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Portfolio OptimizationGerhard-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
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Outline I1 The Mean-Variance Approach in a One-Period Model Introduction2 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
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Outline III4 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
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Outline IV5 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
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Outline1 The Mean-Variance Approach in a One-Period Model 6 / 477
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Outline1 The Mean-Variance Approach in a One-Period Model Introduction 7 / 477
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IntroductionMVA Based on H. M ARKOWITZOPM • 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
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The one-period modelMarket 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
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Securities in a OPMReturns of Securities Pi (T )Ri (T ) := pi (1 ≤ i ≤ d )Estimated Means, Variances and CovariancesE (Ri (T )) = µi , Cov Ri (T ), Rj (T ) = σij (1 ≤ i ≤ d )RemarkThe matrix σ := σij i,j∈{1,...,d}is positive semi-deﬁnite as it is a variance-covariance matrix. 10 / 477
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Securities in a OPMEach 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
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Budget equation and portfolio returnThe Budget EquationInvestor with initial wealth x > 0 holds ϕi ≥ 0 shares of securityi with ϕi · pi = x 1≤i≤dThe Portfolio Vector π := (π1 , . . . , πd )T ϕi · pi πi := (1 ≤ i ≤ d ) xPortfolio Return R π := πi · Ri (T ) = π T R 1≤i≤d 12 / 477
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Budget equation and portfolio returnRemark 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
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Budget equation and portfolio returnRemark (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
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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
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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
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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
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Relations between the formulations (i) and (ii)TheoremAssume: 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 =1Then ∗(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
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The diversiﬁcation effect–exampleHolding different Securities reduces VarianceBoth security prices ﬂuctuate randomly σ11 , σ22 > 0 independent σ12 = σ21 = 0 0.5Then for the Portfolio π = we get 0.5 σ11 σ22 Var (R π ) = Var (0.5 · R1 + 0.5 · R2 ) = + 4 4 19 / 477
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The diversiﬁcation effect–exampleHolding 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
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ExampleMean-Variance CriterionInvesting 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
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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
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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
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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
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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
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Stock price modelOPM No assumption on distribution of security returns Solving MV Problem just needed expectations and covariances 26 / 477
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Stock price modelOPM 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
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Stock price modelOPM 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
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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 returnNeed of Continuous-Time Market Models Discrete-time multi-period models (many periods −→ no fast algorithms) 29 / 477
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Outline2 The Continuous-Time Market Model 30 / 477
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Outline2 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
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Modeling the security pricesMarket 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
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The bond priceAssume: 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
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The stock priceStock 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 1log-linear model for a stock price ln(Pi (t)) = ln(pi ) + bi · t + ”randomness” 34 / 477
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The stock priceRandomness 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
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The stock priceDeviation at time t Y (t) := ln(Pi (t)) − ln(pi ) − bi · twith Y (t) ∼ N (0, σ 2 t)Properties: E (Y (t)) = 0, Y (t) is time-dependent. 36 / 477
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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
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The stock priceExistence and properties of the stochastic process {Y (t)}t∈[0,∞)will be studied in the excursion on the Brownian motion. 38 / 477
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Outline2 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
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General assumptionsGeneral assumptionsLet (Ω, F, P) be a complete probability space with sample space Ω,σ-ﬁeld F and probability measure P. 40 / 477
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FiltrationDeﬁnitionLet {Ft }t∈I be a family of sub-σ-ﬁelds of F, I be an ordered index setwith 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
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Stochastic processDeﬁnitionA set {(Xt , Ft )}t∈I consisting of a ﬁltration {Ft }t∈I and a family ofRn -valued random variables {Xt }t∈I with Xt being Ft -measurable iscalled a stochastic process with ﬁltration {Ft }t∈I . 42 / 477
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RemarkRemark 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
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Sample pathSample pathFor ﬁxed ω ∈ Ω the set X .(ω) := {Xt (ω)}t∈I = {X (t, ω)}t∈Iis called a sample path or a realization of the stochastic process. 44 / 477
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Identiﬁcation of stochastic processesCan two stochastic processes be identiﬁed with each other?DeﬁnitionLet {(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.DeﬁnitionLet {(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
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Identiﬁcation of stochastic processesRemark X , Y indistinguishable ⇒ Y modiﬁcation of X .TheoremLet the stochastic process Y be a modiﬁcation of X . If both processeshave continuous sample paths P-almost surely, then X and Y areindistinguishable. 46 / 477
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Brownian motionDeﬁnitionThe 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
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Brownian motionRemarkBy an n-dimensional Brownian motion we mean the Rn -valued process W (t) = (W1 (t), . . . , Wn (t)),with components Wi being independent one-dimensional Brownianmotions. 48 / 477
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Brownian motion and ﬁltrationBrownian 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
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Brownian motion and ﬁltrationRequirement 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.ConventionIf we consider a Brownian motion {(Wt , Ft )}t≥0 with an arbitraryﬁltration we implicitly assume iii)∗ to be fulﬁlled. 50 / 477
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Existence of the Brownian motionHow can we show the existence of a stochastic process satisfying therequirements 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
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Brownian motion and ﬁltrationTheoremThe Brownian ﬁltration {Ft }t≥0 is right-continuous as well asleft-continuous, i.e., we have Ft = Ft+ := Ft+ε and Ft = Ft− := σ Fs . ε>0 s<tDeﬁnitionA ﬁltration {Gt }t≥0 satiﬁes the usual conditions, if it is right-continuousand G0 contains all P-null sets of F.General assumption for this sectionLet {Ft }t≥0 be a ﬁltration which satisﬁes the usual conditions. 52 / 477
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MartingalesDeﬁnitionThe 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
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Interpretation of the martingale conceptExample: Modeling games of chanceXn : Wealth of a gambler after n-th participation in a fair gameMartingale condition: E (Xn+1 |Fn ) = Xn P-a.s.⇒ "After the game the player is as rich as he was before"favorable game = sub-martingalenon-favorable game = super-martingale 54 / 477
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Interpretation of the martingale conceptExample: Tossing a fair coin"Head": Gambler receives one dollar"Tail": Gambler loses one dollar⇒ Martingale 55 / 477
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Interpretation of the martingale conceptTheoremA 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
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Interpretation of the martingale conceptTheorem(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
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Interpretation of the martingale conceptRemark(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
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Stopping timeDeﬁnitionA 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).TheoremIf τ1 , τ2 are both stopping times then τ1 ∧ τ2 := min{τ1 , τ2 } is also astopping time. 59 / 477
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The stopped processThe stopped processLet {(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 ofgames until he is either bankrupt or has reached a given level ofwealth. 60 / 477
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The stopped ﬁltrationThe stopped ﬁltrationLet 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
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The stopped ﬁltrationWhat will happen if we stop a martingale or a sub-martingale?Theorem: Optional samplingLet {(Xt , Ft )}t∈[0,∞) be a right-continuous sub-martingale (ormartingale). Let τ1 , τ2 be stopping times with τ1 ≤ τ2 . Then for allt ∈ [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
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The stopped ﬁltrationCorollaryLet τ be a stopping time and {(Xt , Ft )}t∈[0,∞) a right-continuoussub-martingale (or martingale). Then the stopped process{(Xt∧τ , Ft )}t∈[0,∞) is also a sub-martingale (or martingale). 63 / 477
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The stopped ﬁltrationTheoremLet {(Xt , Ft )}t∈[0,∞) be a right-continuous process. Then Xt is amartingale if and only if for all bounded stopping times τ we have EXτ = EX0 .→ Characterization of a martingale 64 / 477
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The stopped ﬁltrationDeﬁnitionLet {(Xt , Ft )}t∈[0,∞) be a stochastic process with X0 = 0. If there is anon-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. Thesequence {τn }n∈N is called a localizing sequence corresponding to X . 65 / 477
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The stopped ﬁltrationRemark(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 ].TheoremA non-negative local martingale is a super-martingale. 66 / 477
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The stopped ﬁltrationTheorem: Doob’s inequalityLet {Mt }t≥0 be a martingale with right-continuous paths and 2E (MT ) < ∞ or all T > 0. Then, we have 2 2 E sup |Mt | ≤ 4 · E (MT ). 0≤t≤TTheoremLet {(Xt , Ft )}t∈[0,∞) be a non-negative super-martingale withright-continuous paths. Then, for λ > 0 we obtain λ·P ω sup Xs (ω) ≥ λ ≤ E (X0 ). 0≤s≤t 67 / 477
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Outline2 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
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Continuation: The stock pricelog-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 tomodel the "randomness" 69 / 477
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Continuation: The stock priceMarket with one stock and one bond (d=1) ln(P1 (t)) = ln(p1 ) + b1 · t + σ11 Wt P1 (t) = p1 · exp b1 · t + σ11 WtMarket 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
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Continuation: The stock priceDistribution 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
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Continuation: The stock priceLemma m 1 2Let 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
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Interpretation of the stock price modelThe 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
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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
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Summary: Stock pricesBond 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
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ExtensionExtension: Model with non-constant, time-dependent, and integrablerates 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 tProblem: σij (s) dWj (s) 0 ˆ⇒ Stochastic integral (Ito integral) 76 / 477
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Outline2 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
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ˆThe Ito integralIs it possible to deﬁne the stochastic integral t Xs (ω) dWs (ω) 0ω-wise in a reasonable way? 78 / 477
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ˆThe Ito integralTheoremP-almost all paths of the Brownian motion {Wt }t∈[0,∞) are nowheredifferentiable.⇒ A deﬁnition of the form t t dWs (ω) Xs (ω) dWs (ω) = Xs (ω) ds ds 0 0 is impossible. 79 / 477
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ˆThe Ito integralTheoremWith the deﬁnition 2n Zn (ω) := W i (ω) − W i−1 (ω) , n ∈ N, ω ∈ Ω 2n 2n i=1we have n→∞ Zn (ω) − − ∞ −→ P-a.s. ,i.e., the paths Wt (ω) of the Brownian motion admit inﬁnite variation onthe interval [0, 1] P-almost surely.The paths Wt (ω) of the Brownian motion have inﬁnite variation on eachnon-empty ﬁnite interval [s1 , s2 ] ⊂ [0, ∞) P-almost surely. 80 / 477
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General assumptionsGeneral assumptions for this sectionLet (Ω, F, P) be a complete probability space equipped with a ﬁltration{Ft }t satisfying the usual conditions. Further assume that on thisspace a Brownian motion {(Wt , Ft )}t∈[0,∞) with respect to this ﬁltrationis deﬁned. 81 / 477
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Simple processDeﬁnitionA stochastic process {Xt }t∈[0,T ] is called a simple process if there existreal numbers 0 = t0 < t1 < . . . < tp = T , p ∈ N, and bounded randomvariables Φi : Ω → R, i = 0, 1, . . . , p, with Φ0 F0 -measurable, Φi Fti−1 -measurable, i = 1, . . . , psuch 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=1for each ω ∈ Ω. 82 / 477
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Simple processRemark 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
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Stochastic integralDeﬁnitionFor a simple process {Xt }t∈[0,T ] the stochastic integral I.(X ) fort ∈ (tk , tk +1 ] is deﬁned according to t It (X ) := Xs dWs := Φi (Wti − Wti−1 ) + Φk +1 (Wt − Wtk ), 0 1≤i≤kor more generally for t ∈ [0, T ]: t It (X ) := Xs dWs := Φi (Wti ∧t − Wti−1 ∧t ). 0 1≤i≤p 84 / 477
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Stochastic integralTheorem: Elementary properties of the stochastic integralLet 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
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Stochastic integralRemark(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
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Stochastic integralRemark(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
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MeasurabilityDeﬁnitionA stochastic process {(Xt , Gt )}t∈[0,∞) is called measurable if themapping [0, ∞) × Ω → Rn (s, ω) → Xs (ω)is B([0, ∞)) ⊗ F-B(Rn )-measurable.RemarkMeasurability of the process X implies that X (., ω) isB([0, ∞))-B(Rn )-measurable for a ﬁxed ω ∈ Ω. Thus, for all t ∈ [0, ∞), ti = 1, . . . , n, the integral Xi2 (s) ds is deﬁned. 0 88 / 477
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MeasurabilityDeﬁnitionA stochastic process {(Xt , Gt )}t∈[0,∞) is called progressivelymeasurable 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
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MeasurabilityRemark(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
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MeasurabilityTheoremIf all paths of the stochastic process {(Xt , Gt )}t∈[0,∞) areright-continuous (or left-continuous), then the process is progressivelymeasurable.TheoremLet τ be a stopping time with respect to the ﬁltration {Gt }t∈[0,∞) . If thestochastic process {(Xt , Gt )}t∈[0,∞) is progressively measurable, thenso is the stopped process {(Xt∧τ , Gt )}t∈[0,∞) . In particular, Xt∧τ is Gtand Gt∧τ -measurable. 91 / 477
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Extension of the stochastic integral toL2 [0, T ]-processesDeﬁnition 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 TNorm on L2 [0, T ]: X 2 T := E Xt2 dt . 0 92 / 477
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Extension of the stochastic integral toL2 [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
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Extension of the stochastic integral toL2 [0, T ]-processes ˆIto isometryLet 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 0a norm on the space of stochastic integrals.⇒ I.(X ) linear, norm-preserving (= isometry) ˆ⇒ I.(X ) Ito isometry 94 / 477
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Extension of the stochastic integral toL2 [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
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Extension of the stochastic integral toL2 [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
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Extension of the stochastic integral toL2 [0, T ]-processesTheoremAn arbitrary X ∈ L2 [0, T ] can be approximated by a sequence ofsimple 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
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Extension of the stochastic integral toL2 [0, T ]-processesLemmaLet {(Xt , Gt )}t∈[0,∞) be a martingale where the ﬁltration {Gt }t∈[0,∞)satisﬁes the usual conditions. Then the process Xt possesses aright-continuous modiﬁcation {(Yt , Gt )}t∈[0,∞) such that{(Yt , Gt )}t∈[0,∞) is a martingale. 98 / 477
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Extension of the stochastic integral toL2 [0, T ]-processes ˆConstruction of the Ito integral for processes in L2 [0, T ]There exists a unique linear mapping J from L2 [0, T ] into the space ofcontinuous 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 0Uniqueness: If J, J ′ satisfy (1) and (2), then for all X ∈ L2 [0, T ] theprocesses J ′ (X ) and J(X ) are indistinguishable. 99 / 477
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Extension of the stochastic integral toL2 [0, T ]-processesDeﬁnitionFor X ∈ L2 [0, T ] and J as before we deﬁne by t Xs dWs := Jt (X ) 0 ˆthe stochastic integral (or Ito integral) of X with respect to W . 100 / 477
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Extension of the stochastic integral toL2 [0, T ]-processesTheorem: Special case of Doob’s inequalityLet 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
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Extension of the stochastic integral toL2 [0, T ]-processesMulti-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 ]. ˆIto 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
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Further extension of the stochastic integralDeﬁnition 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
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Further extension of the stochastic integralProcesses 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 timesStopping times (with respect to {Ft }t ): t 2 τn (ω) := T ∧ inf 0 ≤ t ≤ T Xs (ω) ds ≥ n , n ∈ N 0Sequence of stopped processes: (n) Xt (ω) := Xt (ω) · 1{τn (ω)≥t}⇒ X (n) ∈ L2 [0, T ] ⇒ Stochastic integral already deﬁned. 104 / 477
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Further extension of the stochastic integralStochastic integral: It (X ) := It (X (n) ) for 0 ≤ t ≤ τnConsistence 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
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Further extension of the stochastic integralStopping times: n→∞ τn − − +∞ P-a.s. −→⇒ It (X ) local martingale with localizing sequence τn .⇒ Stochastic integral is linear and possesses continuous paths. 106 / 477
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Outline2 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
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ˆThe Ito formulaGeneral assumptions for this sectionLet (Ω, F, P) be a complete probability space equipped with a ﬁltration{Ft }t satisfying the usual conditions. Further, assume that on thisspace a Brownian motion {(Wt , Ft )}t∈[0,∞) with respect to this ﬁltrationis deﬁned. 108 / 477
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ˆThe Ito formulaDeﬁnitionLet {(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
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ˆThe 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
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ˆThe Ito formulaRemark 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
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ˆThe Ito formulaDeﬁnition ˆLet 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 0Quadratic covariation of X and Y : m t X,Y t := Hi (s) · Mi (s) ds. i=1 0 112 / 477
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ˆThe Ito formulaDeﬁnitionQuadratic variation of X X t := X , X t .Notation ˆLet X be a real-valued Ito process, and Y a real-valued, progressivelymeasurable process. We set t t t Y (s) dX (s) := Y (s) · K (s) ds + Y (s) · H(s) dW (s) 0 0 0if all integrals on the right-hand side are deﬁned. 113 / 477
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ˆThe Ito formula ˆTheorem: One-dimensional Ito formula ˆLet Wt be a one-dimensional Brownian motion, and Xt a real-valued Itoprocess with t t Xt = X0 + Ks ds + Hs dWs . 0 0Let 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
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ˆThe Ito formulaRemark 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
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ˆThe Ito formulaLemmaLet 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 ≤mThen 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
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′ ˆSome applications of Ito s formula ′ ˆSome 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
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′ ˆSome 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
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′ ˆSome applications of Ito s formula III ′ ˆSome 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
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ˆThe Ito formula ˆTheorem: Multi-dimensional Ito formula ˆX (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 0where 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
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Product rule or partial integrationCorollary: Product rule or partial integration ˆLet 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 0Then we have t t tXt · 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
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The stock price equationSimple 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
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The stock price equationThe stock price equation t t P(t) = p + P(s) · b ds + P(s) · σ dWs 0 0RemarkThe 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
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The stock price equationThe stock price equation in differential form dP(t) = P(t) b dt + σ dWt P(0) = p 124 / 477
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The stock price equationTheorem: Variation of constantsLet {(W (t), Ft )}t∈[0,∞) be an m-dimensional Brownian motion.Let x ∈ R and A, a, Sj , σj be progressively measurable, real-valuedprocesses 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
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The stock price equationTheorem: Variation of constantsThen 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) = xpossesses 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
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The stock price equationTheorem: Variation of constantsHereby is t t 1 2 Z (t) = exp A(u) − 2 · S(u) du + S(u) dW (u) 0 0the unique solution of the homogeneous equation dZ (t) = Z (t) A(t) dt + S(t)T dW (t) Z (0) = 1. 127 / 477
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The stock price equationRemarkThe process {(X (t), Ft )}t∈[0,∞) solves the stochastic differentialequation 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 0for all t ≥ 0 P-almost surely. 128 / 477
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Outline2 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
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General assumptionsGeneral 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 0for t ∈ [0, T ], T > 0, i = 1, . . . , d . 130 / 477
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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
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Bond and stock pricesBond and stock prices are unique solutions of the stochasticdifferential 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
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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
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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
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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
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Negative bond positions and credit interest ratesNegative 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
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Mathematical realizations of some requirementsMarket 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
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Discrete-time example: self-ﬁnancing strategyMarket with 1 riskless bond and 1 stockTwo-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
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Discrete-time example: self-ﬁnancing strategyt =0Investor uses initial capital to buy shares of bond and stock X (0) = x = ϕ0 (0) · P0 (0) + ϕ1 (0) · P1 (0). 139 / 477
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Discrete-time example: self-ﬁnancing strategyt =1Security prices have changed, investor consumes parts of his wealthCurrent 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 - consumptionInvest remaining capital at the market: X (1) = ϕ0 (1) · P0 (1) + ϕ1 (1) · P1 (1). 140 / 477
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Discrete-time example: self-ﬁnancing strategyt =2Invest remaining capital at the marketWealth: X (2) = ϕ0 (2) · P0 (2) + ϕ1 (2) · P1 (2). Wealth = total wealth of shares heldTotal: 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
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Discrete-time example: self-ﬁnancing strategySelf-ﬁnancing trading strategy: wealth before rebalancing - consumption = wealth after rebalancingCondition: 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
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Discrete-time example: self-ﬁnancing strategyContinuous-time settingWealth 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
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Trading strategy and wealth processesDeﬁnition(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
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Trading strategy and wealth processesDeﬁnition 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
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Trading strategy and wealth processesDeﬁnition(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
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Trading strategy and wealth processesDeﬁnitionA pair (ϕ, c) consisting of a trading strategy ϕ and a consumption rateprocess c is called self-ﬁnancing if the corresponding wealth processX (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
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Trading strategy and wealth processesRemarkWe 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
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Self-ﬁnancing portfolio processDeﬁnitionLet (ϕ, c) be a self-ﬁnancing pair consisting of a trading strategy and aconsumption process with corresponding wealth process X (t) > 0P-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 thepair (ϕ, c). 149 / 477
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Portfolio processesRemark(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
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The wealth equationThe 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
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Alternative deﬁnition of a portfolio processDeﬁnitionThe progressively measurable Rd -valued process π(t) is called aself-ﬁnancing portfolio process corresponding to the consumptionprocess c(t) if the corresponding wealth equation possesses a uniquesolution X (t) = X π,c (t) with T 2 X (t) · πi (t) dt < ∞ P-a.s. for i = 1, . . . , d . 0 152 / 477
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AdmissibilityDeﬁnitionA self-ﬁnancing pair (ϕ, c) or (π, c) consisting of a trading strategy ϕ ora portfolio process π and a consumption process c will be calledadmissible for the initial wealth x > 0, if the corresponding wealthprocess 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
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An examplePortfolio process: π(t) ≡ π ∈ Rd constantConsumption rate: c(t) = γ · X (t), γ > 0Wealth 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
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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
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Outline2 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
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Properties of the continuous-time market modelAssumptions: 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
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General assumption / notationGeneral assumption for this section d =mNotation 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
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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
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Completeness of the marketTheorem: 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
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Completeness of the marketTheorem: 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
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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
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Completeness of the marketRemark 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
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