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Portfolio OptimizationGerhard-Wilhelm Weber1           Erik Kropat2      Zafer-Korcan Görgülü3                   1 Institu...
Outline I1   The Mean-Variance Approach in a One-Period Model      Introduction2   The Continuous-Time Market Model      M...
Outline II3   Option Pricing      Introduction      Examples      The Replication Principle      Arbitrage Opportunity    ...
Outline III4   Pricing of Exotic Options and Numerical Algorithms       Introduction       Examples       Examples       E...
Outline IV5   Optimal Portfolios      Introduction and Formulation of the Problem      The martingale method      Optimal ...
Outline1   The Mean-Variance Approach in a One-Period Model                                                       6 / 477
Outline1   The Mean-Variance Approach in a One-Period Model      Introduction                                             ...
IntroductionMVA     Based on H. M ARKOWITZOPM   • Decisions on investment strategies only at the beginning of the        p...
The one-period modelMarket with d traded securities   d different securities with positive prices p1 , . . . , pd at time ...
Securities in a OPMReturns of Securities             Pi (T )Ri (T ) :=     pi          (1 ≤ i ≤ d )Estimated Means, Varian...
Securities in a OPMEach security perfectly divisable    Hold ϕi ∈ R shares of security i        (1 ≤ i ≤ d )    Negative p...
Budget equation and portfolio returnThe Budget EquationInvestor with initial wealth x > 0 holds ϕi ≥ 0 shares of securityi...
Budget equation and portfolio returnRemark   πi . . . fraction of total wealth invested in security i                     ...
Budget equation and portfolio returnRemark (continued)   Portfolio Return                                                 ...
Selection of a portfolio–criterion (i) Maximize mean return (choose security of highest mean return)    −→ risky, big fluct...
Selection of a portfolio–approach by Markowitz (MVA)Balance Risk (Portfolio Variance) and Return (Portfolio Mean) (i) Maxi...
Solution methods (i) Linear Optimization Problem with quadratic constraint    −→ No standard algorithms, numerical ineffici...
Relations between the formulations (i) and (ii)TheoremAssume:    σ positive definite                    min µi ≤ c2 ≤ max µ...
The diversification effect–exampleHolding different Securities reduces VarianceBoth security prices fluctuate    randomly   ...
The diversification effect–exampleHolding different Securities reduces Variance                                            ...
ExampleMean-Variance CriterionInvesting into seemingly bad security can be optimal. Let be                         1      ...
Example                             1             0.5   Consider Portfolios              and          (does not satify    ...
Example   Ignore expectation constraint and remember   π1 , π2 ≥ 0  π1 + π2 = 1. Hence          min    0.1 · π1 + 0.15 · (...
Example                        1.0                       π2                        0.5                        0.4         ...
Example                                0.15                                    0.1                              Var       ...
Stock price modelOPM   No assumption on distribution of security returns   Solving MV Problem just needed expectations and...
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   ...
Stock price modelOPM with just one security (price p1 at time t = 0 )    After n periods the security has price           ...
Comments on MVA   Only trading at initial time t = 0   No reaction to current price changes possible   ( −→ static model) ...
Outline2   The Continuous-Time Market Model                                       30 / 477
Outline2   The Continuous-Time Market Model      Modeling the Security Prices      Excursion 1: Brownian Motion and Martin...
Modeling the security pricesMarket with d+1 securities    d risky stocks with    prices p1 , p2 , . . . , pd at time t = 0...
The bond priceAssume: Continuous compounding of interest at    a constant rate r :           Bond price: P0 (t) = p0 · er ...
The stock priceStock price = random fluctuations around an intrinsic bond part                       2                     ...
The stock priceRandomness is assumed   to have no tendency, i.e., E("randomness") = 0,   to be time-dependent,   to repres...
The stock priceDeviation at time t                        Y (t) := ln(Pi (t)) − ln(pi ) − bi · twith                      ...
The stock price                Y (t) = Y (δ) + (Y (t) − Y (δ)), δ ∈ (0, t)Distribution of the increments of the deviation ...
The stock priceExistence and properties of the stochastic process                            {Y (t)}t∈[0,∞)will be studied...
Outline2   The Continuous-Time Market Model      Modeling the Security Prices      Excursion 1: Brownian Motion and Martin...
General assumptionsGeneral assumptionsLet (Ω, F, P) be a complete probability space with sample space Ω,σ-field F and proba...
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...
Stochastic processDefinitionA set {(Xt , Ft )}t∈I consisting of a filtration {Ft }t∈I and a family ofRn -valued random varia...
RemarkRemark   I = [0, ∞) or I = [0, T ].   Canoncial filtration (natural filtration) of {Xt }t∈I :                      Ft ...
Sample pathSample pathFor fixed ω ∈ Ω the set                  X .(ω) := {Xt (ω)}t∈I = {X (t, ω)}t∈Iis called a sample path...
Identification of stochastic processesCan two stochastic processes be identified with each other?DefinitionLet {(Xt , Ft )}t∈...
Identification of stochastic processesRemark    X , Y indistinguishable ⇒ Y modification of X .TheoremLet the stochastic pro...
Brownian motionDefinitionThe real-valued process {Wt }t≥0 with continuous sample paths and  i) W0 = 0 P-a.s. ii) Wt − Ws ∼ ...
Brownian motionRemarkBy an n-dimensional Brownian motion we mean the Rn -valued process                    W (t) = (W1 (t)...
Brownian motion and filtrationBrownian motion can be associated with    natural filtration                   FtW := σ{Ws | 0...
Brownian motion and filtrationRequirement iii) of a Brownian motion with respect to a filtration{Ft }t≥0 is often stated as ...
Existence of the Brownian motionHow can we show the existence of a stochastic process satisfying therequirements of a Brow...
Brownian motion and filtrationTheoremThe Brownian filtration {Ft }t≥0 is right-continuous as well asleft-continuous, i.e., w...
MartingalesDefinitionThe real-valued process {(Xt , Ft )}t∈I with E |Xt | < ∞ for all t ∈ I(where I is an ordered index set...
Interpretation of the martingale conceptExample: Modeling games of chanceXn : Wealth of a gambler after n-th participation...
Interpretation of the martingale conceptExample: Tossing a fair coin"Head": Gambler receives one dollar"Tail":   Gambler l...
Interpretation of the martingale conceptTheoremA one-dimensional Brownian motion Wt is a martingale.Remark    Each stochas...
Interpretation of the martingale conceptTheorem(1) Let {(Xt , Ft )}t∈I be a martingale and ϕ : R → R be a convex    functi...
Interpretation of the martingale conceptRemark(1) Typical applications are given by                         ϕ(x) = x 2 ,  ...
Stopping timeDefinitionA stopping time with respect to a filtration {Ft }t∈[0,∞)(or {Fn }n∈N ) is an F-measurable random var...
The stopped processThe stopped processLet {(Xt , Ft )}t∈I be a stochastic process, let I be either N or [0, ∞),and τ a sto...
The stopped filtrationThe stopped filtrationLet τ be a stopping time with respect to a filtration {Ft }t∈[0,∞) .    σ-field of...
The stopped filtrationWhat will happen if we stop a martingale or a sub-martingale?Theorem: Optional samplingLet {(Xt , Ft ...
The stopped filtrationCorollaryLet τ be a stopping time and {(Xt , Ft )}t∈[0,∞) a right-continuoussub-martingale (or martin...
The stopped filtrationTheoremLet {(Xt , Ft )}t∈[0,∞) be a right-continuous process. Then Xt is amartingale if and only if f...
The stopped filtrationDefinitionLet {(Xt , Ft )}t∈[0,∞) be a stochastic process with X0 = 0. If there is anon-decreasing seq...
The stopped filtrationRemark(1) Each martingale is a local martingale.(2) A local martingale with continuous paths is calle...
The stopped filtrationTheorem: Doob’s inequalityLet {Mt }t≥0 be a martingale with right-continuous paths and     2E (MT ) <...
Outline2   The Continuous-Time Market Model      Modeling the Security Prices      Excursion 1: Brownian Motion and Martin...
Continuation: The stock pricelog-linear model for a stock price               ln(Pi (t)) = ln(pi ) + bi · t + ”randomness”...
Continuation: The stock priceMarket with one stock and one bond (d=1)                  ln(P1 (t)) = ln(p1 ) + b1 · t + σ11...
Continuation: The stock priceDistribution of the logarithm of the stock prices                                            ...
Continuation: The stock priceLemma                     m                 1          2Let bi := bi +   2         σij for i ...
Interpretation of the stock price modelThe stock price model                                              m               ...
Interpretation of the stock price model   Vector of mean rates of stock returns                             b = (b1 , . . ...
Summary: Stock pricesBond price and stock prices P0 (t) = p0 · ert                                                        ...
ExtensionExtension: Model with non-constant, time-dependent, and integrablerates of return bi (t) and volatilities σ(t).St...
Outline2   The Continuous-Time Market Model      Modeling the Security Prices      Excursion 1: Brownian Motion and Martin...
ˆThe Ito integralIs it possible to define the stochastic integral                                 t                        ...
ˆThe Ito integralTheoremP-almost all paths of the Brownian motion {Wt }t∈[0,∞) are nowheredifferentiable.⇒ A definition of ...
ˆThe Ito integralTheoremWith the definition                       2n           Zn (ω) :=         W i (ω) − W i−1 (ω) , n ∈ ...
General assumptionsGeneral assumptions for this sectionLet (Ω, F, P) be a complete probability space equipped with a filtra...
Simple processDefinitionA stochastic process {Xt }t∈[0,T ] is called a simple process if there existreal numbers 0 = t0 < t...
Simple processRemark   Xt is Fti−1 -measurable for all t ∈ (ti−1 , ti ].   The paths X (., ω) of the simple process Xt are...
Stochastic integralDefinitionFor a simple process {Xt }t∈[0,T ] the stochastic integral I.(X ) fort ∈ (tk , tk +1 ] is defin...
Stochastic integralTheorem: Elementary properties of the stochastic integralLet X := {Xt }t∈[0,T ] be a simple process. Th...
Stochastic integralRemark(1) By (2) the stochastic integral is a square-integrable stochastic    process.(2) For the simpl...
Stochastic integralRemark(1) Integrals with general boundaries:                    T               T                  t   ...
MeasurabilityDefinitionA stochastic process {(Xt , Gt )}t∈[0,∞) is called measurable if themapping                         ...
MeasurabilityDefinitionA stochastic process {(Xt , Gt )}t∈[0,∞) is called progressivelymeasurable if for all t ≥ 0 the mapp...
MeasurabilityRemark(1) If the real-valued process {(Xt , Gt )}t∈[0,∞) is progressively    measurable and bounded, then for...
MeasurabilityTheoremIf all paths of the stochastic process {(Xt , Gt )}t∈[0,∞) areright-continuous (or left-continuous), t...
Extension of the stochastic integral toL2 [0, T ]-processesDefinition   L2 [0, T ] := L2 [0, T ], Ω, F, {Ft }t∈[0,T ] , P  ...
Extension of the stochastic integral toL2 [0, T ]-processes     ·   2   L2 -norm on the probability space         T       ...
Extension of the stochastic integral toL2 [0, T ]-processes  ˆIto isometryLet X be a simple process. The mapping X → I.(X ...
Extension of the stochastic integral toL2 [0, T ]-processes   Use processes X ∈ L2 [0, T ] approximated by a sequence X (n...
Extension of the stochastic integral toL2 [0, T ]-processes                                 J(.)                          ...
Extension of the stochastic integral toL2 [0, T ]-processesTheoremAn arbitrary X ∈ L2 [0, T ] can be approximated by a seq...
Extension of the stochastic integral toL2 [0, T ]-processesLemmaLet {(Xt , Gt )}t∈[0,∞) be a martingale where the filtratio...
Extension of the stochastic integral toL2 [0, T ]-processes                      ˆConstruction of the Ito integral for pro...
Extension of the stochastic integral toL2 [0, T ]-processesDefinitionFor X ∈ L2 [0, T ] and J as before we define by        ...
Extension of the stochastic integral toL2 [0, T ]-processesTheorem: Special case of Doob’s inequalityLet X ∈ L2 [0, T ]. T...
Extension of the stochastic integral toL2 [0, T ]-processesMulti-dimensional generalization of the stochastic integral{(W ...
Further extension of the stochastic integralDefinition    H 2 [0, T ] := H 2 [0, T ], Ω, F, {Ft }t∈[0,T ] , P              ...
Further extension of the stochastic integralProcesses X ∈ H 2 [0, T ]    do not necessarily have a finite T -norm    → no a...
Further extension of the stochastic integralStochastic integral:                       It (X ) := It (X (n) ) for 0 ≤ t ≤ ...
Further extension of the stochastic integralStopping times:                             n→∞                         τn − −...
Outline2   The Continuous-Time Market Model      Modeling the Security Prices      Excursion 1: Brownian Motion and Martin...
ˆThe Ito formulaGeneral assumptions for this sectionLet (Ω, F, P) be a complete probability space equipped with a filtratio...
ˆThe Ito formulaDefinitionLet {(Wt , Ft )}t∈[0,∞) be an m-dimensional Brownian motion.                                     ...
ˆThe Ito formula    X (0) F0 -measurable,    {K (t)}t∈[0,∞) , {H(t)}t∈[0,∞) progressively measurable with                t...
ˆThe Ito formulaRemark   Hj ∈ H 2 [0, T ] for all T > 0.                                 ˆ   The representation of an Ito ...
ˆThe Ito formulaDefinition                                 ˆLet X and Y be two real-valued Ito processes with              ...
ˆThe Ito formulaDefinitionQuadratic variation of X                                           X   t   := X , X t .Notation  ...
ˆThe Ito formula                           ˆTheorem: One-dimensional Ito formula                                          ...
ˆThe Ito formulaRemark         ˆ   The Ito formula differs from the fundamental theorem of calculus   by the additional te...
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...
′                       ˆSome applications of Ito s formula                          ′                       ˆSome applica...
′                       ˆSome applications of Ito s formula II                       ˆ′Some applications of Ito s formula(...
′                       ˆSome applications of Ito s formula III                                  ′                       ˆ...
ˆThe Ito formula                             ˆTheorem: Multi-dimensional Ito formula                                      ...
Product rule or partial integrationCorollary: Product rule or partial integration                                 ˆLet Xt ...
The stock price equationSimple continuous-time market model (1 bond, one stock).Stock price influenced by a one-dimensional...
The stock price equationThe stock price equation                                   t                       t              ...
The stock price equationThe stock price equation in differential form                    dP(t) = P(t) b dt + σ dWt        ...
The stock price equationTheorem: Variation of constantsLet {(W (t), Ft )}t∈[0,∞) be an m-dimensional Brownian motion.Let x...
The stock price equationTheorem: Variation of constantsThen the stochastic differential equation                          ...
The stock price equationTheorem: Variation of constantsHereby is                         t                                ...
The stock price equationRemarkThe process {(X (t), Ft )}t∈[0,∞) solves the stochastic differentialequation in the sense th...
Outline2   The Continuous-Time Market Model      Modeling the Security Prices      Excursion 1: Brownian Motion and Martin...
General assumptionsGeneral assumptions for this section(Ω, F, P) be a complete probability space,{(W (t), Ft )}t∈[0,∞) m-d...
General assumptions (continued)General assumptions for this section (continued)    r (t), b(t) = (b1 (t), . . . , bd (t))T...
Bond and stock pricesBond and stock prices are unique solutions of the stochasticdifferential equations    dP0 (t) = P0 (t...
Possible actions of investors(1) Investor can rebalance his holdings    → sell some securities    → invest in securities  ...
Requirements on a market model(1) Investor should not be able to foresee events    → no knowledge of future prices.(2) Act...
Requirements on a market model(6) Securities are perfectly divisible.(7) Negative positions in securities are possible    ...
Negative bond positions and credit interest ratesNegative bond positions and credit interest rates    Assume: Interest rat...
Mathematical realizations of some requirementsMarket with 1 bond and d stocks    Time t = 0:   – Initial capital of invest...
Discrete-time example: self-financing strategyMarket with 1 riskless bond and 1 stockTwo-period model for time points t = 0...
Discrete-time example: self-financing strategyt =0Investor uses initial capital to buy shares of bond and stock            ...
Discrete-time example: self-financing strategyt =1Security prices have changed, investor consumes parts of his wealthCurren...
Discrete-time example: self-financing strategyt =2Invest remaining capital at the marketWealth:                 X (2) = ϕ0 ...
Discrete-time example: self-financing strategySelf-financing trading strategy:    wealth before rebalancing - consumption = ...
Discrete-time example: self-financing strategyContinuous-time settingWealth process corresponding to strategy ϕ(t):        ...
Trading strategy and wealth processesDefinition(1) A trading strategy ϕ with                                               ...
Trading strategy and wealth processesDefinition    The value                                    d                          ...
Trading strategy and wealth processesDefinition(3) A non-negative progressively measurable process c(t) with    respect to ...
Trading strategy and wealth processesDefinitionA pair (ϕ, c) consisting of a trading strategy ϕ and a consumption rateproce...
Trading strategy and wealth processesRemarkWe have       t                          t           ϕ0 (s) dP0 (s) =          ...
Self-financing portfolio processDefinitionLet (ϕ, c) be a self-financing pair consisting of a trading strategy and aconsumpti...
Portfolio processesRemark(1) The portfolio process denotes the fractions of total wealth invested    in the different stoc...
The wealth equationThe wealth equation        dX (t) = [r (t) X (t) − c(t)] dt               + X (t) π(t)T (b(t) − r (t) 1...
Alternative definition of a portfolio processDefinitionThe progressively measurable Rd -valued process π(t) is called aself-...
AdmissibilityDefinitionA self-financing pair (ϕ, c) or (π, c) consisting of a trading strategy ϕ ora portfolio process π and...
An examplePortfolio process:                       π(t) ≡ π ∈ Rd constantConsumption rate:                        c(t) = γ...
An example   Wealth equation:                    dX (t) = [r (t) − γ] X (t) dt                           + X (t)π T (b(t) ...
Outline2   The Continuous-Time Market Model      Modeling the Security Prices      Excursion 1: Brownian Motion and Martin...
Properties of the continuous-time market modelAssumptions:    Dimension of the underlying Brownian motion    = number of s...
General assumption / notationGeneral assumption for this section                                        d =mNotation      ...
Properties of the continuous-time market model   b, r uniformly bounded   σσ T uniformly positive definite   ⇒ θ(t) 2 unifo...
Completeness of the marketTheorem: Completeness of the market(1) Let the self-financing pair (π, c) consisting of a portfol...
Completeness of the marketTheorem: Completeness of the market(2) Let B ≥ 0 be an FT -measurable random variable and c(t) a...
Completeness of the market   H(t) can be regarded as the appropriate discounting process that   determines the initial wea...
Completeness of the marketRemark   1/H(t) is the wealth process corresponding to the pair                          π(t), c...
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AACIMP 2011 Summer School. Operational Research stream. Lecture by Gerhard-Wilhelm Weber.

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  1. 1. 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
  2. 2. 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
  3. 3. Outline II3 Option Pricing Introduction Examples The Replication Principle Arbitrage Opportunity Continuation Partial Differential Approach (PDA) Arbitrage & Option Pricing 3 / 477
  4. 4. 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
  5. 5. 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
  6. 6. Outline1 The Mean-Variance Approach in a One-Period Model 6 / 477
  7. 7. Outline1 The Mean-Variance Approach in a One-Period Model Introduction 7 / 477
  8. 8. 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
  9. 9. 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
  10. 10. 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
  11. 11. 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
  12. 12. 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
  13. 13. 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
  14. 14. 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
  15. 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. 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. 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. 18. 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
  19. 19. 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
  20. 20. 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
  21. 21. 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
  22. 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. 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. 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. 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. 26. Stock price modelOPM No assumption on distribution of security returns Solving MV Problem just needed expectations and covariances 26 / 477
  27. 27. 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
  28. 28. 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
  29. 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 returnNeed of Continuous-Time Market Models Discrete-time multi-period models (many periods −→ no fast algorithms) 29 / 477
  30. 30. Outline2 The Continuous-Time Market Model 30 / 477
  31. 31. 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
  32. 32. 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
  33. 33. 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
  34. 34. 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
  35. 35. 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
  36. 36. 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
  37. 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. 38. 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
  39. 39. 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
  40. 40. General assumptionsGeneral assumptionsLet (Ω, F, P) be a complete probability space with sample space Ω,σ-field F and probability measure P. 40 / 477
  41. 41. 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
  42. 42. 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
  43. 43. 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
  44. 44. 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
  45. 45. 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
  46. 46. 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
  47. 47. 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
  48. 48. 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
  49. 49. 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
  50. 50. 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
  51. 51. 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
  52. 52. 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
  53. 53. 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
  54. 54. 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
  55. 55. Interpretation of the martingale conceptExample: Tossing a fair coin"Head": Gambler receives one dollar"Tail": Gambler loses one dollar⇒ Martingale 55 / 477
  56. 56. 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
  57. 57. 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
  58. 58. 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
  59. 59. 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
  60. 60. 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
  61. 61. 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
  62. 62. 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
  63. 63. 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
  64. 64. 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
  65. 65. 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
  66. 66. 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
  67. 67. 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
  68. 68. 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
  69. 69. 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
  70. 70. 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
  71. 71. 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
  72. 72. 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
  73. 73. 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
  74. 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. 75. 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
  76. 76. 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
  77. 77. 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
  78. 78. ˆThe Ito integralIs it possible to define the stochastic integral t Xs (ω) dWs (ω) 0ω-wise in a reasonable way? 78 / 477
  79. 79. 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
  80. 80. 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
  81. 81. 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
  82. 82. 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
  83. 83. 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
  84. 84. 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
  85. 85. 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
  86. 86. 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
  87. 87. 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
  88. 88. 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
  89. 89. 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
  90. 90. 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
  91. 91. 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
  92. 92. 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
  93. 93. 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
  94. 94. 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
  95. 95. 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
  96. 96. 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
  97. 97. 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
  98. 98. 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
  99. 99. 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
  100. 100. 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
  101. 101. 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
  102. 102. 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
  103. 103. 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
  104. 104. 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
  105. 105. 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
  106. 106. 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
  107. 107. 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
  108. 108. 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
  109. 109. 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
  110. 110. 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
  111. 111. 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
  112. 112. 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
  113. 113. 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
  114. 114. 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
  115. 115. 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
  116. 116. 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
  117. 117. ′ 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
  118. 118. ′ 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
  119. 119. ′ 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
  120. 120. 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
  121. 121. 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
  122. 122. 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
  123. 123. 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
  124. 124. The stock price equationThe stock price equation in differential form dP(t) = P(t) b dt + σ dWt P(0) = p 124 / 477
  125. 125. 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
  126. 126. 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
  127. 127. 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
  128. 128. 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
  129. 129. 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
  130. 130. 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
  131. 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. 132. 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
  133. 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. 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. 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. 136. 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
  137. 137. 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
  138. 138. 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
  139. 139. 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
  140. 140. 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
  141. 141. 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
  142. 142. 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
  143. 143. 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
  144. 144. 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
  145. 145. 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
  146. 146. 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
  147. 147. 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
  148. 148. 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
  149. 149. 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
  150. 150. 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
  151. 151. 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
  152. 152. 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
  153. 153. 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
  154. 154. 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
  155. 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. 156. 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
  157. 157. 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
  158. 158. 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
  159. 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. 160. 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
  161. 161. 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
  162. 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. 163. 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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