Doubly reflected BSDEs with jumps and generalized Dynkin games

Transcription

1 Doubly reflected BSDEs with jumps and generalized Dynkin games Roxana DUMITRESCU (University Paris Dauphine, Crest and INRIA) Joint works with M.C. Quenez (Univ. Paris Diderot) and Agnès Sulem (INRIA Paris-Rocquecourt) May 28 th, 2014 Journée Mathématiques en Mouvement

2 Plan BSDEs and Financial application: Pricing of European options.

3 Plan BSDEs and Financial application: Pricing of European options. Reflected BSDEs with jumps.

4 Plan BSDEs and Financial application: Pricing of European options. Reflected BSDEs with jumps. Links with variational inequalities for partial-integro differential equations.

5 Plan BSDEs and Financial application: Pricing of European options. Reflected BSDEs with jumps. Links with variational inequalities for partial-integro differential equations. Financial application: Optimal stopping for dynamic risk measures.

6 Plan BSDEs and Financial application: Pricing of European options. Reflected BSDEs with jumps. Links with variational inequalities for partial-integro differential equations. Financial application: Optimal stopping for dynamic risk measures. Double reflected backward stochastic differential equations with jumps.

7 Plan BSDEs and Financial application: Pricing of European options. Reflected BSDEs with jumps. Links with variational inequalities for partial-integro differential equations. Financial application: Optimal stopping for dynamic risk measures. Double reflected backward stochastic differential equations with jumps. Links with classical Dynkin games.

8 Plan BSDEs and Financial application: Pricing of European options. Reflected BSDEs with jumps. Links with variational inequalities for partial-integro differential equations. Financial application: Optimal stopping for dynamic risk measures. Double reflected backward stochastic differential equations with jumps. Links with classical Dynkin games. Links with generalized Dynkin games.

9 Plan BSDEs and Financial application: Pricing of European options. Reflected BSDEs with jumps. Links with variational inequalities for partial-integro differential equations. Financial application: Optimal stopping for dynamic risk measures. Double reflected backward stochastic differential equations with jumps. Links with classical Dynkin games. Links with generalized Dynkin games. Financial application: Pricing of recallable options.

10 Backward stochastic differential equations Definition A process (Y,Z ) is said to be a solution of the BSDE associated with driver f and terminal condition ξ if dy t = f (t,y t,z t )dt Z t dw t Y T = ξ T In other words... We are given the terminal condition ξ and we want to obtain the value at a time t < T. The solution is a process, i.e. it depends on (t,ω).

11 Backward stochastic differential equations with jumps Definition A process (Y,Z,K ) is said to be a solution of the BSDE associated with driver f and terminal condition ξ if dy t = f (t,y t,z t,k t ( ))dt Z t dw t K t (e)ñ(t,de) R Y T = ξ T (Y ) t T is a RCLL process (right continuous and left limited).

12 BSDEs and Pricing of European options. What is an European option? Definition An European option is a financial contract between two parties, the buyer and the seller of this type of option. The buyer of the option has the right - but not the obligation - to buy or sell a specific financial instrument ( the underlying S) at a specific price ( strike K ) and time ( maturityt ). The seller is obligated to sell the financial instrument if the buyer so decides. The buyer pays a fee (called a premium) for this right.

13 BSDEs and Pricing of European options. Mathematical modelling. A European contingent claim ξ settled at time T is an F T measurable random variable. It is a contract which pays ξ at maturity T. The arbitrage-free pricing principle is the following: if we start with the price of the claim as initial endowment and invest in n assets, the value of the portfolio at time T must be just enough to guarantee ξ. Theorem (N.E.Karoui, M.C. Quenez 97) Assume that the market is complete (the claim is hedgeable). Let ξ be a positive square-integrable contingent claim. There exists a hedging strategy (X,π) against ξ such that dx t = r t X t dt + π t σ t θ t dt + π t σ t dw t, X T = ξ

14 Reflected BSDEs with jumps and RCLL barrier. Definition A process (Y,Z,k(.),A) is said to be a solution of the reflected BSDE with jumps associated with driver g and obstacle ξ. if dy t = g(t,y t,z t,k t ( ))dt + da t Z t dw t k t (u)ñ(dt,du) R Y T = ξ T ξ t Y t, 0 t T a.s., T (Y t ξ t )da c t = 0 a.s. and A d τ = A d τ 1 {Yτ =ξ τ } 0

15 Reflected BSDEs with jumps and obstacle problems for PIDEs. 1. Reflected BSDEs with jumps in the Markovian case For each (t,x) [0,T ] R, let {Xs t,x,t s T } be the unique R-valued solution of the SDE with jumps: s s Xs t,x = x + b(xr t,x )dr + σ(xr t,x )dw r + t t s The obstacle ξ t,x and driver f are of the following form: ξs t,x ξ t,x T := h(s,xs t,x ), s < T t,x := g(x f (s,x t,x s T ) (ω),y,z,k) := ϕ(s,x t,x s t R β(x t,x r,e)ñ(dr,de) (ω),y,z, R k(e)γ(x,e)ν(de))1 s t 2. We set u(t,x) := Y t,x t. (1)

16 Reflected BSDEs with jumps and obstacle problems for PIDEs. Theorem (R.D., M.C.Quenez and A.Sulem) The function u defined by (1) is the unique viscosity solution of the following obstacle problem: min(u(t,x) h(t,x), u u (t,x) Lu(t,x) f (t,x,u(t,x),(σ )(t,x),bu(t,x)) = 0, t x u(t,x) = g(x),x R (2)

17 Reflected BSDEs with jumps and obstacle problems for PIDEs. where L := A + K Aφ(x) := 1 2 σ 2 (x) 2 φ (x) + b(x) φ x 2 x (x), φ C2 (R) K φ(x) := ( R φ(x + β(x,e)) φ(x) φ ) x (x)β(x,e) ν(de) Bφ(x) := R (φ(x + β(x,e)) φ(x))γ(x,e)ν(de).

18 RBSDEs with jumps and optimal stopping. Optimal stopping problem for dynamic risk measures induced by BSDEs with jumps. In the framework of risk measures: the state process X = an index, an interest rate process, an economic factor, an indicator of the market, the value of a portfolio, which has an influence on the risk measure and the position. Dynamic risk measure of the financial position ζ at time t: ρ t (ζ,s) := X t (ζ,s) where X t (ζ,s) = X t denotes the solution of the following BSDE: dx t = f (t,x t,π t,l t ( ))dt π t dw t l t (u)ñ(dt,du); X S = ζ R The minimal risk measure: v(s) := ess inf τ T S ρ S (ξ τ,τ) (3)

19 RBSDEs with jumps and optimal stopping. Proposition 1. The minimal risk measure at time S satisfies v(s) = Y S = u(s,x S ) a.s. (4) where u is the unique viscosity solution of the PIDIV (2). Moreover, the stopping time τs defined by τ S := inf{t S, Y t = ξ t } = inf{t S, u(t,x t ) = h(t,x t )} is optimal for (3), that is v(s) = ρ S (ξ τ S,τ S ) a.s.

20 Double barrier reflected BSDEs with jumps and RCLL barriers. Definition A process (Y,Z,k(.),A,A ) is said to be a solution of the double barrier reflected BSDE with jumps associated with driver g and obstacles ξ.,ζ. if T 0 dy t = g(t,y t,z t,k t ( ))dt + da t da t Z t dw t k t (u)ñ(dt,du) Y T = ξ T (5) R ξ t Y t ζ t, 0 t T a.s., (Y t ξ t )da c t = 0 a.s. and A d τ = A d τ 1 {Yτ =ξ τ } and A d τ = A d τ 1 {Yτ =ζ τ } a.s. T 0 (ζ t Y t )da c t = 0 a.s.

21 Dynkin games. Dynkin games have been introduced by Dynkin (1967) as a generalization of optimal stopping problems. The setting of the problem: There are two players,labeled Player 1 and Player 2,who observe two payoff processes ξ and ζ defined on a probability space (Ω, F, P). Player 1 (resp.,2) chooses a stopping time σ (resp., τ) as control for this optimal stopping problem. At (stopping) time σ τ, the game is over, and Player 2 pays the amount ξ τ 1 {τ σ} + ζ σ 1 {σ<τ} to Player 1. The objective of Player 1 is to maximize the payment, while Player 2 wishes to minimize it.

22 DBBSDEs with jumps and classical Dynkin games. We consider two cases. Case 1: the driver g does not depend on the solution Let us introduce the following stochastic game: Classical Dynkin games Consider the gain (or payoff): σ τ I S (τ,σ) = g(u)du + ξ τ 1 {τ σ} + ζ σ 1 {σ<τ} S The upper and lower value functions at time S are defined respectively by V (S) := ess inf σ ess sup τ E[I S (τ,σ) F S ] V (S) := ess sup τ ess inf σ E[I S (τ,σ) F S ] The game is said to be fair if it admits a value function, i.e. V (S) = V (S).

23 DBBSDEs with jumps and classical Dynkin games. We give the following characterization theorem: Theorem Let ξ and ζ be two adapted RCLL processes with ζ T = ξ T a.s., ξ S 2, ζ S 2, ξ t ζ t, t [0,T ] a.s. Then DBBSDE associated with driver process g(t) admits a unique solution (Y,Z,k(.),α(= A A )) and for each stopping time S, Y S is the common value function of the Dynkin game, that is Y S = V (S) = V (S) a.s.

24 DBBSDEs with jumps and generalized Dynkin games. Case 2: the driver g depends on (y,z,k) Generalized Dynkin games Let I(τ,σ) be the F τ σ -measurable random variable defined by I(τ,σ) = ξ τ 1 τ σ + ζ σ 1 σ<τ. For each stopping time S, the upper and lower value functions at time S are defined respectively by V (S) := ess inf σ T S ess sup τ T S E S,τ σ (I (τ,σ)) V (S) := ess sup τ T S ess inf σ T S E S,τ σ (I (τ,σ)). where E,τ σ (I (τ,σ)) = X τ,σ, with (X τ,σ,π τ,σ,l τ,σ the solution of the BSDE dxs τ,σ dw s R lτ,σ τ,σ s (u)ñ(ds,du);x π τ,σ s = g(s,xs τ,σ τ σ = I(τ,σ).,π τ,σ s ) being )ds,l τ,σ s

25 DBBSDEs with jumps and generalized Dynkin games. We give the following characterization theorem. Theorem (R.D., M.C.Quenez and A. Sulem) Let ξ and ζ be RCLL adapted processes in S 2 such that ξ t ζ t, t [0,T ] a.s. Suppose that Mokobodski s condition is satisfied. Let (Y,Z,k( ),α) be the solution of the DBBSDE (5). Then, for each stopping time S, we have Y S = V (S) = V (S) a.s. (6) We obtain that in the case when α := A A is continuous, there exist optimal stopping times given by τ S = inf{t S : Y t = ξ t } and σ S = inf{t S : Y t = ζ t }

26 DBBSDEs with jumps and pricing of recallable options. Recallable option( american game option): the seller is allowed to cancel the recallable option and the buyer is allowed to exercise it at any stopping time up to the maturity T. If the buyer decides to exercise at σ or the seller to cancel at τ, then the seller pays the amount: I(τ,σ) = ξ σ 1 {σ τ} + ζ τ 1 {τ<σ}. The quantity ξ σ (resp. ζ τ ) is the amount that the buyer obtains (resp. the seller pays) for her decision to exercise (resp., cancel) first at σ (resp., τ). The difference ζ ξ represents the compensation process. The nonarbitrage price process of the recallable option is equal to the value process of the zero-sum Dynkin game associated with ξ and ζ.

27 References D.R., M.C. Quenez and A. Sulem. Double barrier reflected BSDEs with jumps and generalized Dynkin games D.R., Quenez M.-C. and Sulem A., Optimal stopping for dynamic risk measures with jumps, preprint. El Karoui N. and M.C. Quenez: Non-linear Pricing Theory and Backward Stochastic Differential Quenez M.-C. and Sulem A., BSDEs with jumps, optimization and applications to dynamic risk measures, Stochastic Processes and their Applications 123 (2013), pp Quenez M.-C. and Sulem A., Reflected BSDEs and robust optimal stopping for dynamic risk measures with jumps, INRIA Research report RR-8211, January 2013.