X ) LT := E Xs dWs =E Xs ds = X T 0 0 a norm on the space of stochastic integrals. ⇒ I.(X ) linear, norm-preserving (= isometry) ˆ ⇒ I.(X ) Ito isometry 94 / 477
  • 95. Extension of the stochastic integral to L2 [0, T ]-processes Use processes X ∈ L2 [0, T ] approximated by a sequence X (n) of simple processes. I.(X (n) ) is a Cauchy-sequence with respect to · LT . To show: I.(X (n) ) is convergent, limit independent of X (n) . Denote limit by I(X ) = Xs dWs . 95 / 477
  • 96. Extension of the stochastic integral to L2 [0, T ]-processes J(.) C X ∈ L2 [0, T ] _ _ _ _ _ _ _/ J(X ) ∈ M2 O O · T · LT X (n) / I(X (n) ) I(.) simple process stochastic integral for simple processes 96 / 477
  • 97. Extension of the stochastic integral to L2 [0, T ]-processes Theorem An arbitrary X ∈ L2 [0, T ] can be approximated by a sequence of simple processes X (n) . More precisely: There exists a sequence X (n) of simple processes with T (n) 2 lim E Xs − Xs ds = 0. n→∞ 0 97 / 477
  • 98. Extension of the stochastic integral to L2 [0, T ]-processes Lemma Let {(Xt , Gt )}t∈[0,∞) be a martingale where the filtration {Gt }t∈[0,∞) satisfies the usual conditions. Then the process Xt possesses a right-continuous modification {(Yt , Gt )}t∈[0,∞) such that {(Yt , Gt )}t∈[0,∞) is a martingale. 98 / 477
  • 99. Extension of the stochastic integral to L2 [0, T ]-processes ˆ Construction of the Ito integral for processes in L2 [0, T ] There exists a unique linear mapping J from L2 [0, T ] into the space of continuous martingales on [0, T ] with respect to {Ft }t∈[0,T ] satisfying (1) X = {Xt }t∈[0,T ] simple process ⇒ P Jt (X ) = It (X ) for all t ∈ [0, T ] = 1 t (2) E Jt (X )2 = E 2 ˆ Xs ds Ito isometry 0 Uniqueness: If J, J ′ satisfy (1) and (2), then for all X ∈ L2 [0, T ] the processes J ′ (X ) and J(X ) are indistinguishable. 99 / 477
  • 100. Extension of the stochastic integral to L2 [0, T ]-processes Definition For X ∈ L2 [0, T ] and J as before we define by t Xs dWs := Jt (X ) 0 ˆ the stochastic integral (or Ito integral) of X with respect to W . 100 / 477
  • 101. Extension of the stochastic integral to L2 [0, T ]-processes Theorem: Special case of Doob’s inequality Let X ∈ L2 [0, T ]. Then we have t T 2 2 E sup Xs dWs ≤4·E Xs ds . 0≤t≤T 0 0 101 / 477
  • 102. Extension of the stochastic integral to L2 [0, T ]-processes Multi-dimensional generalization of the stochastic integral {(W (t), Ft )}t : m-dimensional Brownian motion with W (t) = (W1 (t), . . . , Wm (t)) {(X (t), Ft )}t : Rn,m -valued progressively measurable process with Xij ∈ L2 [0, T ]. ˆ Ito integral of X with respect to W :  t  m  X1j (s) dWj (s)   t  j=1   0   . .  X (s) dW (s) :=  . , t ∈ [0, T ]   0  m t     Xnj (s) dWj (s) j=1 0 102 / 477
  • 103. Further extension of the stochastic integral Definition H 2 [0, T ] := H 2 [0, T ], Ω, F, {Ft }t∈[0,T ] , P := {(Xt , Ft )}t∈[0,T ] real-valued stochastic process {Xt }t progressively measurable, T Xt2 dt < ∞ P-a.s. 0 103 / 477
  • 104. Further extension of the stochastic integral Processes X ∈ H 2 [0, T ] do not necessarily have a finite T -norm → no approximation by simple processes as for processes in L2 [0, T ] can be localized with suitable sequences of stopping times Stopping times (with respect to {Ft }t ): t 2 τn (ω) := T ∧ inf 0 ≤ t ≤ T Xs (ω) ds ≥ n , n ∈ N 0 Sequence of stopped processes: (n) Xt (ω) := Xt (ω) · 1{τn (ω)≥t} ⇒ X (n) ∈ L2 [0, T ] ⇒ Stochastic integral already defined. 104 / 477
  • 105. Further extension of the stochastic integral Stochastic integral: It (X ) := It (X (n) ) for 0 ≤ t ≤ τn Consistence property: It (X ) = It (X (m) ) for 0 ≤ t ≤ τn (≤ τm ), m ≥ n ⇒ It (X ) well-defined for X ∈ H 2 [0, T ] 105 / 477
  • 106. Further extension of the stochastic integral Stopping times: n→∞ τn − − +∞ P-a.s. −→ ⇒ It (X ) local martingale with localizing sequence τn . ⇒ Stochastic integral is linear and possesses continuous paths. 106 / 477
  • 107. Outline 2 The Continuous-Time Market Model Modeling the Security Prices Excursion 1: Brownian Motion and Martingales Continuation: Modeling the Security prices ˆ Excursion 2: The Ito Integral ˆ Excursion 3: The Ito Formula Trading Strategy and Wealth Process Properties of the Continuous-Time Market Model Excursion 4: The Martingale Representation Theorem 107 / 477
  • 108. ˆ The Ito formula General assumptions for this section Let (Ω, F, P) be a complete probability space equipped with a filtration {Ft }t satisfying the usual conditions. Further, assume that on this space a Brownian motion {(Wt , Ft )}t∈[0,∞) with respect to this filtration is defined. 108 / 477
  • 109. ˆ The Ito formula Definition Let {(Wt , Ft )}t∈[0,∞) be an m-dimensional Brownian motion. ˆ (1) {(X (t), Ft )}t∈[0,∞) is a real-valued Ito process if for all t ≥ 0 it admits the representation t t X (t) = X (0) + K (s) ds + H(s) dW (s) 0 0 t m t = X (0) + K (s) ds + Hj (s) dWj (s) P-a.s. 0 j=1 0 109 / 477
  • 110. ˆ The Ito formula X (0) F0 -measurable, {K (t)}t∈[0,∞) , {H(t)}t∈[0,∞) progressively measurable with t t |K (s)| ds < ∞, Hi2 (s) ds < ∞ P-a.s. 0 0 for all t ≥ 0, i = 1, . . . , m. (2) n-dimensional Ito process X = (X (1) , . . . , X (n) ) ˆ ˆ = vector with components being real-valued Ito processes. 110 / 477
  • 111. ˆ The Ito formula Remark Hj ∈ H 2 [0, T ] for all T > 0. ˆ The representation of an Ito process is unique up to indistinguishability of the representing integrands Kt , Ht . Symbolic differential notation: dXt = Kt dt + Ht dWt 111 / 477
  • 112. ˆ The Ito formula Definition ˆ Let X and Y be two real-valued Ito processes with t t X (t) = X (0) + K (s) ds + H(s) dW (s), 0 0 t t Y (t) = Y (0) + L(s) ds + M(s) dW (s). 0 0 Quadratic covariation of X and Y : m t X,Y t := Hi (s) · Mi (s) ds. i=1 0 112 / 477
  • 113. ˆ The Ito formula Definition Quadratic variation of X X t := X , X t . Notation ˆ Let X be a real-valued Ito process, and Y a real-valued, progressively measurable process. We set t t t Y (s) dX (s) := Y (s) · K (s) ds + Y (s) · H(s) dW (s) 0 0 0 if all integrals on the right-hand side are defined. 113 / 477
  • 114. ˆ The Ito formula ˆ Theorem: One-dimensional Ito formula ˆ Let Wt be a one-dimensional Brownian motion, and Xt a real-valued Ito process with t t Xt = X0 + Ks ds + Hs dWs . 0 0 Let f ∈ C 2 (R). Then, for all t ≥ 0 we have t t ′ 1 f (Xt ) = f (X0 ) + f (Xs ) dXs + f ′′ (Xs ) d X s 2 0 0 t t 1 = f (X0 ) + f (Xs ) · Ks + · f ′′ (Xs ) · Hs ds + ′ 2 f ′ (Xs )Hs dWs 2 0 0 114 / 477
  • 115. ˆ The Ito formula Remark ˆ The Ito formula differs from the fundamental theorem of calculus by the additional term t 1 f ′′ (Xs ) d X s . 2 0 The quadratic variation X t ˆ is an Ito process. Differential notation: 1 ′′ df (Xt ) = f ′ (Xt ) dXt + · f (Xt ) d X t . 2 115 / 477
  • 116. ˆ The Ito formula Lemma Let X be a martingale with |Xs | ≤ C for all s ∈ [0, t] P-a.s. Let π = {t0 , t1 , . . . , tm }, t0 = 0, tm = t, be a partition of [0, t] with π := max |tk − tk −1 |. 1≤k ≤m Then we have m 2 2 (1) E Xtk − Xtk −1 ≤ 48 · C 4 k =1 m 4 π →0 (2) X continuous ⇒ E Xtk − Xtk −1 − − → 0. −− k =1 116 / 477
  • 117. ′ ˆ Some applications of Ito s formula ′ ˆ Some applications of Ito s formula I (1) Xt = t : Representation: t t Xt = 0 + 1 ds + 0 dWs . 0 0 For f ∈ C 2 (R) we have t f (t) = f (0) + f ′ (s) ds. 0 ⇒ Fundamental theorem of calculus is a special case of Ito′ s formula. ˆ 117 / 477
  • 118. ′ ˆ Some applications of Ito s formula II ˆ′ Some applications of Ito s formula (2) Xt = h(t) : For h ∈ C 1 (R) Ito′ s formula implies the chain rule ˆ t t ′ Xt = h(0) + h (s) ds + 0 dWs 0 0 t ⇒ (f ◦ h)(t) = (f ◦ h)(0) + f ′ (h(s)) · h′ (s) ds. 0 118 / 477
  • 119. ′ ˆ Some applications of Ito s formula III ′ ˆ Some applications of Ito s formula (3) Xt = Wt , f (x) = x 2 : Due to t t Wt = 0 + 0 ds + 1 dWs 0 0 we obtain t t t 1 Wt2 = 2 · Ws dWs + · 2 ds = 2 · Ws dWs + t 2 0 0 0 ⇒ Additional term "t" (→ nonvanishing quadratic variation of Wt ). 119 / 477
  • 120. ˆ The Ito formula ˆ Theorem: Multi-dimensional Ito formula ˆ X (t) = X1 (t), . . . , Xn (t) n-dimensional Ito process with t m t Xi (t) = Xi (0) + Ki (s) ds + Hij (s) dWj (s), i = 1, . . . , n 0 j=1 0 where W (t) = W1 (t), . . . , Wm (t) is an m-dimensional Brownian motion. Let f : [0, ∞) × Rn → R be a C 1,2 -function. Then, we have f (t, X1 (t), . . . , Xn (t)) = f (0, X1 (0), . . . , Xn (0)) t n t + ft (s, X1 (s), . . . , Xn (s)) ds + fxi (s, X1 (s), . . . , Xn (s)) dXi (s) 0 i=1 0 n t 1 + · fxi xj (s, X1 (s), . . . , Xn (s)) d Xi , Xj s . 2 i,j=1 0 120 / 477
  • 121. Product rule or partial integration Corollary: Product rule or partial integration ˆ Let Xt , Yt be one-dimensional Ito processes with t t Xt = X0 + Ks ds + Hs dWs , 0 0 t t Yt = Y0 + µs ds + σs dWs . 0 0 Then we have t t t Xt · Yt = X0 · Y0 + Xs dYs + Ys dXs + d X,Y s 0 0 0 t t = X0 · Y0 + Xs µs + Ys Ks + Hs σs ds + Xs σs + Ys Hs dWs . 0 0 121 / 477
  • 122. The stock price equation Simple continuous-time market model (1 bond, one stock). Stock price influenced by a one-dimensional Brownian motion Price of the stock at time t: P(t) = p · exp b − 1 σ 2 t + σWt 2 Choose t t 1 2 Xt = 0 + b− 2σ ds + σ dWs , f (x) = p · ex 0 0 ˆ Ito formula implies t t 1 2 1 2 f (Xt ) = p + f (Xs )(b − 2 σ ) + 2 f (Xs ) ·σ ds + f (Xs ) · σ dWs 0 0 122 / 477
  • 123. The stock price equation The stock price equation t t P(t) = p + P(s) · b ds + P(s) · σ dWs 0 0 Remark The stock price equation is valid for time-dependent b and σ, if t t 1 2 Xt = b(s) − 2 σ (s) ds + σ(s) dWs . 0 0 123 / 477
  • 124. The stock price equation The stock price equation in differential form dP(t) = P(t) b dt + σ dWt P(0) = p 124 / 477
  • 125. The stock price equation Theorem: Variation of constants Let {(W (t), Ft )}t∈[0,∞) be an m-dimensional Brownian motion. Let x ∈ R and A, a, Sj , σj be progressively measurable, real-valued processes with t |A(s)| + |a(s)| ds < ∞ for all t ≥ 0 P-a.s. 0 t Sj2 (s) + σj2 (s) ds < ∞ for all t ≥ 0 P-a.s. . 0 125 / 477
  • 126. The stock price equation Theorem: Variation of constants Then the stochastic differential equation m dX (t) = A(t) · X (t) + a(t) dt + Sj (t)X (t) + σj (t) dWj (t) j=1 X (0) = x possesses a unique solution with respect to λ ⊗ P : t m 1 X (t) = Z (t) · x + a(u) − Sj (u)σj (u) du Z (u) 0 j=1 m t σj (u) + dWj (u) Z (u) j=1 0 126 / 477
  • 127. The stock price equation Theorem: Variation of constants Hereby is t t 1 2 Z (t) = exp A(u) − 2 · S(u) du + S(u) dW (u) 0 0 the unique solution of the homogeneous equation dZ (t) = Z (t) A(t) dt + S(t)T dW (t) Z (0) = 1. 127 / 477
  • 128. The stock price equation Remark The process {(X (t), Ft )}t∈[0,∞) solves the stochastic differential equation in the sense that X (t) satisfies t X (t) = x + A(s) · X (s) + a(s) ds 0 m t + Sj (s) X (s) + σj (s) dWj (s) j=1 0 for all t ≥ 0 P-almost surely. 128 / 477
  • 129. Outline 2 The Continuous-Time Market Model Modeling the Security Prices Excursion 1: Brownian Motion and Martingales Continuation: Modeling the Security prices ˆ Excursion 2: The Ito Integral ˆ Excursion 3: The Ito Formula Trading Strategy and Wealth Process Properties of the Continuous-Time Market Model Excursion 4: The Martingale Representation Theorem 129 / 477
  • 130. General assumptions General assumptions for this section (Ω, F, P) be a complete probability space, {(W (t), Ft )}t∈[0,∞) m-dimensional Brownian motion. Dynamics of bond and stock prices: t P0 (t) = p0 · exp r (s) ds bond 0 t m 1 2 Pi (t) = pi · exp bi (s) − σij (s) ds 2 0 j=1 m t + σij (s) dWj (s) stock j=1 0 for t ∈ [0, T ], T > 0, i = 1, . . . , d . 130 / 477
  • 131. General assumptions (continued) General assumptions for this section (continued) r (t), b(t) = (b1 (t), . . . , bd (t))T , σ(t) = (σij (t))ij progressively measurable processes with respect to {Ft }t , component-wise uniformly bounded in (t, ω). σ(t)σ(t)T uniformly positive definite, i.e., it exists K > 0 with x T σ(t)σ(t)T x ≥ Kx T x for all x ∈ Rd and all t ∈ [0, T ] P-a.s. Deterministic rate of return r (t) is not required r (t) can be a stochastic process ⇒ bond is no longer riskless. 131 / 477
  • 132. Bond and stock prices Bond and stock prices are unique solutions of the stochastic differential equations dP0 (t) = P0 (t) · r (t) dt bond P0 (t) = p0 m dPi (t) = Pi (t) bi (t) dt + σij (t) dWj (t) , i = 1, . . . , d j=1 Pi (0) = pi stock ˆ ⇒ Representations of prices as Ito processes 132 / 477
  • 133. Possible actions of investors (1) Investor can rebalance his holdings → sell some securities → invest in securities ⇒ Portfolio process / trading strategy. (2) Investor is allowed to consume parts of his wealth ⇒ Consumption process. 133 / 477
  • 134. Requirements on a market model (1) Investor should not be able to foresee events → no knowledge of future prices. (2) Actions of a single investor have no impact on the stock prices (small investor hypothesis). (3) Each investor has a fixed initial capital at time t = 0. (4) Money which is not invested into stocks has to be invested in bonds. (5) Investors act in a self-financing way (no secret source or sink for money). 134 / 477
  • 135. Requirements on a market model (6) Securities are perfectly divisible. (7) Negative positions in securities are possible bond → credit stock → we sold some stock short. (8) No transaction costs. 135 / 477
  • 136. Negative bond positions and credit interest rates Negative bond positions and credit interest rates Assume: Interest rate r (t) is constant Negative bond position = it is possible to borrow money for the same rate as we would get for investing in bonds. Interest depends on the market situation ((t, ω) ∈ [0, T ] × Ω), but not on positive or negative bond position. 136 / 477
  • 137. Mathematical realizations of some requirements Market with 1 bond and d stocks Time t = 0: – Initial capital of investor: x > 0 – Buy a selection of securities T ϕ(0) = ϕ0 (0), ϕ1 (0), . . . , ϕd (0) Time t > 0: – Trading strategy: ϕ(t) (1) ⇒ trading strategy is progressively measurable with respect to {Ft }t Decisions on buying and selling are made on basis of information available at time t (→ modelled by {Ft }t ) (5) ⇒ only self-financing trading strategies should be used. 137 / 477
  • 138. Discrete-time example: self-financing strategy Market with 1 riskless bond and 1 stock Two-period model for time points t = 0, 1, 2. Number of shares of bond and stock at time t: (ϕ0 (t), ϕ1 (t))T ∈ R2 Consumption of investor at time t: C(t) Wealth at time t: X (t) Bond/stock prices at time t: P0 (t), P1 (t) Initial conditions: C(0) = 0, X (0) = x 138 / 477
  • 139. Discrete-time example: self-financing strategy t =0 Investor uses initial capital to buy shares of bond and stock X (0) = x = ϕ0 (0) · P0 (0) + ϕ1 (0) · P1 (0). 139 / 477
  • 140. Discrete-time example: self-financing strategy t =1 Security prices have changed, investor consumes parts of his wealth Current wealth: X (1) = ϕ0 (0) · P0 (1) + ϕ1 (0) · P1 (1) − C(1). Total: X (1) = x + ϕ0 (0) · P0 (1) − P0 (0) + ϕ1 (0) · P1 (1) − P1 (0) − C(1) Wealth = initial wealth + gains/losses - consumption Invest remaining capital at the market: X (1) = ϕ0 (1) · P0 (1) + ϕ1 (1) · P1 (1). 140 / 477
  • 141. Discrete-time example: self-financing strategy t =2 Invest remaining capital at the market Wealth: X (2) = ϕ0 (2) · P0 (2) + ϕ1 (2) · P1 (2). Wealth = total wealth of shares held Total: 2 X (2) = x + ϕ0 (i − 1) · (P0 (i) − P0 (i − 1)) i=1 +ϕ1 (i − 1) · (P1 (i) − P1 (i − 1)) 2 − C(i). i=1 141 / 477
  • 142. Discrete-time example: self-financing strategy Self-financing trading strategy: wealth before rebalancing - consumption = wealth after rebalancing Condition: ϕ0 (i) · P0 (i) + ϕ1 (i) · P1 (i) = ϕ0 (i − 1) · P0 (i) + ϕ1 (i − 1) · P1 (i) − C(i) ⇒ Useless in continuous-time setting (securities can be traded at each time instant / "before" and "after" cannot be distinguished) 142 / 477
  • 143. Discrete-time example: self-financing strategy Continuous-time setting Wealth process corresponding to strategy ϕ(t): t t t X (t) = x + ϕ0 (s) dP0 (s) + ϕ1 (s) dP1 (s) − c(s) ds 0 0 0 ˆ ⇒ Price processes are Ito processes. 143 / 477
  • 144. Trading strategy and wealth processes Definition (1) A trading strategy ϕ with T ϕ(t) := ϕ0 (t), ϕ1 (t), . . . , ϕd (t) is an Rd+1 -valued progressively measurable process with respect to {Ft }t∈[0,T ] satisfying T |ϕ0 (t)| dt < ∞ P-a.s. 0 d T 2 ϕi (t) · Pi (t) dt < ∞ P-a.s. for i = 1, . . . , d . j=1 0 144 / 477
  • 145. Trading strategy and wealth processes Definition The value d x := ϕi (0) · pi i=0 is called initial value of ϕ. (2) Let ϕ be a trading strategy with initial value x > 0. The process d X (t) := ϕi (t)Pi (t) i=0 is called wealth process corresponding to ϕ with initial wealth x. 145 / 477
  • 146. Trading strategy and wealth processes Definition (3) A non-negative progressively measurable process c(t) with respect to {Ft }t∈[0,T ] with T c(t) dt < ∞ P-a.s. 0 is called consumption (rate) process. 146 / 477
  • 147. Trading strategy and wealth processes Definition A pair (ϕ, c) consisting of a trading strategy ϕ and a consumption rate process c is called self-financing if the corresponding wealth process X (t) satisfies d t t X (t) = x + ϕi (s) dPi (s) − c(s) ds P-a.s. i=0 0 0 current wealth = initial wealth + gains/losses - consumption 147 / 477
  • 148. Trading strategy and wealth processes Remark We have t t ϕ0 (s) dP0 (s) = ϕ0 (s) P0 (s) r (s) ds 0 0 t t ϕi (s) dPi (s) = ϕi (s) Pi (s) bi (s) ds 0 0 m t + ϕi (s) Pi (s) σij (s) dWj (s), i = 1, . . . , d . j=1 0 148 / 477
  • 149. Self-financing portfolio process Definition Let (ϕ, c) be a self-financing pair consisting of a trading strategy and a consumption process with corresponding wealth process X (t) > 0 P-a.s. for all t ∈ [0, T ]. Then the Rd -valued process T ϕi (t) · Pi (t) π(t) = π1 (t), . . . , πd (t) with πi (t) = X (t) is called a self-financing portfolio process corresponding to the pair (ϕ, c). 149 / 477
  • 150. Portfolio processes Remark (1) The portfolio process denotes the fractions of total wealth invested in the different stocks. (2) The fraction of wealth invested in the bond is given by ϕ0 (t) · P0 (t) 1 − π(t)T 1 = , where 1 := (1, . . . , 1)T ∈ Rd . X (t) (3) Given knowledge of wealth X (t) and prices Pi (t), it is possible for an investor to describe his activities via a self-financing pair (π, c). → Portfolio process and trading strategy are equivalent descriptions of the same action. 150 / 477
  • 151. The wealth equation The wealth equation dX (t) = [r (t) X (t) − c(t)] dt + X (t) π(t)T (b(t) − r (t) 1) dt + σ(t) dW (t) X (0) = x 151 / 477
  • 152. Alternative definition of a portfolio process Definition The progressively measurable Rd -valued process π(t) is called a self-financing portfolio process corresponding to the consumption process c(t) if the corresponding wealth equation possesses a unique solution X (t) = X π,c (t) with T 2 X (t) · πi (t) dt < ∞ P-a.s. for i = 1, . . . , d . 0 152 / 477
  • 153. Admissibility Definition A self-financing pair (ϕ, c) or (π, c) consisting of a trading strategy ϕ or a portfolio process π and a consumption process c will be called admissible for the initial wealth x > 0, if the corresponding wealth process satisfies X (t) ≥ 0 P-a.s. for all t ∈ [0, T ]. The set of admissible pairs will be denoted by A(x). 153 / 477
  • 154. An example Portfolio process: π(t) ≡ π ∈ Rd constant Consumption rate: c(t) = γ · X (t), γ > 0 Wealth process corresponding to (π, c) : X (t) Investor rebalances his holdings in such a way that the fractions of wealth invested in the different stocks and in the bond remain constant over time. Consumption rate is proportional to the current wealth of the investor. 154 / 477
  • 155. An example Wealth equation: dX (t) = [r (t) − γ] X (t) dt + X (t)π T (b(t) − r (t) 1) dt + σ(t) dW (t) X (0) = 0 Wealth process: t 1 T X (t) = x · exp r (s) − γ + π T b(s) − r (s) · 1 − π σ(s) 2 ds 2 0 t + π T σ(s) dW (s) 0 155 / 477
  • 156. Outline 2 The Continuous-Time Market Model Modeling the Security Prices Excursion 1: Brownian Motion and Martingales Continuation: Modeling the Security prices ˆ Excursion 2: The Ito Integral ˆ Excursion 3: The Ito Formula Trading Strategy and Wealth Process Properties of the Continuous-Time Market Model Excursion 4: The Martingale Representation Theorem 156 / 477
  • 157. Properties of the continuous-time market model Assumptions: Dimension of the underlying Brownian motion = number of stocks Past and present prices are the only sources of information for the investors ⇒ Choose Brownian filtration {Ft }t∈[0,T ] Aim: Final wealths X (T ) when starting with initial capital of x. 157 / 477
  • 158. General assumption / notation General assumption for this section d =m Notation t γ(t) := exp − r (s) ds 0 θ(t) := σ −1 (t) b(t) − r (t) 1 t t T 1 2 Z (t) := exp − θ(s) dW (s) − θ(s) ds 2 0 0 H(t) := γ(t) · Z (t) 158 / 477
  • 159. Properties of the continuous-time market model b, r uniformly bounded σσ T uniformly positive definite ⇒ θ(t) 2 uniformly bounded Interpretation of θ(t): Relative risk premium for stock investment. Process H(t) is important for option pricing. H(t) is positive, continuous, and progressively measurable with respect to {Ft }t∈[0,T ] . H(t) is the unique solution of the SDE dH(t) = −H(t) r (t) dt + θ(t)T dW (t) H(0) = 1. 159 / 477
  • 160. Completeness of the market Theorem: Completeness of the market (1) Let the self-financing pair (π, c) consisting of a portfolio process π and a consumption process c be admissible for an initial wealth of x ≥ 0, i.e., (π, c) ∈ A(x). Then the corresponding wealth process X (t) satisfies t E H(t) X (t) + H(s)c(s) ds ≤ x for all t ∈ [0, T ]. 0 160 / 477
  • 161. Completeness of the market Theorem: Completeness of the market (2) Let B ≥ 0 be an FT -measurable random variable and c(t) a consumption process satisfying T x := E H(T ) B + H(s)c(s) ds < ∞. 0 Then there exists a portfolio process π(t) with (π, c) ∈ A(x) and the corresponding wealth process X (t) satisfies X (T ) = B P-a.s. 161 / 477
  • 162. Completeness of the market H(t) can be regarded as the appropriate discounting process that determines the initial wealth at time t = 0 T E H(s) · c(s) ds + E (H(T ) · B) 0 which is necessary to attain future aims. (1) puts bounds on the desires of an investor given his initial capital x ≥ 0. (2) proves that future aims which are feasible in the sense of part (1) can be realized. (2) says that each desired final wealth in t = T can be attained exactly via trading according to an appropriate self-financing pair (π, c) if one possesses sufficient initial capital (completeness/complete model). 162 / 477
  • 163. Completeness of the market Remark 1/H(t) is the wealth process corresponding to the pair π(t), c(t) = σ −1 (t)T θ(t), 0 with initial wealth x := 1/H(0) = 1 and final wealth B:= 1/H(T ). 163 / 477
  • 164. Outline 2 The Continuous-Time Market Model Modeling the Security Prices Excursion 1: Brownian Motion and Martingales Continuation: Modeling the Security prices ˆ Excursion 2: The Ito Integral ˆ Excursion 3: The Ito Formula Trading Strategy and Wealth Process Properties of the Continuous-Time Market Model Excursion 4: The Martingale Representation Theorem 164 / 477
  • 165. Excursion 4: The martingale representation theorem General assumptions (Ω, F, P) complete probability space. {(Wt , Ft )}t∈[0,∞) m-dimensional Brownian motion. {Ft }t Brownian filtration. Definition A real-valued martingale {(Mt , Ft )}t∈[0,T ] with respect to the Brownian filtration {Ft }t is called a Brownian martingale. 165 / 477
  • 166. The martingale representation theorem Martingale representation theorem Let {(Mt , Ft )}t∈[0,T ] be a square-integrable Brownian martingale, i.e., EMt2 < ∞ for all t ∈ [0, T ]. Then there exists a progressively measurable Rm -valued process Ψ(t) with T 2 E Ψ(t) dt <∞ 0 and t Mt = M0 + Ψ(s)T dW (s) P-a.s. . 0 166 / 477
  • 167. The martingale representation theorem Corollary Let {(Mt , Ft )}t∈[0,T ] be a local martingale with respect to the Brownian filtration {Ft }t . Then there exists a progressively measurable Rm -valued process Ψ(t) with T 2 Ψ(t) dt < ∞ 0 and t Mt = M0 + Ψ(s)T dW (s) P-a.s. 0 167 / 477
  • 168. The martingale representation theorem Remark Each local martingale with respect to the Brownian filtration can ˆ be represented as an Ito process. ˆ Each Brownian martingale can be represented as an Ito process. ⇒ Quadratic variation and quadratic covariation are defined. 168 / 477
  • 169. Outline 3 Option Pricing 169 / 477
  • 170. Outline 3 Option Pricing Introduction Examples The Replication Principle Arbitrage Opportunity Continuation Partial Differential Approach (PDA) Arbitrage & Option Pricing 170 / 477
  • 171. Introduction Option derivative security (i.e., from underlying assets) call buy fixed amount of asset at fixed time in future for fixed strike price put sell fixed amount of asset for strike price American Option Sell/buy asset during timespan of contract European Option Act at maturity ( =⇒ expiry) 171 / 477
  • 172. Outline 3 Option Pricing Introduction Examples The Replication Principle Arbitrage Opportunity Continuation Partial Differential Approach (PDA) Arbitrage & Option Pricing 172 / 477
  • 173. European call European call Right to buy security at time t = T for strike price K > 0 fixed at time t = 0 P(T ) > K Gain (P(T ) − K )+ P(T ) ≤ K No gain (Holder will buy in market) 173 / 477
  • 174. European put European put Right to sell security at time t = T for price K > 0 fixed at time t = 0 P(T ) ≥ K No gain (Holder will sell in market) P(T ) < K Gain (K − P(T ))+ 174 / 477
  • 175. The payoff diagram The payoff diagram Graph of final gain (through the option) as function of the stock price at time T 175 / 477
  • 176. Payoff diagram of a call + (P1(T)−K) K 0 K P1(T) 176 / 477
  • 177. Payoff diagram of a put + (K−P1(T)) K 0 K P1(T) 177 / 477
  • 178. The sense of options Hedge against price fluctuations of underlying asset Bound risks of future cash flows Risks of option speculation High losses not unusual 178 / 477
  • 179. Outline 3 Option Pricing Introduction Examples The Replication Principle Arbitrage Opportunity Continuation Partial Differential Approach (PDA) Arbitrage & Option Pricing 179 / 477
  • 180. Option pricing Option Pricing Bond Interest Rate (BIR) r constant Fixed deterministic payment of B at time T has present value of e−rT · B Sum to be invested at t = 0 to obtain amount of B at maturity. 180 / 477
  • 181. Option pricing Option Pricing BIR r (t) time dependent and random variable Expected value T − r (s) ds E e 0 ·B 181 / 477
  • 182. Examples of option pricing Pricing in discrete-time Market with bond ( P0 (t) ) and stock ( P1 (t) ) and BIR r = 0 , European Call with strike K = 1 P0 (0) = 1 , P (P0 (T ) = 1) = 1 P (P1 (T ) = 3) = a P1 (0) = 1 , P (P1 (T ) = 0.5) = 1 − a (a ∈ (0, 1)) 182 / 477
  • 183. Examples of option pricing E (P1 (T ) − K )+ = (3 − 1) · a + 0 · (1 − a) = 2a Parameter a unkown Final payment of option can be obtained by following a self-financing trading strategy in stock and bond ( −→ Replication Principle) 183 / 477
  • 184. Examples of option pricing Determine (ϕ0 (0), ϕ1 (0)) such that X (T ) = ϕ0 (0) · P0 (T ) + ϕ1 (0) · P1 (T ) = (P1 (T ) − K )+ 184 / 477
  • 185. Examples of option pricing Option Price ˆ p = ϕ0 (0) · P0 (0) + ϕ1 (0) · P1 (0) equals capital in t = 0 to buy replication strategy (ϕ0 (0), ϕ1 (0)) For all other choices riskless gains are possible without initial capital −→ arbitrage opportunities 185 / 477
  • 186. Examples of option pricing Proof– Part 1 ˜ ˆ ˜ (i) Be option price p < p. Then buy option for p and sell ˆ (ϕ0 (0), ϕ1 (0)) for p (Hold position (−ϕ0 (0), −ϕ1 (0)) ) ˆ ˜ =⇒ Initial gain of p − p without own capital 186 / 477
  • 187. Examples of option pricing Proof– Part 2 ˜ ˆ (ii) Be option price p > p. Then sell call and hold position ˆ (ϕ0 (0), ϕ1 (0)) for p ˜ ˆ =⇒ Initial gain of p − p without own capital 187 / 477
  • 188. Examples of option pricing Our example yields the system of equations ϕ0 (0) · 1 + ϕ1 (0) · 3 =2 ϕ0 (0) · 1 + ϕ1 (0) · 0.5 = 0 with unique solution (ϕ0 (0), ϕ1 (0)) = (−0.4, 0.8) ˆ =⇒ Option price p = −0.4 · 1 + 0.8 · 1 = 0.4 188 / 477
  • 189. Examples of option pricing Option price independent of unknown probability a No arbitrage opportunities in market ˆ Calculated call price p equals expected discounted terminal payment of call ⇐⇒ a = 0.2 −→ P1 (t) is martingale 189 / 477
  • 190. General assumptions Assumptions for this section Self-financing pair (π, c) ∈ A(x) admissible for initial capital x ≥ 0 190 / 477
  • 191. Outline 3 Option Pricing Introduction Examples The Replication Principle Arbitrage Opportunity Continuation Partial Differential Approach (PDA) Arbitrage & Option Pricing 191 / 477
  • 192. Arbitrage opportunity Definition Be (ϕ, c) a self-financing and admissible pair ϕ a trading strategy c a consumption process (ϕ, c) is called arbitrage opportunity ⇐⇒ corresponding wealth process satisfies 192 / 477
  • 193. Arbitrage opportunity Definition continued X (0) = 0 , X (T ) ≥ 0 P-a.s.  T  P (X (T ) > 0) > 0 or P  c(t) dt > 0 > 0 0 193 / 477
  • 194. Arbitrage opportunity Corollary In the complete continuous-time market model there is no arbitrage opportunity. 194 / 477
  • 195. Contingent claim Definition A contingent claim (g, B) consists of an {Ft }t -progressively measurable payout rate process g(t), t ∈ [0, T ], g(t) ≥ 0, and a FT -measurable terminal payment B ≥ 0 at T with  T µ  E  g(t) dt + B   < ∞ 0 for some µ > 1. 195 / 477
  • 196. Replication strategy Definition (π, c) is called replication strategy for the contingent claim (g, B) if g(t) = c(t) P-a.s. ∀ t ∈ [0, T ] X (T ) = B P-a.s. where X (t) is wealth process corresponding to (π, c). 196 / 477
  • 197. Set of replication strategies of price x Definition D(x) := D (x; (g, B)) := {(π, c) ∈ A(x)| (π, c) replication strategy for (g, B)} 197 / 477
  • 198. Fair price Definition The fair price of the contingent claim (g, B) is defined as ˆ p := inf {p | D(p) = ∅ } . 198 / 477
  • 199. Fair price of contingent claim (g, B) Theorem The fair price of the contingent claim (g, B) is given by  T  p = E H(T )B + ˆ H(t)g(t) dt  < ∞, 0 and there exists a unique (with respect to P ⊗ λ) replication strategy π, c ∈ D p . ˆ Its wealth process X (t) (also called the valuation process of (g, B)) is given by  T  ˆ 1 X (t) = E H(T )B + H(s)g(s) ds Ft  . H(t) t 199 / 477
  • 200. The valuation process Remark The above theorem gives us the fair price of the contingent claim ˆ (g, B) at time t as this price p(t) has to coincide with X (t). Otherwise, there would be arbitrage opportunities in the market consisting of stock, bond, and contingent claim. 200 / 477
  • 201. Black-Scholes formula Theorem Consider a market model with just one stock and a bond with constant market coefficients, i.e., d =m=1 and r (t) ≡ r , b(t) ≡ b , σ(t) ≡ σ > 0 for all t ∈ [0, T ], T > 0, r , b, σ ∈ R. 201 / 477
  • 202. Black-Scholes formula (i) The price XC (t) of a European call with strike price K > 0 and maturity T is given by XC (t) = P1 (t)Φ(d1 (t)) − Ke−r (T −t) Φ(d2 (t)) P1 (t) ln K + (r + 0.5σ 2 )(T − t) d1 (t) = √ σ T −t P1 (t) ln K + (r − 0.5σ 2 )(T − t) d2 (t) = √ σ T −t √ = d1 (t) − σ T − t where Φ is the distribution function of the standard normal distribution. 202 / 477
  • 203. Black-Scholes formula (ii) Price XP (t) of European put with strike K > 0 XP (t) = Ke−r (T −t) Φ(−d2 (t)) − P1 (t)Φ(−d1 (t)). 203 / 477
  • 204. The change of measure W Q (t) := W (t) + θt =⇒ p = XC (0) = EQ e−rT (P1 (T ) − K )+ ˆ where EQ (·) denotes expected value with respect to measure Q given by Radon-Nikodym derivative dQ 2 = e−0.5θ T −θW (T ) . dP 204 / 477
  • 205. General assumptions of this section General Assumptions of this Section Be {(X (t), Ft )}t≥0 an m-dimensional progressively measurable process, {Ft } the Brownian filtration with t Xi2 (s) ds < ∞ P-a.s. for all t ≥ 0, i = 1, . . . , m. 0 Let further m t t 1 2 Z (t, X ) := exp − Xi (s) dWi (s) − X (s) ds . 2 i=1 0 0 205 / 477
  • 206. Consequences ˆ The argument in Z (t, X ) is an Ito process and we have m t Z (t, X ) = 1 − Z (s, X )Xi (s) dWi (s). i=1 0 Z (t, X ) is a continuous local martingale with Z (0, X ) = 1. 206 / 477
  • 207. Girsanov’s theorem Girsanov’s theorem Be process Z (t, X ) a martingale and define process W Q (t), Ft t≥0 by t WiQ (t) := Wi (t) + Xi (s) ds (1 ≤ i ≤ m, t ≥ 0) 0 Then, for each fixed T ∈ [0, ∞) the process W Q (t), Ft t∈[0,T ] is an m-dimensional Brownian motion on (Ω, FT , QT ) with probability measure QT (A) := E (1A · Z (T , X )) ∀ A ∈ FT 207 / 477
  • 208. The Novikov condition Z (t, X ) martingale −→ apply Girsanov’s theorem Sufficient condition is the Novikov condition: t 0.5 X (s) 2 ds E e 0 <∞ 208 / 477
  • 209. The Novikov condition Proposition T If we have X (s) 2 ds < K for some constant K > 0, then 0 Z (t, X ) is a martingale. 209 / 477
  • 210. Outline 3 Option Pricing Introduction Examples The Replication Principle Arbitrage Opportunity Continuation Partial Differential Approach (PDA) Arbitrage & Option Pricing 210 / 477
  • 211. Girsanov’s theorem & Option Pricing Lemma Be Q a probability measure which is equivalent to P restricted on FT . Then the density process {Dt }t∈[0,T ] defined by dQ Dt := , t ∈ [0, T ] dP Ft satisfies: {(Dt , Ft )}t∈[0,T ] is a positive Brownian martingale with respect to P satisfying t Dt = 1 + Ψ(s)T dW (s) 0 for a progressively measurable d -dimensional process Ψ with T 2 Ψ(s) ds < ∞ P-a.s. 0 211 / 477
  • 212. Uniqueness of the equivalent martingale measure Theorem In the complete market model QT is the unique equivalent martingale measure on {Ft }t∈[0,T ] for the price processes Pi (t), 0 ≤ i ≤ d . Remark The existence of an equivalent martingale measure implies the absence of arbitrage opportunities in the market. The absence of arbitrage implies the existence of an equivalent martingale measure. 212 / 477
  • 213. Option pricing & equivalent martingale measure Corollary Be (g, B) a contingent claim such that g(s) is uniformly bounded on ˆ [0, T ]. Then its price process X (t) satisfies  T T  − r (s) ds s ˆ X (t) = EQ e t B+ e− t r (u) du g(s) ds Ft  t for 0 ≤ t ≤ T with EQ = EQ,T . 213 / 477
  • 214. Independence of the option price In the case of g ≡ 0 we have  T  − r (s) ds p = EQ e ˆ 0 B ˆ Thus, p equals the natural price with respect to a new (uniquely determined) probability measure Q. 214 / 477
  • 215. European digital call If the shock price P1 (t) exceeds a certain boundary K at time T , the owner of the call is paid B ∗ , otherwise he gets nothing. Choose B ∗ = 1 Final payment B = 1{P1 (T )≥K } Black-Scholes model (d = m = 1, b, r , σ constant, σ > 0) and last Corollary yields ˆ X (t) = EQ e−r (T −t) · 1{P1 (T )≥K } |Ft = e−r (T −t) · Q (P1 (T ) ≥ K |P1 (t) ) . 215 / 477
  • 216. European digital call For fixed t we have P1 (T ) ≥ K ⇐⇒ K σ2 ln P1 (t) − r− 2 (T − t) Q Q W (T ) − W (t) ≥ σ :=K As W Q (T ) − W Q (t) is normally distributed with expectation 0 and variance T − t, we obtain 216 / 477
  • 217. European digital call ∞ 2 ˆ 1 x − 2(T −t) X (t) = e−r (T −t) e dx ˆ K 2π(T − t)  P1 (t) 2  ln K + r−σ2 (T − t) = e−r (T −t) Φ  √  σ T −t 217 / 477
  • 218. Outline 3 Option Pricing Introduction Examples The Replication Principle Arbitrage Opportunity Continuation Partial Differential Approach (PDA) Arbitrage & Option Pricing 218 / 477
  • 219. Option pricing by PDA General Assumptions now Consider a Black-Scholes model, i.e., d =m=1 and b, r , σ constant with σ > 0. 219 / 477
  • 220. Option pricing by PDA Proposition-Part I Let there exist a polynomially bounded solution f : [0, T ] × (0, ∞)d −→ R, i.e. k max |f (t, p)| ≤ M 1 + p 0≤t≤T for a fixed M > 0, k ∈ N, p ∈ (0, ∞)d , for the Cauchy problem 1 ft + aij pi pj fpi pj + rpi fpi − rf = 0 2 1≤i,j≤d 1≤i≤d on [0, T ) × Rd , 220 / 477
  • 221. Option pricing by PDA Proposition–Part II f (T , p1 , . . . , pd ) = g(p1 , . . . , pd ) for p ∈ Rd such that f is continuous and f ∈ C 1,2 [0, T ] × (0, ∞)d . Further, let EQ (g(P1 (T ), . . . , Pd (T ))) < ∞, where EQ = EQ,T . Then the price XB (t) of the contingent claim B = g(P1 (T ), . . . , Pd (T )) in the d -dimensional Black-Scholes model is given by XB (t) = f (t, P1 (t), . . . , Pd (t)). 221 / 477
  • 222. Option pricing by PDA Proposition–Part III Further for 1 ≤ i ≤ d , Ψi (t) = fpi (t, P1 (t), . . . , Pd (t)) f (t, P1 (t), . . . , Pd (t)) − Ψi (t)Pi (t) 1≤i≤d Ψ0 (t) = P0 (t) is a replication strategy for B. 222 / 477
  • 223. SDE Definition–Part I If on (Ω, F, P) there exists a d -dimensional continuous process {(X (t), Ft )}t≥0 with X (0) = x, x ∈ Rd fixed, t t Xi (t) = xi + bi (s, X (s)) ds + σij (s, X (s)) dWj (s) 0 0≤j≤m 0 P-a.s. for all t ≥ 0, 1 ≤ i ≤ d , satisfying t   |bi (s, X (s))| + 2 σij (s, X (s)) ds < ∞ 0 1≤j≤m P-a.s. for all t ≥ 0, 1 ≤ i ≤ d , 223 / 477
  • 224. SDE Definition–Part II then X (t) is called a strong solution of the stochastic differential equation (SDE) dX (t) = b(t, X (t)) dt + σ(t, X (t)) dW (t) X (0) = x where b : [0, ∞) × Rd −→ Rd , σ : [0, ∞) × Rd −→ Rd,m are given functions (m dimension of Brownian motion). 224 / 477
  • 225. Existence and uniqueness of solutions of SDEs Theorem–Part I Let the coefficients b(t, x), σ(t, x) of the SDE be continuous functions with b(t, x) − b(t, y) + σ(t, x) − σ(t, y) ≤ K x −y 2 2 b(t, x) + σ(t, x) ≤ K2 1 + x 2 for all t ≥ 0, x, y ∈ Rd and a constant K > 0. 225 / 477
  • 226. Existence and uniqueness of solutions of SDEs Theorem–Part II Then there exists a continuous, strong solution {(X (t), Ft )}t≥0 of the SDE with 2 2 E X (t) ≤C 1+ x eCT ∀ t ∈ [0, T ] for some constant C = C(K , T ) and T > 0. Further, X (t) is unique up to indistinguishability. 226 / 477
  • 227. Existence and uniqueness of solutions of SDEs Proof plan Uniqueness Existence – some estimates Existence – convergence of the iteration Solution property 227 / 477
  • 228. Existence and uniqueness of solutions of SDEs Lemma The solution X of the SDE satisfies for m ≥ 1 and fixed T > 0 2m 2m E max X (s) ≤C 1+ x eCt 0≤s≤t for all t ∈ [0, T ] and suitable constant C = C(T , K , m, d ). Notation Solution of the SDE with X (t) = x denote X t,s (s) E (. . . X t,s (s) . . .) = E t,s (. . . X (s) . . .) 228 / 477
  • 229. SDEs Definition Be X (t) unique solution of the SDE under the introduced conditions. For f : Rd −→ Rd , f ∈ C 2 Rd , the operator At defined by 1 ∂2f (At f )(x) := aik (t, x) (x) 2 ∂xi ∂xk 1≤i≤d 1≤k ≤d ∂f + bi (t, x) (x) ∂xi 1≤i≤d with aik (t, x) := σij (t, x)σkj (t, x) 1≤j≤m is called the characteristic operator corresponding to X (t). 229 / 477
  • 230. The Cauchy problem Description of the Cauchy problem I Be T > 0 fixed. Consider the following Cauchy problem corresponding to operator At : Find a function v(t, x) : [0, T ] × Rd −→ R with −vt + kv = At v + g on [0, T ] × Rd v(T , x) = f (x) for x ∈ Rd where f : Rd −→ R , g : [0, T ] × Rd −→ R , k : [0, T ] × Rd −→ (0, ∞). 230 / 477
  • 231. The Cauchy problem Description of the Cauchy problem II To ensure the uniqueness of a solution we require that v obeys a polynomial growth condition: 2µ max |v(t, x)| ≤ M 1 + x 0≤t≤T with M > 0 , µ ≥ 1. 231 / 477
  • 232. The Cauchy problem Description of the Cauchy problem III Usually we assume that for suitable constants L, λ the functions f , g, k are continuous with 2λ |f (x)| ≤ L 1 + x , L > 0, λ ≥ 1 or f (x) ≥ 0 2λ |g(t, x)| ≤ L 1 + x , L > 0, λ ≥ 1 or g(t, x) ≥ 0. 232 / 477
  • 233. The Feynman-Kac representation Theorem–Part I Let the inequalities for f and g be satisfied. Let further v(t, x) : [0, T ] × Rd −→ R be continuous solution of the Cauchy problem with v ∈ C 1,2 ([0, T ) × Rd ). Denote by At the characteristic operator corresponding to the unique solution X (t) of the SDE with continuous coefficients b, σ with bi (t, x), σ(t, x) : [0, ∞) × Rd −→ R for 1 ≤ i ≤ d , 1 ≤ j ≤ m. 233 / 477
  • 234. The Feynman-Kac representation Theorem–Part II If v(t, x) satisfies the polynomial growth condition we have the representation  T − k (θ,X (θ)) dθ v(t, x) = E t,x f (X (T )) e t + T s  − k (θ,X (θ)) dθ + g(s, X (s)) e t ds . t In particular, v(t, x) is unique solution. 234 / 477
  • 235. Outline 3 Option Pricing Introduction Examples The Replication Principle Arbitrage Opportunity Continuation Partial Differential Approach (PDA) Arbitrage & Option Pricing 235 / 477
  • 236. General assumptions General Assumptions Consider market with d + 1 traded securities with positive prices ˆ P0 (t), . . . , Pd (t). Prices shall be Ito processes with respect to an m-dimensional Brownian motion {(Wt , Ft )}t∈[0,∞) with m ≥ d where {Ft }t∈[0,∞) is the Brownian filtration. 236 / 477
  • 237. General assumptions Notations As trading strategy ϕ(t) = (ϕ0 (t), . . . , ϕd (t))T , t ≥ 0, define a (d + 1)-dimensional progressively measurable process such that the stochastic integrals T T ϕi (s) dPi (s) , ˆ ϕi (s) d Pi (s), 0≤i≤d 0 0 exist for all T ≥ 0 where P (t) Pi (t) := i ˆ P0 (t) denotes the discounted price process. 237 / 477
  • 238. General assumptions Notations Wealth process X (t) corresponding to the trading strategy ϕ(t) and the self-financing condition are defined by X (t) = ϕi (t)Pi (t) 0≤i≤d t = x+ ϕi (s) dPi (s) 0≤i≤d 0 P-a.s., for all t ≥ 0. 238 / 477
  • 239. Equivalent martingale measures Definition A probability measure Q defined on (Ω, FT ) equivalent to P (P and Q have same zero sets) is called an equivalent martingale measure for P0 (t), . . . , Pd (t) if the discounted prices ˆ Pi (t) Pi (t) = (1 ≤ i ≤ m), t ∈ [0, T ] P0 (t) are martingales with respect to Q. 239 / 477
  • 240. Equivalent martingale measures Theorem All martingale measures Q for P0 (t), . . . , Pd (t) equivalent to P can be obtained from P by a Girsanov transformation with an m-dimensional progressively measurable stochastic process {(θ(t), Ft )}t≥0 where for all t ≥ 0 we have t θi2 (s) ds < ∞ P-a.s. (1 ≤ i ≤ m) 0 and where Z (t, θ) is a martingale with respect to P. In particular, Q is given as Q(A) := QT (A) := E (1A · Z (T , θ)) ∀ A ∈ FT . 240 / 477
  • 241. Equivalent martingale measures =⇒ no arbitrage Theorem If there exists an equivalent martingale measure then the market given by the price process P0 (t), . . . , Pd (t) contains no arbitrage opportunity. 241 / 477
  • 242. Completeness of the market Definition (i) A contingent claim B is a non-negative FT -measureable random variable with 1 EQ B <∞ P0 (T ) for all equivalent martingale measures Q. (ii) B is called attainable if there exists an admissible trading strategy ϕ(t) with wealth process X (t) and B = X (T ) P-a.s. ˆ such that X (t) = X (t)/P0 (t) is martingale with respect to some equivalent martingale measure Q. 242 / 477
  • 243. Completeness of the market Theorem The security market under examination is complete (i.e. each contingent claim is attainable) if and only if there exists a unique equivalent martingale measure Q. 243 / 477
  • 244. Option pricing in incomplete markets Definition A market in which not every contingent claim is attainable is called incomplete. Possible reasons for incomplete markets can be trading constraints to invest into particular stock. additional random fluctuations in the market coefficients. In an incomplete market, typically the σ-algebra FT is bigger than the one generated by final wealths produced by admissible trading strategies T X (T ) = x + ϕi (s) dPi (s). 0≤i≤d 0 244 / 477
  • 245. Prices of attainable contingent claims Theorem The unique price process X ∗ (t) of an attainable contingent claim B is given by P0 (t) X ∗ (t) = EQ B Ft (t ∈ [0, T ]), P0 (T ) where Q is an equivalent martingale measure. 245 / 477
  • 246. Option price and equivalent martingale measure Theorem Let Q be an equivalent martingale measure to P. Let B be an arbitrary not necessary attainable contingent claim. If we choose Q P0 (t) XB (t) := EQ B Ft P0 (T ) as price of the contingent claim then the extended security market consisting of d + 1 securities and the contingent claim contains no arbitrage opportunity. 246 / 477
  • 247. Market numeraire and numeraire invariance Definition ˆ Let {(Y (t), Ft )}t∈[0,T ] be a strictly positive Ito process and discount process. Then we call such a discount process a numeraire. Questions: Does change of numeraire (i.e., the choice of a numeraire different from P0 (t)) affect the option price or its calculation? Does there exist a numeraire such that the option price is given as expected value of final payment B with respect to original measure P when discounted by this numeraire? 247 / 477
  • 248. Market numeraire and numeraire invariance General Assumptions Consider the complete market model with d = m and t t 1 1 2 = exp r (s) + θ(s) ds + θ(s)T dW (s) . H(t) 2 0 0 By using the product rule and the SDE of the stock prices, we can verify that H(t) · Pi (t) are P-martingales for 0 ≤ i ≤ d . 1 H(t) forms the wealth process corresponding to the admissible pair (π, c) = (σ(t)−1 (b(t) − r (t) 1), 0) ∈ A(1). This numeraire can be replicated at market by trading in suitable way (market numeraire/numeraire portfolio). 248 / 477
  • 249. Option price and equivalent martingale measure Theorem 1 In complete market H(t) is the unique numeraire such that the corresponding discounted price processes H(t) Pi (t), 0 ≤ i ≤ d , are martingales with respect to P. 249 / 477
  • 250. Numeraire invariance in the complete market Theorem Consider the complete market model with constant market coefficients r (t) ≡ r , b(t) ≡ b, σ(t) ≡ σ > 0. Then we have (i) The process ZY (t) = H(t) Y π (t) is P-martingale for all constant portfolio processes π(t) ≡ π, where Y π (t) is wealth process corresponding to π. (ii) The corresponding probability measure QY with ZY (T ) = dQY is dP unique equivalent martingale measure for price processes discounted by Y π (t). (iii) Fair price p of a contingent claim B with E (B µ ) < ∞ for some ˆ µ > 1 is given as 1 ˆ p = E (H(T )B) = E B Yπ (T ) if Y π (t) is numeraire of part (i)/(ii). 250 / 477
  • 251. Outline 4 Pricing of Exotic Options and Numerical Algorithms 251 / 477
  • 252. Outline 4 Pricing of Exotic Options and Numerical Algorithms Introduction Examples Examples Equivalent Martingale Measure Exotic Options with Explicit Pricing Formulae Weak Convergence of Stochastic Processes Monte-Carlo Simulation Approximation via Binomial Trees The Pathwise Binomial Approach of Rogers and Stapleton 252 / 477
  • 253. General assumptions General assumptions Consider a Black-Scholes model with d = m and constant coefficients b, r , σ with σ > 0 (or σ regular if d > 1). All options are assumed to be of European type. 253 / 477
  • 254. Outline 4 Pricing of Exotic Options and Numerical Algorithms Introduction Examples Examples Equivalent Martingale Measure Exotic Options with Explicit Pricing Formulae Weak Convergence of Stochastic Processes Monte-Carlo Simulation Approximation via Binomial Trees The Pathwise Binomial Approach of Rogers and Stapleton 254 / 477
  • 255. Exotic options Options on Minimum/Maximum of the Stock Price European Call on Maximum given by the terminal payment + B= max P1 (t) − K . 0≤t≤T Barrier Options They have a zero value if stock price exceeds certain barrier or have a positive payment at time T if a certain barrier is reached. 255 / 477
  • 256. Outline 4 Pricing of Exotic Options and Numerical Algorithms Introduction Examples Examples Equivalent Martingale Measure Exotic Options with Explicit Pricing Formulae Weak Convergence of Stochastic Processes Monte-Carlo Simulation Approximation via Binomial Trees The Pathwise Binomial Approach of Rogers and Stapleton 256 / 477
  • 257. Some examples for exotic options Down-and-out Call B = (P1 (T ) − K1 )+ 1 min P1 (t)>K2 0≤t≤T Down-and-in Call B = (P1 (T ) − K1 )+ 1 min P1 (t)≤K2 0≤t≤T Double-barrier Call B = (P1 (T ) − K1 )+ 1 min P1 (t)>K2 , max P1 (t)<K3 0≤t≤T 0≤t≤T with K2 < K1 < K3 . 257 / 477
  • 258. Outline 4 Pricing of Exotic Options and Numerical Algorithms Introduction Examples Examples Equivalent Martingale Measure Exotic Options with Explicit Pricing Formulae Weak Convergence of Stochastic Processes Monte-Carlo Simulation Approximation via Binomial Trees The Pathwise Binomial Approach of Rogers and Stapleton 258 / 477
  • 259. Price of European option The price of a European option is given by p = EQ e−rT B , ˆ where Q is unique equivalent martingale measure and EQ its expectation. Assume Stock prices are given as solutions of the SDE (1 ≤ i ≤ d )   dPi (t) = Pi (t) r dt + σij dWj (t) . 1≤j≤d 259 / 477
  • 260. Outline 4 Pricing of Exotic Options and Numerical Algorithms Introduction Examples Examples Equivalent Martingale Measure Exotic Options with Explicit Pricing Formulae Weak Convergence of Stochastic Processes Monte-Carlo Simulation Approximation via Binomial Trees The Pathwise Binomial Approach of Rogers and Stapleton 260 / 477
  • 261. Path independent options on one stock Binary Option–Part I Terminal payment in T of binary option with bound K given by Call Put Bd = 1{P1 (T )>K } , Bd = 1{P1 (T )<K } if the final price P1 (T ) exceeds K (in the case of a call) or is smaller than K (in the case of a put). 261 / 477
  • 262. Path independent options on one stock Binary Option–Part II Pricing of these digital options Xd (t) = e−t(T −t) Φ d2 (t) , Call Xd (t) = e−t(T −t) Φ −d2 (t) Put with P1 (t) 2 ln K + r−σ2 (T − t) d2 (t) = √ , σ T −t where Φ is distribution function of standard normal distribution. 262 / 477
  • 263. Path independent options on one stock Paylater Options–Part I Final payoffs Call BPL = P1 (T ) − K + D Call · 1{P1 (T )≥ K } Put BPL = K − D Put − P1 (T ) · 1{P1 (T )≤K } where D Call , D Put determined in such a way that prices of paylater options are zero at initial time. 263 / 477
  • 264. Path independent options on one stock Paylater Options–Part II Final payoffs decomposed Call BPL = B Call − D Call Bd Call Put BPL = B Put − D Put Bd . Put With Call Put XPL (0) = 0 = XPL (0) X Call (0) X Put (0) D Call = Call , D Put = Put Xd (0) Xd (0) we get 264 / 477
  • 265. Path independent options on one stock Paylater Options–Part III Call p1 Φ(d1 (0)) XPL = P1 (t)Φ(d1 (t)) − Φ(d2 (t)) ert Φ(d2 (0)) Put p1 Φ(−d1 (0)) XPL = −P1 (t)Φ(−d1 (t)) + Φ(−d2 (t)) ert Φ(−d2 (0)) 265 / 477
  • 266. Path independent options on one stock Proposition–Part I (i) Given K > 0, maturity T1 and strike K1 there exists a unique p ∗ > 0 for T ≤ T1 such that for P1 (T ) = p ∗ we have X Call (T ) = X Call (T , p ∗ ) = K . (ii) With P1 (t) 2 ln p∗+ r+σ 2 (T − t) g1 (t) = √ , σ T −t √ g2 (t) = g1 (t) − σ T − t, 266 / 477
  • 267. Path independent options on one stock Proposition–Part II P1 (t) 2 ln K1 + r + σ (T1 − t) 2 h1 (t) = √ , σ T1 − t h2 (t) = h1 (t) − σ T1 − t, we get the price of a call on a call Xcom (t) = P1 (t)Φ(ρ1 ) (g1 (t), h1 (t)) CC − K1 e−r (T1 −t) Φ(ρ1 ) (g2 (t), h2 (t)) − Ke−r (T −t) Φ(g2 (t)) 267 / 477
  • 268. Path independent options on one stock Proposition–Part III for t ∈ [0, T ], where Φ(ρ) (x, y) is distribution function of bivariate standard normal distribution with correlation coefficient ρ and with T −t ρ1 := , T1 − t X 0 1 ρ1 ∼ N , . Y 0 ρ1 1 268 / 477
  • 269. Path independent options on one stock Lemma–Part I If X and Y independent random variables with X ∼ N (µ, σ 2 ) , Y ∼ N (0, 1) ˜ then for x , α, β ∈ R , α > 0 we have ∞ ϕµ,σ2 (x) Φ(αx + β) dx ˜ = P(X ≥ x , Y ≤ αX + β) ˜ x ˜ = P(X ≥ x , Z ≤ β), 269 / 477
  • 270. Path independent options on one stock Lemma–Part II where µ σ2 −ασ 2 (X , Z ) ∼ N , −αµ −ασ 2 1 + α2 σ 2 =⇒ ϕµ,σ2 is density function of the normal distribution with mean µ and variance σ 2 . 270 / 477
  • 271. Options on more than one underlying Stock Indexed Options Consider 2-dimensional Black-Scholes model with given stock prices dPi (t) = Pi (bi dt + σi1 dW1 (t) + σi2 dW2 (t)) , Pi (0) = pi (i = 1, 2) Be a1 , a2 ∈ R+ . An indexed option with parameters a1 , a2 is then given by final payment Bind = (a1 P1 (T ) − a2 P2 (T ))+ . 271 / 477
  • 272. Options on minimum/maximum of 2 stocks Call on minimum/maximum Call + Bmin = min P1 (T ), P2 (T ) − K Call + Bmax = max P1 (T ), P2 (T ) − K Put on minimum/maximum Put + Bmin = K − min P1 (T ), P2 (T ) Put + Bmax = K − max P1 (T ), P2 (T ) 272 / 477
  • 273. Options on minimum/maximum of 2 stocks Proposition–Part I The prices of the minimum/maximum options are given by ˜ Xmin (0) = p1 Φ(˜) (d1 , d3 ) + p2 Φ(ρ) (d2 , d4 ) Call ρ ˜ √ √ −Ke−rT Φ(ρ) d1 − σ1 T , d2 − σ2 T , ˜ ˜ Xmin (0) = Xmin (0) + Ke−rT − p1 Φ (d3 ) − p2 Φ (d4 ) , Put Call Call Call Call Call Xmax (0) = X(1) (0) + X(2) (0) − Xmin (0), Put Put Put Put Xmax (0) = X(1) (0) + X(2) (0) − Xmin (0), 273 / 477
  • 274. Options on minimum/maximum of 2 stocks Proposition–Part II Call Put where X(i) , X(i) denote prices of ordinary European calls/puts on stock i with strike K (i = 1, 2) and σi ˜ := 2 σi1 + σi2 2 (i = 1, 2), σ11 σ21 + σ12 σ22 ρ := , σ1 σ2 ˜ ˜ ρ σ2 − σ1 ˜ ˜ ρ := ˜ , σ ˜ ρ˜1 − σ2 σ ˜ ρ := ˜ , σ 274 / 477
  • 275. Options on minimum/maximum of 2 stocks Proposition–Part III pi ln K + r + 0.5˜i2 T σ di := √ , σi T ˜ p1 (−1)i ln p2 − 0.5σ 2 T di+2 := √ (i = 1, 2). σ T 275 / 477
  • 276. Path dependent options One-sided barrier options Owner receives final payoff of European call/put if stock price does not exceed/does exceed given barrier before time T . Look at down-and-out call and down-and-in call: Call Bdo = (P1 (T ) − K )+ · 1{P1 (t)>b ∀ t∈[0,T ]} , Call Bdi = (P1 (T ) − K )+ · 1{∃ t∈[0,T ]: P1 (t)≤b} . 276 / 477
  • 277. Path dependent options Lemma Be M(t) := max0≤s≤t W (s) running maximum of 1-dimensional Brownian motion W (t). Then for x ≥ 0, x ≥ w , we have w 2x−w (i) P (W (t) ≤ w , M(t) < x) = Φ √ t −1+Φ √ t ˜ ˜ ˜ (ii) For µ ∈ R be W (t) := W (t) + µ · t, M(t) := max0≤s≤t W (s). Thus we have ˜ ˜ w − µt w − 2x − µt P W (t) ≤ w , M(t) < x = Φ √ − e2µ x Φ √ t t 277 / 477
  • 278. Outline 4 Pricing of Exotic Options and Numerical Algorithms Introduction Examples Examples Equivalent Martingale Measure Exotic Options with Explicit Pricing Formulae Weak Convergence of Stochastic Processes Monte-Carlo Simulation Approximation via Binomial Trees The Pathwise Binomial Approach of Rogers and Stapleton 278 / 477
  • 279. General assumptions General assumptions for this section We consider the probability space (Ω, F, P) = (C[0, 1], B(C[0, 1]), P), i.e., the space of continuous, real-valued functions on [0,1] equipped with the corresponding Borel σ-field and a probability measure P. 279 / 477
  • 280. Stochastic process with distribution P The function valued random variable X on (Ω, F, P) given by X (ω) := ω, ω ∈ C[0, 1] defines a real-valued stochastic process with distribution P. Value of the process at time t ∈ [0, 1]: X (t, ω) := πt ◦ X (ω) := ω(t). → projection on the "t-th coordinate" of ω. 280 / 477
  • 281. Notion of convergence The notion of convergence of stochastic process Xn via the usual weak convergence of random variables n→∞ Xn (t) − − X (t) for all t ∈ [0, 1] in distribution −→ is too weak. ⇒ Consider weak convergence of probability measures on metric spaces 281 / 477
  • 282. Weak convergence Definition Let (S, B(S)) be a metric space with metric ρ and the Borel-σ-field B(S) over S. Let further P, Pn , n ∈ N, be probability measures on (S, B(S)). Then we say that the sequence Pn converges weakly (or converges in distribution) to P if for every continuous and bounded real-valued function f on S we have n→∞ f dPn − − −→ f dP. S S Special case: Weak convergence of stochastic processes with continuous paths. Remark: (C[0, 1], B(C[0, 1])) is a metric space with the metric ρ(x, y) = sup |x(t) − y(t)|. 0≤t≤1 282 / 477
  • 283. Weak convergence Definition The sequence of continuous stochastic processes {Xn (t)}t∈[0,1] converges weakly (or in distribution) to X if for all f ∈ C(C[0, 1], R) we have n→∞ Ef (Xn ) − − Ef (X ). −→ Remark: C(C[0, 1], R) is the space of the uniformly continuous, bounded functionals on C[0, 1]. 283 / 477
  • 284. Weak convergence Stochastic process Xn is defined on (C[0, 1], B(C[0, 1]), Pn ). Process X is defined on (C[0, 1], B(C[0, 1]), P). P, Pn are probability measures on (C[0, 1], B(C[0, 1])). n→∞ Ef (Xn ) = f (Xn ) dP = f dPn − − −→ f dP = f (X ) dP = Ef (X ). ⇒ The weak convergence of stochastic processes is represented as the weak convergence of the probability measures Pn → P. 284 / 477
  • 285. Weak convergence Theorem Let P, Pn , n ∈ N, be probability measures on the metric space (S, B(S)) endowed with the metric ρ. Further, let h : S → S ′ be a measurable mapping into a metric space S ′ with metric ρ′ and Borel-σ-field B(S ′ ). If for the set Dh of points of discontinuity of h we have P(Dh ) = 0, then we get n→∞ n→∞ Pn − − P in distribution ⇒ Pn · h−1 − − P · h−1 in distribution. −→ −→ ⇒ Weak convergence is preserved under continuous mappings. 285 / 477
  • 286. Weak convergence Corollary If the sequence Xn of continuous stochastic processes converges weakly to the continuous process X , then for every fixed t ∈ [0, 1] the random variables Xn (t) converge in distribution to X (t). 286 / 477
  • 287. Weak convergence Remark Define the projections πt1 ,...,tk : C[0, 1] → Rk by πt1 ,...,tk (ω) = (ω(t1 ), . . . , ω(tk )) for 0 ≤ t1 < . . . < tk ≤ 1. Then, we have n→∞ Xn − − X in distribution −→ n→∞ ⇒ (Xn (t1 ), . . . , Xn (tk )) − − (X (t1 ), . . . , X (tk )) in distribution. −→ 287 / 477
  • 288. Weak convergence Remark Weak convergence of stochastic processes implies convergence of the finite-dimensional distributions. Convergence of all finite-dimensional distributions Pn · πt−1 k 1 ,...,t does not in general imply convergence of the distributions Pn of the corresponding processes. If the sequence of the Pn is relatively compact (i.e., each subsequence contains a weakly convergent subsequence), then the convergence of the finite-dimensional subsequences implies weak convergence of the Pn . 288 / 477
  • 289. Approximation of the one-dimensional Brownian motion Algorithm (1) Choose a sequence {ξn }n∈N of i.i.d. random variables of a simple form with E (ξi ) = 0, Var (ξi ) = σ 2 < ∞ and set n S0 := 0, Sn := ξi . i=1 Example: ξi = Yi − q with Yi ∼ B(1, q). 289 / 477
  • 290. Approximation of the one-dimensional Brownian motion Algorithm (2) By means of linear interpolation construct a stochastic process Xn (t) with continuous paths of that sequence 1 1 Xn (t, ω) = √ S[nt] (ω) + (nt − [nt]) √ ξ[nt]+1 (ω) σ n σ n for t ∈ [0, 1], n ∈ N, i.e., we have k 1 Xn n,ω = √ Sk (ω), σ n k k +1 and for t ∈ n, n we obtain Xn (t) by linear interpolation. 290 / 477
  • 291. Approximation of the one-dimensional Brownian motion Algorithm (3) The finite-dimensional distributions of Xn converge in distribution to that of a Brownian motion. [ns] n→∞ • From n −− −→ s and the central limit theorem, we obtain 1 n→∞ √ S[ns] − − W (s) in distribution. −→ σ n • Chebychev’s inequality yields 1 1 n→∞ Xn (s) − √ S[ns] ≤ √ |ξ[ns]+1 | − − 0 in probability. −→ σ n σ n Hence, n→∞ Xn (s) − − W (s) −→ in distribution. 291 / 477
  • 292. Approximation of the one-dimensional Brownian motion Algorithm • Due to the independence of the ξi and the theorem of Slut- sky this results in 1 1 n→∞ √ S[ns] , √ S[nt] − S[ns] − − (Ws , Wt − Ws ) −→ σ n σ n for s < t. From this we get n→∞ (Xn (s), Xn (t) − Xn (s)) − − (Ws , Wt − Ws ) in distribution. −→ Slutsky’s theorem implies n→∞ (Xn (s), Xn (t)) − − (Ws , Wt ) in distribution. −→ Analogously: Convergence of finite tuples of Xn (ti )-components. 292 / 477
  • 293. Approximation of the one-dimensional Brownian motion Algorithm (4) Show that the sequence of the distributions Pn on (C[0, 1], B(C[0, 1])) corresponding to Xn is relatively compact. 293 / 477
  • 294. Donsker’s theorem Donsker’s theorem Let {ξn }n∈N be an i.i.d. sequence with E (ξi ) = 0 and 0 < Var (ξi ) = σ 2 < ∞. Then, the sequence Xn of stochastic processes defined by 1 1 Xn (t, ω) = √ S[nt] (ω) + (nt − [nt]) √ ξ[nt]+1 (ω), t ∈ [0, 1], n ∈ N σ n σ n converges weakly to the one-dimensional Brownian motion W (t), t ∈ [0, 1]. 294 / 477
  • 295. Donsker’s theorem Remark The convergence assertion and the limiting distribution are independent of the exact choice of ξi (Donsker’s invariance principle). Donsker’s theorem can be viewed as the "process version" of the central limit theorem. Donsker’s theorem can be assumed valid for arbitrary intervals [0, T ]. 295 / 477
  • 296. Donsker’s theorem for triangular schemes Donsker’s theorem for triangular schemes The random variables ξn1 , . . . , ξnkn , n ∈ N, kn ∈ N, are assumed to be 2 i.i.d. with E (ξn1 ) = 0 and 0 < Var (ξn1 ) = σn1 ≤ c, where c > 0. Let Sni := ξn1 + . . . + ξni , 1 ≤ i ≤ kn , 2 2 2 2 sni := σn1 + . . . + σni = i · σn1 , 2 2 2 sn := snkn = kn · σn1 . Define the process Xn (t), t ∈ [0, 1], by Xn (0) := 0, 2 sn Xn /sn i 2 := Sni/sn , i = 1, . . . , kn , 2 2 2 2 and via linear interpolation on the intervals sni−1 /sn , sni /sn . If kn → ∞ and sn → ∞ for n → ∞, then Xn converges weakly to the Brownian motion W . 296 / 477
  • 297. Donsker’s theorem Remark We have n→∞ E (h(Xn )) − − E (h(X )) −→ for continuous and bounded functionals h : C[0, T ] → R. Not sufficient for practical applications. If in the Black-Scholes model we approximate the Brownian motion by a sequence of processes Xn , Donsker’s theorem would not directly imply n→∞ E eb·T +σ·Xn (T ) − − E eb·T +σ·W (T ) −→ as the exponential functional is not bounded. Additionally, we need the uniform integrability of the sequence exp(σXn (t)). 297 / 477
  • 298. Weak convergence Theorem Let the sequence of random variables {Xn }n∈N be uniformly integrable. Assume further that we have n→∞ Xn − − X −→ in distribution. Then this implies n→∞ E (Xn ) − − E (X ). −→ 298 / 477
  • 299. Outline 4 Pricing of Exotic Options and Numerical Algorithms Introduction Examples Examples Equivalent Martingale Measure Exotic Options with Explicit Pricing Formulae Weak Convergence of Stochastic Processes Monte-Carlo Simulation Approximation via Binomial Trees The Pathwise Binomial Approach of Rogers and Stapleton 299 / 477
  • 300. The basic idea Description of the idea The basis of Monte-Carlo simulation is the strong law of large numbers ⇒ Arithmetic mean of independent, identically distributed ran- dom variables converges towards their mean almost surely. Computing an option price = Computing the discounted expectation (with respect to the equivalent martingale measure) of the payoff B. 300 / 477
  • 301. The basic idea Algorithm: Determine the option price via Monte-Carlo simulation (1) Simulate n independent realizations Bi of the final payoff B. n 1 (2) Choose Bi · e−rT as an approximation for the option price n i=1 EQ (e−rT B). 301 / 477
  • 302. Simulation of the payoff B Assume that the payment B is a functional of the price process P1 (t), t ∈ [0, T ]. Simulate a path P1 (t) of the stock price process with respect to the equivalent martingale measure Q. A path can only be simulated approximately (it is given by infinitely many values). 302 / 477
  • 303. Simulation of the payoff B Procedure (1) Divide the interval [0, T ] into N ≫ 1 equidistant parts. (2) Generate N independent random numbers Yi which are N (0, 1)-distributed. (3) From those, simulate an (approximate) path W (t) of the Brownian motion on [0, T ]: W (0) = 0, T T T W j· N = W (j − 1) · N + N · Yj , j = 1, . . . , N, T W (t) = W (j − 1) · N + (t − (j − 1) · N ) · N T T · W j· T N − W (j − 1) · T N , T T for t ∈ (j − 1) · N ,j·N 303 / 477
  • 304. Simulation of the payoff B Procedure (4) Use this to generate an (approximate) path of the price process P1 (t): 1 r − 2 σ2 t P1 (t) = p1 · e · eσ·W (t) , t ∈ [0, T ]. (5) With this simulated path of the price process compute an estimate for the payoff B. Example (European call): Bi = (P1 (T ) − K )+ . 304 / 477
  • 305. Simulation of the payoff B Remark For the practical realization of the computation of Bi in step 5 it often proves to be more convenient to do the interpolation in step 4 and not in step 3: P1 (0) = p1 , 1 T T T T r − 2 σ2 N σ· N ·Yj , P1 j · N = P1 (j − 1) · N ·e ·e j = 1, . . . , N, T P1 (t) = P1 (j − 1) · N + (t − (j − 1) · N ) · N T T · P1 j · T N − P1 (j − 1) · T N , T T for t ∈ (j − 1) · N , j · N . For large N the differences between both methods are negligible. 305 / 477
  • 306. Convergence of the method (N) Let P1 (t), t ∈ [0, T ] be the approximate price process. If B is a continuous and bounded functional on C([0, T ]) then by Donsker’s theorem we have the convergence (N) N→∞ EQ B P1 (t), t ∈ [0, T ] − − → EQ B(P1 (t), t ∈ [0, T ]) . −− If B is a continuous functional on C([0, T ]), then the convergence of the option price is guaranteed if the family (N) B P1 (t), t ∈ [0, T ] N∈N is uniformly integrable. 306 / 477
  • 307. Convergence of the method Showing the uniformly integrability can be tricky for specific choices of B. Then for a given N the expected value (N) EQ B P1 (t), t ∈ [0, T ] will be approximated by the arithmetic mean n 1 (N) B P1,i (t), t ∈ [0, T ] n i=1 (N) by the strong law of large numbers. Here, P1,i (t) are different paths which have been generated according to the above procedure. 307 / 477
  • 308. Advantages/Disadvantages of the Monte-Carlo method Advantages Easy to implement. Reasonable number generators are available. Every exotic option can be approximated. Refinements to obtain faster convergence are available. Disadvantages Method is time-consuming (n and N have to be very large). Frequently n and N have to be so large that the whole reservoir of pseudo-random numbers is used and an already used sequence of pseudo-random numbers has to be reused (→ independence assumption of the different simulations is no longer true). Strong dependence of the method on the quality of the random number generator. 308 / 477
  • 309. Outline 4 Pricing of Exotic Options and Numerical Algorithms Introduction Examples Examples Equivalent Martingale Measure Exotic Options with Explicit Pricing Formulae Weak Convergence of Stochastic Processes Monte-Carlo Simulation Approximation via Binomial Trees The Pathwise Binomial Approach of Rogers and Stapleton 309 / 477
  • 310. The basic idea Description of the idea Monte-Carlo simulation relies on the strong law of large numbers. The approximation method via binomial trees can be motivated by the central limit theorem. 310 / 477
  • 311. Cox-Ross-Rubinstein model (n) Stock price process: P1 (i), i = 0, 1, . . . , n Possible paths given by binomial tree → price process starts in p at time t = 0 (n) → in each node the price P1 (i) can increase by the factor u with probability q or by the factor d with probability (1 − q) (where d < u). The probability of a price increase by the factor u and the possible values of the relative price change (n) P1 (i) (n) P1 (i − 1) are the same in each node. The factors u and d satisfy T d < er ∆t < u with ∆t := . n → avoid riskless gains 311 / 477
  • 312. Binomial tree t =0 1 · T /n 2 · T /n T ”time” unp pp ppp ppp pppp ppp NNN NNN NNN NNN NN u2p u n−1 dp mm mmm mmmmm mmm mmm up QQ nnnnn QQQ QQQ nnn QQQ nnn QQQ nnn QQ nnn p PPP udp PPP PPP mmmmm PPP mmm PPP mmmmm PP mmm dp QQQ QQQ QQQ QQQ QQQ Q d 2p ud n−1 p pp ppp ppp ppp ppp NNN NNN NNN NNN NNN d np ”price” 312 / 477
  • 313. Cox-Ross-Rubinstein model (n) Number of "up-moves" of P1 (i): Xn Properties: Xn ∼ B(n, q) (n) P1 (n) = p · u Xn · d n−Xn = p · eXn ·ln(u/d)+n·ln(d) . 313 / 477
  • 314. Cox-Ross-Rubinstein model Special case: 1 q= , b∈R 2 and √ √ u = eb∆t+σ ∆t , d = eb∆t−σ ∆t 1 ln(u) + ln(d ) 1 ln(u) − ln(d ) b= , σ= 2 ∆t 2 ∆t Convergence: (n) √ 2Xn − n P1 (n) = p · exp b · T + σ T √ n n→∞ − − p · exp b · T + σ · W (T ) −→ = P1 (T ) in distribution 314 / 477
  • 315. Cox-Ross-Rubinstein model Note that 2Xn − n √ n has zero mean and variance 1. 2Xn is the sum of n independent, double Bernoulli variables. Important: Convergence of the discrete to the continuous price process. Discounted expected payoff of the option in the discrete model can be computed easily. With increasing degree of fineness of the time discretization, the discounted expected final payment in the discrete-time model will converge to that of the continuous-time model if the family (n) Bn := B P1 (i), i = 0, 1, . . . , n is uniformly integrable. 315 / 477
  • 316. Approximation via binomial trees Algorithm (1) For n ≫ 1 set up a suitable binomial tree for the price process (n) P1 (i) in discrete time. (2) Compute the discounted expected payoff E (n) e−rt Bn in the discrete-time model as an approximation for EQ e−rt B . Remark: The choice of n (i.e., the fineness of the (space and) time discretization) is the essential factor for the accuracy of the approximation. Therefore, the algorithm is performed iteratively for different (increasing) values of n and will be stopped if the sequence of approximations for the option price converges. 316 / 477
  • 317. Choice of parameters in the binomial trees Independence of the identically distributed increments (n) P1 (i) (n) P1 (i − 1) in the binomial tree implies weak convergence of (n) P1 (i), i = 0, 1, . . . , n to P1 (t), t ∈ [0, T ] if the first two moments of the logarithm of increments (n) P1 (i) P1 (i · T ) n ln (n) and ln T P1 (i − 1) P1 (i − 1) · n of the discrete and continuous price processes coincide at times i · T . n 317 / 477
  • 318. Choice of parameters in the binomial trees (s,n) If we define a continuous process P1 (t) via linear interpolation between (n) (n) ln P1 (i − 1) and ln P1 (i) i.e., (s,n) (n) T n ln P1 (t) = ln P1 (i − 1) + t − (i − 1) · n · T (s,n) (n) · ln P1 (i) − ln P1 (i − 1) for t ∈ (i − 1) · T , i · n T n , then this continuous process converges weakly to the stochastic price process P1 (t) if the moment conditions hold. 318 / 477
  • 319. Choice of parameters in the binomial trees Consider the price process with respect to the equivalent martingale measure Q and assume that P1 (t) is a martingale. P0 (t) Binomial measure with respect to time discretization n: Q (n) . Expectation with respect to that measure: E (n) . (n) Fi i∈{0,1,...,n} filtration generated by the price process (n) P1 (i) i∈{0,1,...,n} . 319 / 477
  • 320. Choice of parameters in the binomial trees Moment conditions: (n) 1 2 P1 (∆t) (n) P1 (1) r− 2σ ∆t = EQ ln =E ln (n) P1 (0) P1 (0) = ln(u) · q + ln(d ) · (1 − q), 2 2 P1 (∆t) r − 1 σ 2 (∆t)2 + σ 2 ∆t = EQ ln 2 P1 (0) (n) 2 (n) P1 (1) =E ln (n) P1 (0) = ln(u)2 · q + ln(d )2 · (1 − q). 320 / 477
  • 321. Choice of parameters in the binomial trees Moment conditions: (n) 1 2 P1 (∆t) (n) P1 (1) r− 2σ ∆t = EQ ln =E ln (n) P1 (0) P1 (0) = ln(u) · q + ln(d ) · (1 − q), 2 2 P1 (∆t) r − 1 σ 2 (∆t)2 + σ 2 ∆t = EQ ln 2 P1 (0) (n) 2 (n) P1 (1) =E ln (n) P1 (0) = ln(u)2 · q + ln(d )2 · (1 − q). 321 / 477
  • 322. Choice of parameters in the binomial trees Unknown parameters: u, d "incremental factors", q "probability of an upwards movement" Moment conditions allow free choice of one of the parameters if u, d > 0 and q ∈ (0, 1). Popular choices: 1 1 u= , d < 1 or q = . d 2 322 / 477
  • 323. Choice of parameters in the binomial trees 1 Special case: q = 2 Moment conditions: ln(u · d ) = 2 r − 1 σ 2 ∆t 2 ln(u)2 + ln(d )2 = 2 r − 1 σ 2 (∆t)2 + 2σ 2 ∆t 2 Symmetric in u and d Ansatz: u = eB+C , d = eB−C . √ ⇒ B = r − 1 σ 2 ∆t, C = |σ| · 2 ∆t. 1 √ 1 √ r − 2 σ2 ∆t+|σ|· ∆t r − 2 σ2 ∆t−|σ|· ∆t ⇒u=e , d =e . 323 / 477
  • 324. Choice of parameters in the binomial trees 1 Special case: q = 2 For r > 0 we have 0<d <u and d < er ∆t . To obtain er ∆t < u we must have √ |σ| · ∆t − 1 σ 2 ∆t > 0. 2 Time discretization must be sufficiently fine: T · σ2 n> . 4 324 / 477
  • 325. The option price in the binomial model The stock price model given by a binomial tree together with the possibility of a bond investment at times i · T (with a bond price n P0 (t) = ert ) constitute a complete market. Price of an option = discounted expectation of the payoff B in t = T with respect to the unique equivalent martingale measure Qn . Qn is given by the "upwards probability" q = qn . 325 / 477
  • 326. The option price in the binomial model For given u and d with 0 < d < er ∆t < u we obtain q from the martingale requirement (n) (n) P1 (i) P1 (i − 1) (n) 0 = EQn T − T Fi−1 P0 i · n P0 (i − 1) · n (n) P1 (i − 1) T = · (q · u + (1 − q) · d ) e−r n − 1 P0 (i − 1) · T n as T er n − d q= . u−d 326 / 477
  • 327. The option price in the binomial model For given u and d with 1 √ 1 √ r − 2 σ2 ∆t+|σ|· ∆t r − 2 σ2 ∆t−|σ|· ∆t u=e and d = e q differs from 1/2. The value E (n) e−rT Bn computed as an approximation for the option price EQ e−rT B in the continuous model will in general not coincide with the option price EQn e−rT Bn in the binomial model. 327 / 477
  • 328. The option price in the binomial model The use of binomial trees serves us only as a method of numerical approximation of the expectation EQ e−rT B . The fact that this expectation is an option price has no meaning for the numerical method. There is no reason why the approximating sequence for this expectation should be option prices in some discretized models. 328 / 477
  • 329. The option price in the binomial model Another method of approximation of EQ e−rT B is to determine q, u, d via requiring the equality of the first two moments of the increments of the discrete- and the continuous-time price process. Equality of the first moment of the increments with the independence and identical distribution of all increments in the binomial model imply the martingale condition. Choose u, d such that the second moment of the increments of both price models coincide. The option price in the binomial model (given by n, q, u, d ) is computed as an approximation for the option price in the continuous model. 329 / 477
  • 330. The option price in the binomial model The concept "approximate the option price in the continuous model by the discrete-time model option price" cannot be justified by weak convergence arguments. 330 / 477
  • 331. Computation of the expected discounted payoff in the binomial tree The possibility of an efficient calculation of the expectation E (n) e−rT Bn depends on the type of the functional B (resp. its discretized variant Bn ). Two examples for n = 2: European call Double-barrier knockout call 331 / 477
  • 332. Computation of the expected discounted payoff in the binomial tree The European call 1 Choose q = 2 and market parameters r = 0, σ = 0.5, T = 2, p = 1. 332 / 477
  • 333. Computation of the expected discounted payoff in the binomial tree (2) Binomial tree for P1 (i): t =0 1 2 ”time” 2.117 nn 1/2nnnn n nnn nnn n 1.455 PP 1/2 nnnn PPP nn PPP nnn P nn 1/2 PPPP nnn 1 PPPP n 0.779 PPP 1/2nnnnn PPP n PPP nnn 1/2 PP nnn 0.535 PP PPP PPP P 1/2 PPPP 0.287 ”price” 333 / 477
  • 334. Computation of the expected discounted payoff in the binomial tree The European call Aim: Value a European option with a payoff of the form B = f (P1 (T )). Approximate its value by the two-period binomial tree and the discretized variant of the payoff (2) B2 = f P1 (2) . Discounted expected payoff (discretized model) 1 1 E (2) (B2 ) = f (2.117) + f (0.779) 2 2 1 1 + f (0.779) + f (0.287) 2 2 334 / 477
  • 335. Computation of the expected discounted payoff in the binomial tree European call option with strike K = 0.5 Approximate value 1 1 E (2) (B2 ) = · 1.617 + · 0.279 = 0.54375 4 2 Black-Scholes value 0.5416 335 / 477
  • 336. Computation of the expected discounted payoff in the binomial tree Backwards induction principle Payoff: B = f P1 (T ) . Let (n) Bn = f P1 (n) and T (n) T (n) V (n) i · , P1 (i) := E (n) e−r T −i· n · Bn P1 (i) n be the expected payoff in t = T discounted back to t = i · T , if the n stock price in the binomial model at time t = i · T attains the value n (n) P1 (i). 336 / 477
  • 337. Computation of the expected discounted payoff in the binomial tree Backwards induction principle Expected discounted payoff of the option in the binomial model (n) (n) V (n) T , P1 (n) = f P1 (n) , T (n) V (n) i · , P (i) n 1 1 (n) T (n) T (n) = V (i + 1) · , uP1 (i) + V (n) (i + 1) · , dP1 (i) 2 n n T · e−r n , for i = n − 1, . . . , 0 E (n) e−rT Bn = V (n) (0, p). 337 / 477
  • 338. Backwards induction principle Advantage of the induction scheme: Calculate only n · (n − 1) 2 arithmetic means (although the stock price can follow 2n different paths). Binomial tree = recombining tree 338 / 477
  • 339. Backwards induction principle (modification) How about path-dependent options? In case of path-dependent options path dependency has to be taken into account ⇒ Modification of backward induction recursion. 339 / 477
  • 340. Backwards induction principle (modification) Example: Double-barrier knockout call Payoff: BDB = (P1 (T ) − 0.5)+ · 1 Call P1 (t)∈[0.4, 1.4] for all t∈[0,T ] Payoff (discrete): Call (2) + BDB = P1 (2) − 0.5 ·1 (2) 2 P1 (i)∈[0.4, 1.4], i=0,1,2 = 0.279 · 1 (2) (2) P1 (2)=0.779, P1 (1)=0.535 Double-barrier knockout call price (discrete approximation): 1 E (2) (B2 ) = · 0.279 = 0.06975. 4 340 / 477
  • 341. Backwards induction principle (modification) Example: Double-barrier knockout call Call (2) The final payoff BDB at T = 2 in the state P1 (2) = 0.779 can 2 attain two possible values 0 and 0.279 → typical for path-dependent options. Path-dependent final payoff can lead to a situation that each path of the stock price yields a different final payment. Maximum number of different values of final payments is 2n → complexity of computations and storage. 341 / 477
  • 342. Backwards induction principle (modification) A non-recombining tree: t =0 1 2 ”time” 2.117 1/2nnnn nn nnnn nnn 1.455 PP } PPP }} PPP }} PPP }} 1/2 PP 1/2 }} }} } 0.779 }} }} }} }} 1 AA AA AA AA AA AA 0.779 1/2 AAA n AA 1/2nnnnn AA n nnn A nnn 0.535 PP PPP PPP PPP 1/2 PP 0.287 ”price” 342 / 477
  • 343. Backwards induction principle (modification) Example: Double-barrier knockout call Simplifications to keep backward induction algorithm efficient: The binomial tree which corresponds to the above computation of E (2) BDB Call has the usual recombining form of a binomial tree. 2 (2) The option prices in the node P1 (2) = 0.779 are not unique. The payoff depends on the paths reaching this node 343 / 477
  • 344. Backwards induction principle (modification) Principle of backward induction stays valid. For a general path-dependent option we have to calculate up to 2i−1 arithmetic means at time i (i + 1 in the non-path-dependent case). In the double-barrier knockout case, we can simply proceed backwards in the recombining tree, but setting the option price to zero in all nodes where the knockout-condition is satisfied). Computational complexity is comparable to that of a European non-path-dependent option. 344 / 477
  • 345. Backwards induction principle (modification) In other cases such as an average option with final payment of T + Call 1 BAv = P1 (t) dt − K T 0 respectively its discrete variant n + Call 1 (n) BAv = P1 (i) − K n n+1 i=0 the full non-recombining tree has to be considered for the approximative calculation of the option price. 345 / 477
  • 346. Convergence of the model (s,n) If the moment conditions are satisfied the process P1 (t), (n) obtained from P1 (i) by linear interpolation, converges weakly to the process P1 (t). If the family (s,n) Bs,n := B P1 (t), t ∈ [0, T ] is uniformly integrable then we obtain the convergence n→∞ E (n) e−rt Bs,n − − EQ e−rT B . −→ (s,n) Q (n) is defined on the paths of P1 (t) by identifying them with (n) the corresponding paths of P1 (i). 346 / 477
  • 347. Convergence of the model We obtain n→∞ E (n) e−rt Bn − − EQ e−rT B , −→ if lim E (n) e−rt (Bs,n − Bn ) = 0. n→∞ Remark: The latter convergence has to be proved for each type of option explicitly. It is satisfied if (Bs,n − Bn ) converges to zero uniformly. It is fulfilled for European lookbacks, barrier and double barrier options, and Asian options. 347 / 477
  • 348. Advantages/Disadvantages of the method Advantages Easy to implement (but their efficiency can depend strongly on the option type). Binomial methods converge faster than Monte-Carlo-simulation. Hybrid method via combination of binomial tree and Monte-Carlo method for very big binomial trees available. Disadvantages Slow and irregular convergence behaviour for double barrier options. Accuracy of approximation does not necessarily increase with fineness n. 348 / 477
  • 349. Trinomial trees and explicit finite-difference methods Option prices can be obtained as solutions of corresponding Cauchy problems under certain assumptions. In particular for options with a final payment of the form B = f (P1 (T )). If the Cauchy problem does not admit an explicit solution we must solve it numerically. Numerical methods: Connection between stochastic methods and PDE methods similar to Feynman-Kac theorem. 349 / 477
  • 350. Trinomial trees and explicit finite-difference methods Approximation of the Black-Scholes model by a recombining trinomial tree (n) Discrete-time stock price process P1 (i), i = 0, 1, . . . , n, with possible paths in a trinomial tree. Assume 1 T < er · n < u. u Probability for an upwards movement: q1 Probability for an downwards movement: q2 Probability for the stock price to rest at the same level: q3 = 1 − (q1 + q2 ). 350 / 477
  • 351. Trinomial trees and explicit finite-difference methods Trinomial tree: t =0 1 · T /n 2 · T /n T ”time” unp r rr rr rr rrr rr rr u2p u n−1 p nnn nnn nnn nn nnn nnn up PP up up ooo o PPP nn PPP nnnnn ooo oooo nnPPPPP nn ooo nnn PPP ooo nnn PP p OO p nPP P nn p OOO OOO PPP PPP nnn nnn OOO n P OOO nn nnn PPPPP OOO nnn PPP 1 n P u p PPP rr PPP rr PPP rr PPP rr rr PPP rr P rr 1 p rr LL 1 p u2 LL u n−1 LL LL LL LL LL 1 ”price” un p 351 / 477
  • 352. Trinomial trees and explicit finite-difference methods Approximation of the Black-Scholes model by a recombining trinomial tree Assume 0 < q1 , q2 < 1, q1 + q2 ≤ 1. q1 + q2 = 1 ⇒ binomial situation Assume: q1 + q2 < 1. (n) Donsker’s theorem ⇒ weak convergence of P1 (i) to the stock price process (in risk-neutral market) 1 P1 (t) = p · exp r − σ 2 t + σW (t) , 2 if the first two moments of the increments of ln(P1 (t)) between (i − 1) · T and i · T n n T P1 i · n ln T P1 (i − 1) · n (n) coincide with the corresponding increments of ln(P1 (i)). 352 / 477
  • 353. Trinomial trees and explicit finite-difference methods Approximation of the Black-Scholes model by a recombining trinomial tree This leads to 1 1 r − σ 2 ∆t = ln(u) · q1 + ln · q2 , 2 u 2 1 2 2 1 r− σ (∆t)2 + σ 2 ∆t = ln(u)2 · q1 + ln · q2 . 2 u For given u > 0 we can determine q1 , q2 (compare to the binomial model). 353 / 477
  • 354. Trinomial trees and explicit finite-difference methods Approximation of the Black-Scholes model by a recombining trinomial tree Method of Cox-Ross-Rubinstein: √ u = eλσ ∆t := e∆x for some λ ∈ [1, ∞) (neglect terms of higher order than ∆t above). Error negligible for small ∆t. √ λσ ∆t(q1 − q2 ) = r − 1 σ 2 ∆t 2 λ2 σ 2 ∆t(q1 + q2 ) = σ 2 ∆t. 354 / 477
  • 355. Trinomial trees and explicit finite-difference methods Approximation of the Black-Scholes model by a recombining trinomial tree Solutions: 1 1 1 √ 1 q1 = r − σ2 ∆t + 2 , 2 2 λσ λ 1 1 1 1 √ q2 = 2 − r − σ2 ∆t . 2 λ 2 λσ q1 , q2 , q3 ∈ (0, 1) for small ∆t (i.e., large n). λ = 1 ⇒ binomial model. 355 / 477
  • 356. Trinomial trees and explicit finite-difference methods Algorithm: Approximation by trinomial trees (1) For n ≫ 1 set up a suitable trinomial tree for the discrete-time (n) price process P1 (i). (2) Compute the expected discounted final payment E (n) e−rT Bn in the discrete-time model as an approximation for EQ e−rT B . 356 / 477
  • 357. Computation of E (n) e−rT Bn Backward induction Let (n) (n) X1 (i) := ln P1 (i) , i = 0, 1, . . . , n, and (n) (n) V (n) i · ∆t, X1 (i) := E (n) e−r T −i·∆t · Bn P1 (i) 357 / 477
  • 358. Computation of E (n) e−rT Bn Backward induction Compute recursively (n) (n) V (n) T , X1 (n) = f exp X1 (n) , (n) V (n) i · ∆t, X1 (i) (n) = q1 V (n) (i + 1) · ∆t, X1 (i) + ∆x (n) + q3 V (n) (i + 1) · ∆t, X1 (i) (n) + q2 V (n) (i + 1) · ∆t, X1 (i) − ∆x · e−r ∆t , for i = n − 1, . . . , 0 E (n) e−rT Bn = V (n) (0, p). 358 / 477
  • 359. The option price in the trinomial model In general the final payment of a European call cannot be replicated by a trading strategy in stock and bond in the trinomial model. For this model exists a whole family of equivalent martingale measures. The alternative method "compute the option price in an approximating model" cannot be performed without further modifications. Until now we have not developed a method to compute an option price in incomplete markets. 359 / 477
  • 360. Relations between trinomial trees and explicit finite-difference methods The option price solves the Cauchy problem 1 2 2 Vt + σ p Vpp + rpVp − rV = 0, (t, p) ∈ [0, T ] × (0, ∞) 2 V (T , p) = f (p), p > 0. Substitution: x = ln(p). Notation: V (t, x) := V (t, p). Transformed problem: 1 Vt + σ 2 Vxx + r − 1 σ 2 Vx − r V = 0, (t, p) ∈ [0, T ] × R 2 2 V (T , x) = f (ex ), x ∈ R. 360 / 477
  • 361. Explicit finite-difference method Time discretization: 0, ∆t, 2∆t, . . . , T Space discretization: ln(p1 ), ln(p1 ) ± ∆x, ln(p1 ) ± 2∆x, . . . V (n) (t + ∆t, x) − V (n) (t, x) ∆t V (n) (t, x) := , ∆t V (n) (t + ∆t, x + ∆x) − V (n) (t + ∆t, x − ∆x) ∆x V (n) (t, x) := , 2∆x ∆xx V (n) (t, x) V (n) (t + ∆t, x + ∆x) − 2V (n) (t + ∆t, x) + V (n) (t + ∆t, x − ∆x) := . (∆x)2 361 / 477
  • 362. Explicit finite-difference method Notation: ti := i · ∆t, i = 0, 1, . . . , n, X (j) := ln(p1 ) + j · ∆x, j ∈ Z. V (n) (ti , X (j)) 1 1 2 ∆t 1 σ2 ∆t = σ + r− V (n) (ti + ∆t, X (j) + ∆x) 1 + r ∆t 2 (∆x)2 2 2 ∆x ∆t + 1 − σ2 V (n) (ti + ∆t, X (j)) (∆x)2 1 2 ∆t 1 σ2 ∆t + σ + r− V (n) (ti + ∆t, X (j) − ∆x) . 2 (∆x)2 2 2 ∆x 362 / 477
  • 363. Explicit finite-difference method At time T we know all values of V (n) (T , x). We obtain V (n) (T − ∆t, X (j)) from the explicit representation. Via backward induction with step size ∆t we reach the starting time t = 0 in n steps and obtain V (n) (0, x) as an approximation for the option price V (0, x). The recursion in the trinomial tree can be seen as a finite difference scheme. V (n) (0, x) converges to V (0, x) if the stability condition ∆t 1 0< 2 ≤ 2 (∆x) σ is satisfied. 363 / 477
  • 364. Outline 4 Pricing of Exotic Options and Numerical Algorithms Introduction Examples Examples Equivalent Martingale Measure Exotic Options with Explicit Pricing Formulae Weak Convergence of Stochastic Processes Monte-Carlo Simulation Approximation via Binomial Trees The Pathwise Binomial Approach of Rogers and Stapleton 364 / 477
  • 365. The pathwise binomial approach of Rogers and Stapleton Description of the basic idea Binomial method: Only the distribution of P1 (t) is approximated by a simpler, discrete distribution. Method of Rogers and Stapleton: Approximate each single path of P1 (t) by a step function. 365 / 477
  • 366. Description of the basic idea (continued) The step function is only allowed to attain values in a given discrete set and should at most deviate by a given accuracy ε from the corresponding path of P1 (t). Idea: Interprete the set of all paths of a step function as an infinite binomial tree. Compute the (approximate) discounted expected payment of an option in the infinite tree as an approximation for the option price in the Black-Scholes model. 366 / 477
  • 367. The pathwise binomial approach of Rogers and Stapleton Algorithm: Pathwise binomial approach of Rogers and Stapleton (1) For a given accuracy ∆y and starting point y = ln(p1 ) set up an infinite binomial tree. (2) Compute the discounted payoff E (∆y ) e−rT B∆y of the option in the infinite binomial tree as an approximation for EQ e−rT B . 367 / 477
  • 368. Construction of the infinite binomial tree i) Approximation Logarithm of the stock price: Y (t) = ln(P1 (t)) = ln(p1 ) +σ · W (t) + r − 1 σ 2 · t. 2 =:y For a given accuracy ∆y > 0 and for each ω ∈ Ω, t ∈ [0, T ], we define an approximating step function Z (t) via τ0 (ω) := 0, τn (ω) := inf t ∈ [0, T ] | t > τn−1 (ω), |Y (t, ω) − Y (τn−1 (ω), ω)| > ∆y , n = 1, 2, . . . , ξ0 (ω) := y, ξn (ω) := Y (τn , ω), ∞ Z (t, ω) := ξn (ω) · 1[τn ,τn+1 ) (t). n=0 368 / 477
  • 369. Construction of the infinite binomial tree i) Approximation (continued) This means: as soon as Y (t) deviates from the current value of the step function Z (t) by ∆y, the step function will be set at exactly this value of Y (t). By construction of Z (t) we have sup |Y (t) − Z (t)| ≤ ∆y. 0≤t≤T For given y and ∆y the step function Z (t) only attains values in the set {y ± i∆y | i ∈ N}. 369 / 477
  • 370. Construction of the infinite binomial tree i) Approximation (continued) Z (t) can only jump to adjacent values Z (t) ± ∆y Y(t),Z(t) y+2∆ y+∆ y y−∆ y−2∆ τ1 τ2 τ3 τ4 τ5 τ6 τ t 7 370 / 477
  • 371. Construction of the infinite binomial tree i) Approximation (continued) There is no a priori upper bound for the number of values that Z (t, ω) can attain on [0, T ]. ⇒ Identify the sequence of values of the step function Z (t, ω) on [0, T ] with a finite path in the infinite binomial tree. y + 3∆y jjjj y + 2∆y TTTT jjjj T y + ∆y T y + ∆y TTTT j ooo o TTT jjjjjjjj y oO OO y TTTT ... OO jjjj TTTT jjjj T y − ∆y T y − ∆y TTT j jjjj y − 2∆y T TTT y − 3∆y 371 / 477
  • 372. Construction of the infinite binomial tree ii) Computation of the transition probabilities Example: Double-barrier knockout call for Y (t) = ln(P1 (t)). Final option payment: B = (P1 (T ) − K )+ · 1{ln(P1 (t))∈(b∗ ,b∗ ) for all t∈[0,T ]} , where K ≥ 0 is the strike price. The real numbers b∗ < y < b ∗ define the interval in which Y (t) has to stay so that the call is still valid in t = T . If Y (t) leaves the interval (b∗ , b ∗ ) before T then the option runs out worthless. 372 / 477
  • 373. Construction of the infinite binomial tree ii) Computation of the transition probabilities (continued) Price of the call: xB = EQ e−r (T ) (P1 (T ) − K )+ · 1{ln(P1 (t))∈(b∗ ,b∗ ) for all t∈[0,T ]} . Calculate the price of the call approximately with the infinite binomial tree. 373 / 477
  • 374. Construction of the infinite binomial tree ii) Computation of the transition probabilities (continued) Decompose the infinite tree in finite subtrees. For a fixed n ∈ N the possible paths of the step function Z (t) containing exactly n jumps on [0, T ] will be identified with an n-period binomial tree. Calculate the expected discounted payment of the option in this finite tree if the transition probabilities from a node to its successors in the tree are determined. As we have coincidence of the values of Z (t) and Y (t) in both the times τn−1 and τn , the transition probabilities in the tree coincide with those of Y (t) to Y (t) ± ∆y. 374 / 477
  • 375. The pathwise binomial approach of Rogers and Stapleton Theorem For a given ∆y > 0 the probability for an upwards movement of Z (t) in τn , n ∈ N, is given by s(0) − s(−∆y) q= s(∆y) − s(−∆y) with r − 1 σ2 s(x) = −e−2cx and c = 2 . σ2 The probability for a downwards movement of Z (t) in τn , n ∈ N, is (1 − q). 375 / 477
  • 376. Continuation: Double-barrier knockout call For the double-barrier knockout call all paths in the tree which exceed b∗ by below and b ∗ by above have zero value. If the process Z (t) differs from b∗ or b ∗ by a value less than ∆y we have to take extra care. To prevent the option being knocked out by the Y (.)-process before it is knocked out by the Z (.)-process, we must modify the definition of the Z (.)-process. 376 / 477
  • 377. Continuation: Double-barrier knockout call Y (t) can reach the boundary value b∗ or b ∗ (knocking out the option), but it is possible that Y (t) never reaches a value that would cause Z (t) to jump again. To avoid this, choose the corresponding node in the tree such that it is exactly b∗ or b ∗ . The step function Z (t) jumps exactly when b∗ or b ∗ is reached and not when Z (t) − ∆y or Z (t) + ∆y is reached. 377 / 477
  • 378. Continuation: Double-barrier knockout call Consequences of the modification of Z (t): Z (t) only attains values in the modified binomial tree. The final payment of the double-barrier knockout call in the modified binomial tree is given by + B∆y = eZ (t) − K · 1{Z (t)∈(b∗ ,b∗ ) for all t∈[0,T ]} . Z (t) reaches one of the barriers b∗ or b ∗ if and only if Y (t) reaches the same barrier, i.e., 1{Z (t)∈(b∗ ,b∗ ) for all t∈[0,T ]} = 1{Y (t)∈(b∗ ,b∗ ) for all t∈[0,T ]} The option is knocked out before T in the original model if it is knocked out before T in the modified binomial model. 378 / 477
  • 379. Continuation: Double-barrier knockout call For the pricing of the knockout call, it plays no role if we define Z (t) to be constant after reaching b∗ or b ∗ or extend it as it was originally defined. It is important for the pricing purpose that the transition probabilities in the modified tree change for Z (t) ∈ (b ∗ − ∆y, b ∗ ) or Z (t) ∈ (b∗ , b∗ + ∆y). 379 / 477
  • 380. Continuation: Double-barrier knockout call Theorem (1) If we have Y (τn ) = y ∗ with y ∗ ∈ (b ∗ − ∆y, b ∗ ) then we obtain s(b ∗ ) − s(y ∗ ) q ∗ = P Z (τn+1 ) = y ∗ − ∆y | Z (τn ) = y ∗ = . s(b ∗ ) − s(y ∗ − ∆y) Z (τn+1 ) reaches with probability 1 − q ∗ the value b ∗ and the option runs out worthless. (2) In case of Y (τn ) = y∗ with y∗ ∈ (b∗ , b∗ + ∆y), we obtain s(y∗ ) − s(b∗ ) q∗ = P Z (τn+1 ) = y∗ + ∆y | Z (τn ) = y∗ = . s(y∗ + ∆y) − s(b∗ ) Z (τn+1 ) reaches with probability 1 − q∗ the value b∗ and the option runs out worthless. 380 / 477
  • 381. Continuation: Double-barrier knockout call Proposition Let Ψ(k, y) be the expected final payment of the option in the binomial model for an initial value of Z (0) = y ∈ (b∗ , b ∗ ) and a given number k ∈ N ∪ {0} of upwards and downwards movements of Z (t) on [0, T ]. Then Ψ(k, y) can be computed inductively according to the following algorithm Ψ(0, y) = (ey − K )+ Ψ(n + 1, y) = q(y) · Ψ(n, y + ∆y) + q(y) · Ψ(n, y − ∆y), n = 0, 1, 2, . . . 381 / 477
  • 382. Continuation: Double-barrier knockout call Proposition (continued) Here the probabilities q(y) for y ∈ (b∗ + ∆y, b ∗ − ∆y) are given by s(0) − s(−∆y) q= s(∆y) − s(−∆y) with r − 1 σ2 s(x) = −e−2cx and c = 2 σ2 and we have q(y) = 1 − q(y). 382 / 477
  • 383. Continuation: Double-barrier knockout call Proposition (continued) For y ∈ (b∗ , b∗ + ∆y) we have q(y) = 0 and s(y) − s(b∗ ) q(y) = . s(y + ∆y) − s(b∗ ) For y ∈ (b ∗ − ∆y, b ∗ ) we have q(y) = 0 and s(b ∗ ) − s(y ) q(y) = . s(b ∗ ) − s(y − ∆y) 383 / 477
  • 384. Continuation: Double-barrier knockout call Discounted expected final payment of the option in the modified binomial tree: ∞ E (∆y ) e−rT B∆y = P(v = n) · Ψ(n, y) · e−rT . n=0 For the computation we need the probability distribution of the sum v of upwards and downwards movements of the stock price in the binomial model. Due to {ω | v(ω) ≥ n} = {ω | τn (ω) ≤ T } this distribution can be obtained from that of τn : P(v = n) = P(v ≤ n) − P(v ≤ n − 1) = P(τn ≥ T ) − P(τn−1 ≥ T ). 384 / 477
  • 385. Continuation: Double-barrier knockout call Theorem (1) The random variables {τn+1 − τn }n∈N∪{0} are independent and identically distributed. Their Laplace transform ϕ(λ) is given by cosh µσ −2 ∆y ϕ(λ) = E e−λτ1 = cosh γ∆y with 1 2 µ2 + 2λσ 2 µ=r− σ , γ= , λ > 0. 2 σ2 (2) ∆y µ E (τ1 ) = · tanh 2 · ∆y for µ = 0, µ σ 2 σ 2 ∆y µ ∆y E (τ1 ) = 2(E (τ1 ))2 + 2 3 · tanh 2 ∆y − for µ = 0. µ σ µ (3) τn+1 − τn is independent of ξn+1 . 385 / 477
  • 386. Continuation: Double-barrier knockout call We have n τn = (τi − τi−1 ). i=1 Summands are independent and identically distributed. Central limit theorem ⇒ τn − n · E (τ1 ) n→∞ − − N(0, 1) in distribution. −→ n · Var (τ1 ) Determine the distribution of τn approximately for large n. For small n this approximation is not accurate enough. 386 / 477
  • 387. Continuation: Double-barrier knockout call Theorem We have x2 τn − n · E (τ1 ) α3 (1 − x 2 )e− 2 1 P ≤x = Φ(x) + √ + o n− 2 n · Var (τ1 ) 72πn with 3 τ1 − E (τ1 ) α3 = E , Var (τ1 ) where Φ is the distribution function of the standard normal distribution. 387 / 477
  • 388. Continuation: Double-barrier knockout call Use Laplace transform ϕ(λ) to calculate α3 : ∆y · (A + B − C) α3 = 5 , µ 3 σ2 σ 6 s(∆y) − 1 µ A = 12 · ∆y s(2∆y) + s(∆y) , σ2 µ 2 B =8· (∆y)2 s(∆y) − s(2∆y) , σ2 C = 3 · 1 + s(∆y) − s(2∆y) − s(3∆y) . 388 / 477
  • 389. Method of Rogers and Stapleton Algorithm: Method of Rogers and Stapleton (1) For a given initial value of y = ln(P1 (0)) and a given accuracy of ∆y, compute "all" values of Ψ(k, y), k ∈ N ∪ {0}. (2) Compute P(v = n) = P(τn ≥ T ) − P(τn−1 ≥ T ) approximately from the distribution of {τn }n (by neglecting the o n−1/2 -terms). (3) Determine ∞ (∆y ) −rT E e B∆y = P(v = n) · Ψ(n, y) · e−rT n=0 as an approximation for EQ e−rT B . 389 / 477
  • 390. Convergence of the model As for fixed ∆y > 0 we have sup |Y (t) − Z (t)| < ∆y 0≤t≤T we get uniform convergence of Z (t) to Y (t) for ∆y → 0. ⇒ Estimates for the approximation error (depending on the type of option). 390 / 477
  • 391. Convergence of the model Error estimate for the double-barrier knockout call (b ∗ > ln(K ) > b∗ ) Call BDB − B∆y + + = eY (T ) − K − eZ (T ) − K · 1{Y (t)∈(b∗ ,b∗ ) for all t∈[0,T ]} + + ≤ eY (T ) − K − eZ (T ) − K · 1{Y (T )∈(b∗ ,b∗ )} + + ≤ max max eY (T ) − K − eZ (T ) − K , Y (T )∈[ln(K ),b ∗ ) + + max eY (T ) − K − eZ (T ) − K Y (T )∈(b∗ ,ln(K )] ... 391 / 477
  • 392. Convergence of the model Error estimate for the double-barrier knockout call (b ∗ > ln(K ) > b∗ ) Call BDB − B∆y ≤ . . . ∗ ∗ −∆y ≤ max eb − eb , K · e∆y − K ∗ = max eb (1 − e−∆y ), K (e∆y − 1) 392 / 477
  • 393. Advantages/Disadvantages of the method Advantages Paths of the approximating process Z (t) converge uniformly to the paths Y (t) of the logarithm of the price process. Explicit estimate of the convergence error. Flexibility which allows for a choice of the nodes in the binomial tree ensuring that in the double-barrier knockout call case the option runs out worthless in the modified binomial tree if and only if it runs out worthless in the Black-Scholes model. Disadvantages Concept requires a deeper understanding as in the case of the binomial model. Bigger computational complexity. 393 / 477
  • 394. Outline 5 Optimal Portfolios 394 / 477
  • 395. Outline 5 Optimal Portfolios Introduction and Formulation of the Problem The martingale method Optimal Option Portfolios Excursion 8: Stochastic Control Maximize expected value in presence of quadratic control costs Introduction Portfolio Optimization via Stochastic Control Method 395 / 477
  • 396. Introduction Until now Trading strategies that generate a given payoff profile (replication) or a lower bound for a payment (hedging strategy). Costs for replication strategy determined the price of the payoff profile. Now Given a fixed initial capital and search for an admissible self-financing pair of portfolio and consumption processes which yields a payment stream as lucrative as possible. 396 / 477
  • 397. Formulation of the portfolio problem The continuous-time portfolio problem Initial capital x > 0. Determine an optimal consumption and investment strategy. The investor has to determine how many shares of which security he has to hold at which time instant and how much of his wealth he is allowed to consume to maximize his utility from consumption during the period [0, T ] and/or from the terminal wealth at the time horizon t = T . 397 / 477
  • 398. Formulation of the portfolio problem (continued) The continuous-time portfolio problem The portfolio problem contains a choice problem ("which" security) a problem of volumes ("how many" shares, "how much",...) a dynamic component with respect to time ("which time"). The investor can decide on his actions at each time instant t ∈ [0, T ] has only the information of past and present prices should not have any knowledge of future security prices or insider information. 398 / 477
  • 399. General assumptions General assumptions for this section (Ω, F, P) be a complete probability space, {(W (t), Ft )}t∈[0,∞) m-dimensional Brownian motion. Dynamics of bond and stock prices: t P0 (t) = p0 · exp r (s) ds bond 0 t m 1 2 Pi (t) = pi · exp bi (s) − σij (s) ds 2 0 j=1 m t + σij (s) dWj (s) stock j=1 0 for t ∈ [0, T ], T > 0, i = 1, . . . , d . 399 / 477
  • 400. General assumptions (continued) General assumptions for this section (continued) r (t), b(t) = (b1 (t), . . . , bd (t))T , σ(t) = (σij (t))ij progressively measurable processes with respect to {Ft }t , component-wise uniformly bounded in (t, ω). σ(t)σ(t)T uniformly positive definite, i.e., it exists K > 0 with x T σ(t)σ(t)T x ≥ Kx T x for all x ∈ Rd and all t ∈ [0, T ] P-a.s. (π, c) is a self-financing pair consisting of a portfolio process π and a consumption process c to be admissible with initial wealth x > 0: (π, c) ∈ A(x). 400 / 477
  • 401. Solution approaches (continuous-time market model) Two approaches: Stochastic control approach (Robert/Merton) Martingale method (Cox/Huang, Karatzas/Lehoczky/Shreve, Pliska) 401 / 477
  • 402. Formulation of the problem Introduce a functional J which measures the utility of a payment stream (large value = good payment stream). For a given initial wealth x > 0 an investor looks for a self-financing pair (π, c) ∈ A(x) which maximizes the expected utility from consumption and / or terminal wealth T J(x; π, c) = E U1 t, c(t) dt + U2 X (T ) 0 Notation: X (t) wealth corresponding to x and (π, c). U1 , U2 utility functions. 402 / 477
  • 403. Utility functions Definition (1) Let U : (0, ∞) → R be a strictly concave and continuously differentiable function with U ′ (0) := lim U ′ (x) = +∞, U ′ (∞) := lim U ′ (x) = 0. x↓0 x→∞ Then U is called a utility function. (2) A continuous function U : [0, T ] × (0, ∞) → R such that for all t ∈ [0, T ] the function U(t, ·) is a utility function in the sense of (1) is called a utility function. 403 / 477
  • 404. Examples of utility functions Examples: (1) U(x) = ln(x) √ (2) U(x) = x (3) U(x) = x α for 0 < α < 1 (4) U(t, x) = e−ρt · U1 (x), ρ>0 with U1 as in (1) or (2). 404 / 477
  • 405. Utility functions Remark A utility function is strictly increasing ⇒ each additional wealth leads to additional utility. A utility function is strictly concave ⇒ U ′ (x) is strictly decreasing ⇒ decreasing marginal utility ⇒ the gain of utility from one additional unit of money decreases with increasing x. 405 / 477
  • 406. Utility functions Remark (continued) The marginal utility in x = 0 is infinite "a tiny bit is much more than nothing" Vanishing marginal utility in x = ∞ models a saturation effect. A wider class of utility functions can be considered. In particular the (popular but criticized) quadratic utility function 1 U(x) = − (x − a)2 . 2 406 / 477
  • 407. Formulation of the problem Remark For an arbitrary (π, c) ∈ A(x) the expectation in J(x; π, c) is not necessarily defined. We can restrict the class of self-financing pairs (π, c) to all those for which the expectation in (π, c) ∈ A(x) is finite. We require only a weak integrability condition for a feasible pair (π, c). 407 / 477
  • 408. Continuous-time portfolio problem Continuous-time portfolio problem max J(x; π, c) (π,c)∈A′ (x) with T ′ − − A (x) = (π, c) ∈ A(x) E U1 t, c(t) dt + U2 X (T ) <∞ 0 408 / 477
  • 409. Continuous-time portfolio problem Remark (1) By restricting to the set A′ (x), the integral in J is defined. The expectation exists but can be equal to infinity. (2) In the case of positive utility functions, U1 (t, ·) > 0 and U2 (t, ·) > 0, the equality A(x) = A′ (x) is satisfied. 409 / 477
  • 410. Outline 5 Optimal Portfolios Introduction and Formulation of the Problem The martingale method Optimal Option Portfolios Excursion 8: Stochastic Control Maximize expected value in presence of quadratic control costs Introduction Portfolio Optimization via Stochastic Control Method 410 / 477
  • 411. General assumption / notation → completeness of the market General assumption for this section d =m Notation t γ(t) := exp − r (s) ds 0 −1 θ(t) := σ (t) b(t) − r (t) 1 t t T 1 2 Z (t) := exp − θ(s) dW (s) − θ(s) ds 2 0 0 H(t) := γ(t) · Z (t) 411 / 477
  • 412. The main idea The martingale method is based on a separation of the dynamical problem into a static optimization problem (determination of the optimal payoff profile) a representation problem (compute the portfolio process corresponding to the optimal payoff profile). 412 / 477
  • 413. Motivation Portfolio problem without consumption (c ≡ 0, U1 ≡ 0) The self-financing pair (π, 0) be admissible for an initial wealth of x > 0. By the completeness of the market we have for each corresponding wealth process X π (T ) E H(T ) X π (T ) ≤ x for T ≥ 0. The final payment B ≥ 0 be FT -measurable with E (H(T ) B) = x. Example: x B := . E (H(T )) There exists a portfolio process (π, 0) ∈ A(x) with B = X π (T ) P-a.s. 413 / 477
  • 414. Motivation Portfolio problem without consumption (c ≡ 0, U1 ≡ 0) Define B(x) := {B ≥ 0 | B FT -measurable, E (H(T ) B) ≤ x, E (U2 (B)− ) < ∞}. B(x) represents the set of all final wealths with E (U2 (B)− ) < ∞ that can be generated by trading in the securities starting with some initial wealth y ∈ (0, x] and satisfying E (U2 (B)− ) < ∞. 414 / 477
  • 415. Motivation Portfolio problem without consumption (c ≡ 0, U1 ≡ 0) For determining the optimal final wealth X π (T ) in the portfolio problem max E U2 X π (T ) (π,0)∈A′ (x) it is enough to maximize over all random variables B ∈ B(x), i.e., it is sufficient to solve the static optimization problem max E U2 (B) B∈B(x) 415 / 477
  • 416. Motivation Portfolio problem without consumption (c ≡ 0, U1 ≡ 0) If B ∗ is an optimal final wealth in the static optimization problem, then to solve the portfolio problem we have to solve the representation problem Find a (π, 0) ∈ A′ (x) with X π (T ) = B ∗ P-a.s. 416 / 477
  • 417. Lagrangian method f : Rn → R strictly convex g: Rn → Rk convex f , g ∈ C 1. x solves the optimization problem max f (x) n x∈R subject to g(x) = 0 ⇔ there exists a λ such that x, λ ∈ Rn+k satisfies k ∂ ∂ f (x) − λj g(x) = 0, i = 1, . . . , n ∂xi ∂xi j=1 gi (x) = 0, i = 1, . . . , k. 417 / 477
  • 418. Lagrangian method In words: x, λ ∈ Rn+k is a zero of the derivative of the Lagrangian function L(x, λ) = f (x) − λT g(x) 418 / 477
  • 419. The Lagrangian method for the portfolio problem Consider the static optimization problem max E U2 (B) . B∈B(x) Lagrangian function L(B, y) := E U2 (B) − y · H(T ) B − x , y > 0. Formally differentiating L with respect to B and y and interchanging the expectation with the differentiating process yields ′ 0 = LB (B, y) = E U2 (B) − y H(T ) , 0 = Ly (B, y) = x − E H(T ) B . 419 / 477
  • 420. The Lagrangian method for the portfolio problem A random variable B satisfying ′ U2 (B) − y H(T ) = 0 P-a.s. solves the first equation. The range of U2 (.) equals R+ and U2 (.) is strictly decreasing. ′ ′ Therefore, it can be inverted on R + and we obtain ′ −1 B = U2 (y · H(T )). Putting this in the second equation yields ′ −1 0 = x − E H(T ) · U2 (y · H(T )) . =:χ(y ) If we can solve this equation uniquely for y then we have found a candidate for the optimal final wealth. 420 / 477
  • 421. The Lagrangian method for the portfolio problem Define −1 Y (u) := χ−1 (u), ′ I2 := U2 . Candidate for the optimal final wealth B ∗ = I2 Y (x) · H(T ) > 0. Prove that B ∗ is the optimal final wealth. 421 / 477
  • 422. Notation Notation ′ −1 I2 (y) := U2 (y) for y ∈ (0, ∞) ′ −1 I1 (t, y) := U1 (t, y) for y ∈ (0, ∞) T χ(y) := E H(t) I1 t, y·H(t) dt + H(T ) I2 y · H(T ) 0 422 / 477
  • 423. Properties of χ(y ) Lemma Assume χ(y) < ∞ for all y > 0. Then, χ is continuous on (0, ∞), strictly decreasing and satisfies χ(0) := lim χ(y) = ∞ y ↓0 χ(∞) := lim χ(y) = 0. y →∞ 423 / 477
  • 424. Properties of χ(y ) Remark The previous lemma implies the existence of Y (x) := χ−1 (x) on (0, ∞) with Y (0) := lim Y (x) = ∞ x↓0 Y (∞) := lim Y (x) = 0. x→∞ 424 / 477
  • 425. Utility functions Lemma −1 Let U be a utility function with I := U ′ . Then we have U I(y) ≥ U(x) + y(I(y) − x) for 0 < y, x < ∞. 425 / 477
  • 426. Optimal consumption / optimal terminal wealth Theorem: Optimal consumption and optimal terminal wealth Consider the portfolio problem. Let x > 0 and χ(y) < ∞ for all y > 0. Set Y (x) := χ−1 (x). Then, for B ∗ := I2 Y (x) · H(T ) optimal terminal wealth c ∗ (t) := I1 t, Y (x) · H(T ) optimal consumption there exists a self-financing portfolio process π ∗ (t), t ∈ [0, T ], such that ∗ ,c ∗ π ∗ , c ∗ ∈ A′ (x), X x,π (T ) = B ∗ P-a.s. ∗ ∗ and such that π ∗ , c ∗ solves the portfolio problem. Here, X x,π ,c (t) is the wealth process corresponding to the pair π ∗ , c ∗ and the initial wealth x. 426 / 477
  • 427. Example: Logarithmic utility Example: Logarithmic utility U1 (t, x) = U2 (x) = ln(x) 1 ⇒ I1 (t, y) = I2 (y) = y T 1 1 1 ⇒ χ(y) = E H(t) · dt + H(T ) · = (T + 1) y · H(t) y · H(T ) y 0 1 ⇒ Y (x) = χ−1 (x) = (T + 1). x 427 / 477
  • 428. Example: Logarithmic utility Optimal consumption x 1 c ∗ (t) = I1 t, Y (x) · H(t) = · T + 1 H(t) Optimal final wealth x 1 B ∗ = I2 Y (x) · H(t) = · T + 1 H(T ) 428 / 477
  • 429. Example: Logarithmic utility Calculation of the portfolio process We have T x,π ∗ ,c ∗ H(t) · X (t) = E H(s) c ∗ (s) ds + H(T ) B ∗ | Ft t T −t +1 =x· . T +1 This implies T −t +1 t x =x· +x · T +1 T +1 t x,π ∗ ,c ∗ = H(t) · X (t) + H(s) c ∗ (s) ds. 0 429 / 477
  • 430. Example: Logarithmic utility Calculation of the portfolio process ∗ ,c ∗ Application of Ito’s formula to the product H(t) · X x,π ˆ (t) yields t ∗ ,c ∗ x =x+ H(s) · X x,π (s) π ∗ (s)T σ(s) − θ(s)T dW (s). 0 =: f (s) Hence we must have f (s) = 0 P-a.s. for all s ∈ [0, T ]. As H(s) · X x,π ∗ ,c ∗ (s) is positive we thus obtain −1 π ∗ (t) = σ(t)T θ(t) for all t ∈ [0, T ]. In the special case of d = 1 and constant coefficients r , b, σ we get b−r π ∗ (t) = local risk for stock investment σ2 430 / 477
  • 431. Example: Logarithmic utility Calculation of the portfolio process Representation of the consumption rate 1 ∗ ∗ c ∗ (t) = X x,π ,c (t). T −t +1 The consumption rate is proportional to the current wealth of the investor and inversely proportional to the remaining time T − t. 431 / 477
  • 432. Solution of the representation problem Theorem: Solution of the representation problem Consider the portfolio problem. Let x > 0 and assume χ(y) < ∞ for all y > 0. Let B ∗ := I2 Y (x) · H(T ) , c ∗ (t) := I1 t, Y (x) · H(T ) . If there exists a function f ∈ C 1,2 [0, T ] × Rd with f (0, 0, . . . , 0) = x and T 1 ·E H(s) c ∗ (s) ds + H(T ) B ∗ | Ft = f t, W1 (t), . . . , Wd (t) , H(T ) t then for all t ∈ [0, t] we have 1 π ∗ (t) = σ −1 (t) ∇x f t, W1 (t), . . . , Wd (t) . X x,π∗ ,c ∗ (t) 432 / 477
  • 433. Optimal consumption / optimal terminal wealth Corollary (1) The optimal terminal wealth B ∗ of the problem max E U2 X x,π (T ) (π,0)∈A′ (x) is given by B ∗ := I2 Y (x) · H(T ) where in the definition of χ(y) we have to set I1 (t, y) ≡ 0. 433 / 477
  • 434. Optimal consumption / optimal terminal wealth Corollary (continued) (2) The optimal consumption process c ∗ (t) of the problem T max E U1 t, c(t) dt (π,c)∈A′ (x) 0 is given by c ∗ (t) := I1 t, Y (x) · H(T ) where in the definition of χ(y) we have to set I2 (y) ≡ 0. 434 / 477
  • 435. Outline 5 Optimal Portfolios Introduction and Formulation of the Problem The martingale method Optimal Option Portfolios Excursion 8: Stochastic Control Maximize expected value in presence of quadratic control costs Introduction Portfolio Optimization via Stochastic Control Method 435 / 477
  • 436. General assumption / notation General assumption for this section d =m Notation t γ(t) := exp − r (s) ds 0 θ(t) := σ −1 (t) b(t) − r (t) 1 t t T 1 2 Z (t) := exp − θ(s) dW (s) − θ(s) ds 2 0 0 H(t) := γ(t) · Z (t) 436 / 477
  • 437. Description of the market model Restriction The market coefficients r , b, σ are constant. 437 / 477
  • 438. Description of the market model Consider a market, where a bond, d stocks, and d options on these stocks are traded a Portfolio consisting of the bond and the options. Assume The options have a price process of the form f (i) t, P1 (t), . . . , Pd (t) , i = 1, . . . , d , f ∈ C 1,2 Option prices satisfy certain requirements (particularly satisfied for European puts and calls in the Black-Scholes model). 438 / 477
  • 439. Description of the market model Admissible trading strategy in bond and options: ϕ(t) = ϕ0 (t), ϕ1 (t), . . . , ϕd (t) The integrals t ϕ0 (s) dP0 (s) 0 t ϕi (s) df (i) s, P1 (s), . . . , Pd (s) 0 are assumed to be defined. ϕ(t) is assumed to be Ft -progressively measurable. Wealth progress d X (t) = ϕ0 (t) P0 (t) + ϕi (t) f (i) t, P1 (t), . . . , Pd (t) . i=1 439 / 477
  • 440. The problem Consider the problem max E U(X (T )) ϕ where U is a utility function. 440 / 477
  • 441. Motivation of the solution of the problem To motivate the solution of the problem, we take a look at the following diagrams representing the solution of the option pricing problem via the replication approach and the solution of the portfolio problem via the martingale approach. The third diagram will show the solution of the problem. 441 / 477
  • 442. Option pricing option price o ? terminal payoff EQ e−rT B B O = = cost of replication o π∗ (t) replication strategy ∗ X ∗ (0) X π (T ) Starting out from the final payment B of an option we replicate the final payment via following a suitable portfolio strategy leading to a final wealth that coincides with final option payment. Then, the costs of setting up the cheapest replication strategy yield the option price. 442 / 477
  • 443. Portfolio optimization with stocks initial wealth ? / optimal terminal wealth ∗ x X x,π O (T ) optimal final payment / replication strategy B∗ π ∗ (t) The given inital wealth x will be invested according to a portfolio process π ∗ (t) with the aim to obtain a terminal wealth which promises the highest possible utility (we ignore the possibility of consumption). To do so, we first determine an optimal final payoff B ∗ and then look for a replication strategy for B ∗ . 443 / 477
  • 444. Portfolio optimization with options optimal terminal initial wealth ? / wealth x X ∗ (T ) O ϕ=(ϕ0 (t),ϕ(t)) optimal final inversion of the payment replication strategy B∗ QQQ m6 ϕ = Ψ−1 ξ QQQ mm QQQ mmm QQ( mmmm replication strategy in bond and stocks ξ(t) = (ξ0 (t), . . . , ξd (t)) We look for an final wealth starting with an initial capital x. To do so, we first determine an optimal final payoff B ∗ and then a replication strategy ξ(t) = (ξ0 (t), . . . , ξd (t)) in bond and stocks for the payoff B ∗ . As stocks should not appear in our portfolio, we have to replicate the stock position by bond and options. This bond and option strategy yields the optimal terminal wealth X ∗ (T ). 444 / 477
  • 445. Motivation of the solution of the problem Theorem Let the Delta matrix Ψ(t) = Ψij (t) ij , i, j = 1, . . . , d with (i) Ψij := fpj t, P1 (t), . . . , Pd (t) be regular for all t ∈ [0, T ]. Then, the option portfolio problem possesses the following explicit solution: (1) The optimal terminal wealth B ∗ coincides with the optimal terminal wealth of the corresponding stock portfolio problem. 445 / 477
  • 446. Motivation of the solution of the problem Theorem (continued) (2) Let ξ(t) = ξ0 (t), . . . , ξd (t) be the optimal trading strategy in the corresponding stock portfolio problem. Then, the optimal trading strategy ϕ(t) = ϕ0 (t), ϕ1 (t), . . . , ϕd (t) in the option portfolio problem is given by −1 ϕ(t) = Ψ(t)T · ξ(t), d X (t) − ϕi (t) f (i) t, P1 (t), . . . , Pd (t) i=1 ϕ0 (t) = , P0 (t) with ϕ(t) := ϕ1 (t), . . . , ϕd (t) and ξ(t) := ξ1 (t), . . . , ξd (t) . 446 / 477
  • 447. Motivation of the solution of the problem Remark (1) Under the given assumptions, the optimal final wealth only depends on the utility functions but not on the choice of the tradable securities. (2) The optimal trading strategy depends on the traded options via the delta matrix (more precisely: via the replication strategy for the options). 447 / 477
  • 448. Example: Logarithmic utility Utility function U(x) = ln(x). Consider Black-Scholes model with d = 1. Stock position in the optimal trading strategy π ∗ (t) · X (t) b − r X (t) ξ1 (t) = = · . P1 (t) σ2 P1 (t) Optimal trading strategy in bond and options b−r X (t) ϕ1 (t) = 2 · . σ Ψ1 (t) · P1 (t) 448 / 477
  • 449. Example: Logarithmic utility Optimal portfolio process in the stock portfolio problem πstock (t) Option portfolio process ϕ1 (t) · f (1) t, P1 (t) πopt (t) : = X (t) b − r X (t) · f (1) t, P1 (t) = · σ2 X (t) · Ψ1 (t) · P1 (t) b−r f (1) t, P1 (t) = · (1) σ2 fp1 t, P1 (t) · P1 (t) f (1) t, P1 (t) = πstock (t)· (1) fp1 t, P1 (t) · P1 (t) 449 / 477
  • 450. Example: Logarithmic utility Theorem With the choice of U(x) = ln(x), in the Black-Scholes model with d = 1 we have (1) πopt (t) = πstock (t) for all t ∈ [0, T ] ⇔ f (1) t, P1 (t) = k · P1 (t) for a constant k ∈ R {0}. (2) In the case of a European call option we have πopt (t) πstock (t) for all t ∈ [0, T ]. 450 / 477
  • 451. Example: Logarithmic utility Remark (1) states that in the Black-Scholes model, πopt (t) is constant if and only if the payoff of the contingent claims is a multiple of the underlying stock price. (2) says that with the choice of a European call option, in the option portfolio problem the optimal capital which is invested in the risky asset is always smaller than the corresponding risky position in the stock portfolio problem. 451 / 477
  • 452. Example: European call Market coefficients r = 0, b = 0.05, σ = 0.25, T = 1, K = 100, P0 (0) = 1. p 1 pstock(0) 0.8 0.6 p (0) opt 0.4 0.2 0 50 150 250 350 450 P1(0) The deeper the option is in the money (i.e., P1 (0) K ), the closer πopt (0) gets to the optimal stock portfolio component πstock (0). The more the option is out of the money (i.e., P1 (0) K ), the smaller πopt (0) gets. 452 / 477
  • 453. Outline 5 Optimal Portfolios Introduction and Formulation of the Problem The martingale method Optimal Option Portfolios Excursion 8: Stochastic Control Maximize expected value in presence of quadratic control costs Introduction Portfolio Optimization via Stochastic Control Method 453 / 477
  • 454. General Assumptions General assumptions for this section ˆ Let X (t) be an n-dimensional Ito process. Controlled SDE Controlled stochastic differential equation: dX (t) = µ(t, X (t), u(t)) dt + σ(t, X (t), u(t)) dW (t) where W (t) is an m-dimensional Brownian motion and u(t) a d -dimensional stochastic process (the control). 454 / 477
  • 455. Outline 5 Optimal Portfolios Introduction and Formulation of the Problem The martingale method Optimal Option Portfolios Excursion 8: Stochastic Control Maximize expected value in presence of quadratic control costs Introduction Portfolio Optimization via Stochastic Control Method 455 / 477
  • 456. Example t Be X (t) = x + W (t) + u(s) ds controlled process with 1-dimensional 0 Brownian motion W (t), as control action choose intensity u(t) of drift process at each time instant t ∈ [0, T ]. Consider the problem to minimize (a, b 0):  T  E au 2 (t) dt − bX (T ) 0 Under suitable requirements on u(t) we get  T  E (X (T )) = x + E  u(s) ds and hence 0  T  E (au 2 (t) − bu(t)) dt − bx  0 456 / 477
  • 457. Example Minimizing function u(t) under the integral leads to the optimal choice of b u ∗ (t) = . 2a 457 / 477
  • 458. Outline 5 Optimal Portfolios Introduction and Formulation of the Problem The martingale method Optimal Option Portfolios Excursion 8: Stochastic Control Maximize expected value in presence of quadratic control costs Introduction Portfolio Optimization via Stochastic Control Method 458 / 477
  • 459. General assumptions General assumptions for this section For n, d ∈ N, 0 ≤ t0 t1 ∞, U ⊂ Rd closed, a constant C 0 be Q0 := [t0 , t1 ) × Rn , ¯ Q0 := [t0 , t1 ] × Rn ¯ µ : Q0 × U → R n , ¯ σ : Q0 × U → Rn,m ¯ µ(., ., u), σ(., ., u) ∈ C 1 (Q0 ) for u ∈ U |µt | + |µx | ≤ C , |σt | + |σx | ≤ C |µ(t, x, u)| + |σ(t, x, u)| ≤ C (1 + |x| + |u|) 459 / 477
  • 460. Notations If the process X (t) solves the controlled SDE with an initial value of x at time t, then we indicate this by denoting its expectation at time t by E t,x (X (s)). 460 / 477
  • 461. Notations Define for an open set O ⊂ Rn Q := [t0 , t1 ) × O, ¯ ¯ Q := [t0 , t1 ] × O, τ := inf{t ≥ t0 | (t, X (t)) ∈ Q}. / Define the cost function  τ  J(t, x, u) = E t,x  L(s, X (s), u(s)) ds + Ψ(τ, X (τ )) t 461 / 477
  • 462. Notations L, Ψ are continuous functions with |L(t, x, u)| ≤ C 1 + |x|k + |u|k , |Ψ(t, x)| ≤ C 1 + |x|k , ¯ ¯ for some k ∈ N on Q × U or on Q. L(s, X (s), u(s)) is called running costs and Ψ(τ, X (τ )) is called terminal costs. 462 / 477
  • 463. Notations V (t, x) := inf J(t, x; u) (t, x) ∈ Q u∈A(t,x) is called value function, where A(t, x) denotes the set of all admissible controls u(·). 463 / 477
  • 464. Notations (i) For G ∈ C 1,2 (Q), (t, x) ∈ Q, a := σσ T , u ∈ U let Au G(t, x) := Gt (t, x) + 0.5 aij (t, x, u)Gxi xj + 1≤i,j≤ n + µi (t, x, u)Gxi (t, x) 1≤i≤n ¯ (ii) ∂ ∗ Q := ([t0 , t1 ) × ∂O) ∪ ({t1 } × O). 464 / 477
  • 465. Contolled SDE Definition–Part I Be (Ω, F, {Ft }t∈[t0 ,t1 ] , P) probability space with filtration. A U-valued progressively measurable process u(t), t ∈ [t0 , t1 ] is called admissible control, if for all values x ∈ Rn the SDE with initial condition X (t0 ) = x possess a unique solution {X (t)}t∈[t0 ,t1 ] and if we have 465 / 477
  • 466. Contolled SDE Definition–Part II   t1   E |u(s)|k ds ∞, t0 E t,x ||X (·)| |k := E t,x sup |X (s)|k ∞ s∈[t,t1 ] for all k ∈ N. 466 / 477
  • 467. Controlled SDE Theorem–Part I ¯ Be G ∈ C 1,2 (Q) ∩ C(Q) with |G(t, x)| ≤ K 1 + |x|k for some suitable constants K 0, k ∈ N, a solution to the Hamilton-Jacobi-Bellman equation: inf (Au G(t, x) + L(t, x, u)) = 0 (t, x) ∈ Q, u∈U G(t, x) = Ψ(t, x) (t, x) ∈ ∂ ∗ Q. Then we have 467 / 477
  • 468. Controlled SDE Theorem–Part II (i) G(t, x) ≤ J(t, x, u) ∀ (t, x) ∈ Q, u(·) ∈ A(t, x). (ii) If for all (t, x) ∈ Q there exists a u ∗ (·) ∈ A(t, x) with u ∗ (s) = arg min(Au G(s, X ∗ (s)) + L(s, X ∗ (s), u)) u∈U for all s ∈ [t, τ ], where X ∗ is controlled process corresponding to u ∗ , then G(t, x) = V (t, x) = J(t, x; u ∗ ). 468 / 477
  • 469. Outline 5 Optimal Portfolios Introduction and Formulation of the Problem The martingale method Optimal Option Portfolios Excursion 8: Stochastic Control Maximize expected value in presence of quadratic control costs Introduction Portfolio Optimization via Stochastic Control Method 469 / 477
  • 470. Main idea General assumptions for this section Market with constant market coefficients r , b, σ. Assume m ≥ d and σ ∈ Rd,m has full rank. Main idea–Part I Identify wealth equation of investor with strategy (π, c) as controlled SDE of form dX u (t) = µ(t, X u (t), u(t)) dt + +σ(t, X u (t), u(t)) dW (t), 470 / 477
  • 471. Main idea Main idea–Part II where µ, σ, u have special form u = (u1 , u2 ) := (π, c), T µ(t, x, u) = (r + u1 (b − r 1))x − u2 , T σ(t, x, u) = xu1 σ. 471 / 477
  • 472. Optimal consumption wealth with finite horizon Maximize utility functional  T  J(t, x, u) := E t,x  U1 (t, u2 (t)) dt + U2 (X u (T )) t Be V (t, x) = supu∈A(t,x) J(t, x; u) be value function of the portfolio problem. The corresponding Hamilton-Jacobi-Bellman (HJB) equation has the form max 0.5u1 σσ T u1 x 2 Vxx (t, x)+ T u1 ∈[α1 ,α2 ]d ,u2 ∈[0,∞) T + (r + u1 (b − r 1))x − u2 Vx (t, x)+ +U1 (t, u2 ) + Vt (t, x)} = 0 472 / 477
  • 473. Optimal consumption wealth with finite horizon V (T , x) = U2 (x) for given constants α1 , α2 ∈ R, α1 ≤ α2 . Solution of the corresponding HJB equation Solve the HJB equation for the special choice of 1 −βt γ 1 U1 (t, c) = e c , U2 (x) = x γ γ γ with β 0 , γ ∈ (0, 1). 473 / 477
  • 474. Optimal consumption wealth with finite horizon Proposition–Part I The portfolio problem  T  1 1 max E e−β t c(t)γ dt + X (T )γ  (π,c)∈A(x) γ γ 0 is solved by strategy (π ∗ , c ∗ ) according to 1 π ∗ (t) = (σσ T )−1 (b − r 1), 1−γ 1 γ−1 c ∗ (t) = eβt f (t) X (t), 474 / 477
  • 475. Optimal consumption wealth with finite horizon Proposition–Part II where f (t) is given by 1 a1 := −0.5(b − r 1)T (σσ T )−1 (b − r 1) + r, γ−1 1 − γ γ−1 βt a2 := e , γ a1 g(t) = e 1−γ (T −t) + 1−γ a1 −β a1 −β a1 + e 1−γ T −e 1−γ t e 1−γ (T −t) , γ(a1 − β) γ g(t) = f (t) 1−γ . 475 / 477
  • 476. For Further Reading R. Korn and E. Korn Option Pricing and Portfolio Optimization - Modern methods of financial mathematics, AMS, 2001. Gerhard-Wilhelm Weber Further documents distributed in the course. 476 / 477

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