A General Theory of Backward Stochastic Difference Equations Robert - PowerPoint PPT Presentation

A General Theory of Backward Stochastic Difference Equations A General Theory of Backward Stochastic Difference Equations Robert J. Elliott and Samuel N. Cohen University of Adelaide and University of Calgary July 2, 2009

A General Theory of Backward Stochastic Difference Equations Outline Dynamic Nonlinear Expectations Discrete BSDEs Binomial Pricing Existence & Uniqueness A Comparison Theorem BSDEs and Nonlinear Expectations Conclusions

A General Theory of Backward Stochastic Difference Equations Risk and Pricing ◮ A key question in Mathematical Finance is: Given a future random payoff X, what are you willing to pay today for X? ◮ One could also ask “How risky is X?” ◮ Various attempts have been made to answer this question. (Expected utility, CAPM, Convex Risk Measures, etc...) ◮ While giving an axiomatic approach to answering this question, we shall outline the theory of “Backward Stochastic Difference Equations.”

A General Theory of Backward Stochastic Difference Equations Recall: A filtered probability space consists of (Ω , F , {F t } , P ) , where ◮ Ω is the set of all outcomes ◮ F is the set of all events ◮ F t is the set of all events known at time t ◮ P gives the probability of each event The set L 2 ( F t ) is those random variables, with finite variance, that are known at time t . A statement about outcomes is an event A ∈ F . It is said to hold P -a.s. if P ( statement is true ) = P ( ω : ω ∈ A ) = 1 .

A General Theory of Backward Stochastic Difference Equations Dynamic Nonlinear Expectations Nonlinear Expectations For some terminal time T , we define an ‘ F t -consistent nonlinear expectation’ E to be a family of operators E ( ·|F t ) : L 2 ( F T ) → L 2 ( F t ); t ≤ T with 1. (Monotonicity) If Q 1 ≥ Q 2 P -a.s., E ( Q 1 |F t ) ≥ E ( Q 2 |F t ) 2. (Constants) For all F t -measurable Q , E ( Q |F t ) = Q

A General Theory of Backward Stochastic Difference Equations Dynamic Nonlinear Expectations Nonlinear Expectations 3. (Recursivity) For s ≤ t , E ( E ( Q |F t ) |F s ) = E ( Q |F s ) 4. (Zero-One law) For any A ∈ F t , E ( I A Q |F t ) = I A E ( Q |F t ) .

A General Theory of Backward Stochastic Difference Equations Dynamic Nonlinear Expectations Nonlinear Expectations Two other properties are desirable 5. (Translation invariance) For any q ∈ L 2 ( F t ) , E ( Q + q |F t ) = E ( Q |F t ) + q . 6. (Concavity) For any λ ∈ [ 0 , 1 ] , E ( λ Q 1 + ( 1 − λ ) Q 2 |F t ) ≥ λ E ( Q 1 |F t ) + ( 1 − λ ) E ( Q 2 |F t )

A General Theory of Backward Stochastic Difference Equations Dynamic Nonlinear Expectations Nonlinear Expectations There is a relation between nonlinear expectations and convex risk measures: ◮ If (1)-(6) are satisfied, then for each t , ρ t ( X ) := −E ( X |F t ) defines a dynamic convex risk measure. These risk measures are time consistent . ◮ For simplicity, this presentation will discuss nonlinear expectations. ◮ How could we construct such a family of operators?

A General Theory of Backward Stochastic Difference Equations Dynamic Nonlinear Expectations Recall: ◮ A process M is called adapted if M t is known at time t (i.e. M t is F t measurable). ◮ A martingale difference process is an adapted process M such that, for all t , E [ M t |F t − 1 ] = 0 . and E � M t � < + ∞ .

A General Theory of Backward Stochastic Difference Equations Discrete BSDEs ◮ In this presentation, we shall consider discrete time processes satisfying ‘Backward Stochastic Difference Equations’. ◮ These are the natural extension of Backward Stochastic Differential Equations in continuous time. ◮ We shall see that every nonlinear expectation satisfying Axioms (1-5) solves a BSDE with certain properties, and conversely. ◮ We also establish necessary and sufficient conditions for concavity (Axiom 6). ◮ To do this, we first need to set up our probability space.

A General Theory of Backward Stochastic Difference Equations Discrete BSDEs A probabilistic setting ◮ Let X be a discrete time, finite state process. Without loss of generality, X takes values from the unit vectors in R N . ◮ Let {F t } be the filtration generated by X , that is F t consists of every event that can be known from watching X up to time t . ◮ Let M t = X t − E [ X t |F t − 1 ] . Then M is a martingale difference process, that is E [ M t |F t − 1 ] = 0 ∈ R N .

A General Theory of Backward Stochastic Difference Equations Discrete BSDEs Discrete BSDEs (‘D=Difference’) A BSDE is an equation of the form: � � Y t − F ( ω, u , Y u , Z u ) + Z u M u + 1 = Q t ≤ u < T t ≤ u < T ◮ Q is the terminal condition (in R ) ◮ F is a (stochastic) ‘driver’ function, with F ( ω, u , · , · ) known at time u . ◮ A solution is an adapted pair ( Y , Z ) of processes, Y t ∈ R and Z t ∈ R N × 1 . ◮ All quantities are assumed to be P -a.s. finite.

A General Theory of Backward Stochastic Difference Equations Discrete BSDEs Discrete BSDEs (‘D=Difference’) Equivalently, we can write this in a differenced form: Y t − F ( ω, t , Y t , Z t ) + Z t M t + 1 = Y t + 1 with terminal condition Y T = Q . The important detail is that ◮ The terminal condition is fixed, and the dynamics are given in reverse. ◮ The solution ( Y , Z ) is adapted, that is, at time t it depends only on what has happened up to time t .

A General Theory of Backward Stochastic Difference Equations Discrete BSDEs Binomial Pricing A Special Case: Binomial Pricing ◮ Suppose we have a market with two assets: a stock Y following a simple binomial price process, and a risk free Bond B . ◮ Let r t denote the one-step interest rate at time t . ◮ From each time t , there are two possible states for the stock price the following day, Y ( t + 1 , ↑ ) and Y ( t + 1 , ↓ ) . ◮ Suppose these two states occur with (real world) probabilities p and 1 − p respectively.

A General Theory of Backward Stochastic Difference Equations Discrete BSDEs Binomial Pricing It is easy to show that there exists a unique ‘no-arbitrage’ price 1 Y ( t ) = [ π Y ( t + 1 , ↑ ) + ( 1 − π ) Y ( t + 1 , ↓ )] 1 + r t 1 = E π [ Y ( t + 1 ) |F t ] . 1 + r t Here π the ‘risk-neutral probability’, that is, the price today is the average discounted price tomorrow; when π is the probability of a price increase. We know that π is not dependent on the real world – it is simply a mathematical object which we calculate using the stock price, π = ( 1 + r t ) Y ( t ) − Y ( t + 1 , ↓ ) Y ( t + 1 , ↑ ) − Y ( t + 1 , ↓ ) .

A General Theory of Backward Stochastic Difference Equations Discrete BSDEs Binomial Pricing Writing Y t = Y ( t ) etc... we also know, Y t + 1 = E p ( Y t + 1 |F t ) + L t + 1 where E p ( Y t + 1 |F t ) is the (real-world) conditional mean value of Y t + 1 , and L t + 1 is a random variable with conditional mean value zero ( L t + 1 = Y t + 1 − E p ( Y t + 1 |F t )) .

A General Theory of Backward Stochastic Difference Equations Discrete BSDEs Binomial Pricing In the notation we established before, we can define a Martingale difference process M � 1 − p � − p � � M t + 1 ( ↑ ) = , M t + 1 ( ↓ ) = . p − 1 p And it is easy to show that L t + 1 can be written as Z t · M t + 1 , for some row vector Z t known at time t .

A General Theory of Backward Stochastic Difference Equations Discrete BSDEs Binomial Pricing We can then do some basic algebra: Y t + 1 = E p ( Y t + 1 |F t ) + L t + 1 = Y t + r t Y t − ( 1 + r t ) Y t + E p ( Y t + 1 |F t ) + Z t M t + 1 1 = Y t + r t Y t − ( 1 + r t ) E π ( Y t + 1 |F t ) + E p ( Y t + 1 |F t ) + Z t M t + 1 1 + r t = Y t + r t Y t − E π ( Y t + 1 − E p ( Y t + 1 ) |F t ) + Z t M t + 1 = Y t + r t Y t − E π ( L t + 1 |F t ) + Z t M t + 1 � � = Y t − − r t Y t + Z t E π ( M t + 1 |F t ) + Z t M t + 1 = Y t − F ( Y t , Z t ) + Z t M t + 1

A General Theory of Backward Stochastic Difference Equations Discrete BSDEs Binomial Pricing So our one-step pricing formula is equivalent to the equation Y t + 1 = Y t − F ( Y t , Z t ) + Z t M t + 1 where F ( Y t , Z t ) = − r t Y t + Z t · E ( M t + 1 |F t ) � π − 0 . 5 � = − r t Y t + Z t . 0 . 5 − π This is a special case of a BSDE.

A General Theory of Backward Stochastic Difference Equations Discrete BSDEs Existence & Uniqueness Before giving general existence properties of BSDEs, we need the following. Definition t M t + 1 P -a.s. for all t , then we write Z 1 ∼ M Z 2 . If Z 1 t M t + 1 = Z 2 Note this is an equivalence relation for Z t ∈ R N × 1 . Theorem For any F t + 1 -measurable random variable W ∈ R with E [ W |F t ] = 0 , there exists a F t -measurable Z t ∈ R N × 1 with W = Z t M t + 1 .

A General Theory of Backward Stochastic Difference Equations Discrete BSDEs Existence & Uniqueness An Existence Theorem Theorem (Cohen & Elliott, forthcoming) Suppose (i) F ( ω, t , Y t , Z t ) is invariant under equivalence ∼ M (ii) For all Z t , the map Y t �→ Y t − F ( ω, t , Y t , Z t ) is a bijection Then a BSDE with driver F has a unique solution. Corollary These conditions are necessary and sufficient.