Nonlinear pricing in imperfect financial markets

Transcription

1 Nonlinear pricing in imperfect financial markets Agnès Sulem INRIA, Paris International summer school in mathematical finance on Informational and Imperfect Financial Markets, University of Alberta in Edmonton, Canada, June 25-30, 2016

2 Abstract The aim of the lectures is to study pricing and hedging issues for various options (European, American, game options) in the case of imperfections on the market. These imperfections are taken into account via the nonlinearity of the wealth dynamics. In this setting, the pricing system is expressed as a nonlinear g-expectation/g-evaluation induced by a nonlinear BSDE. We shall address in particular superhedging issues for American and game options in this context and their links with generalized optimal stopping problems and Dynkin games.

3 Overview 1. The perfect market model (linear case) Linear pricing system and linear BSDEs 2. Market with imperfections Nonlinear pricing system and nonlinear BSDEs (g-evaluation). Links with dynamic risk measures 3. Nonlinear pricing of American Contingent claims - Introduction to reflected BSDEs. Links with optimal stopping. 4. Game options in an imperfect market. Links with generalized Dynkin games and doubly reflected nonlinear BSDEs. 5. Market model with default

4 I. Linear pricing theory Market model Let (Ω, (F t ) t 0, P)) be a filtered probability space, where (F t ) t 0 is generated by the Wiener process W and completed. Let T > 0 be fixed. Consider a complete financial market with one risk free and one risky investment possibility, with prices S 0, S 1 per unit given by, respectively { dst 0 = St 0 r t dt ; S0 0 (1) = 1 dst 1 = St 1 [µ t dt + σ t dw t ] ; S0 1 > 0; where r t, µ t, σ t are bounded predictable processes. Assume σ t 0 and σt 1 bounded.

5 Linear pricing theory Strategy Wealth/Portfolio Definition (Strategy) A strategy is given by a couple of predictable processes V t the value of the wealth of the investor (or the value of his portfolio) ϕ t the portfolio process equal to the amount of total wealth invested in the risky asset at time t. A strategy is self-financing if dv t = (V t ϕ t ) ds0 t dst 1 St 0 + ϕ t St 1 = (V t ϕ t )rdt + ϕ t (µ t dt + σ t dw t ] = (rv t + θ t σ t ϕ t )dt + σ t ϕ t dw t where θ t := µ t r t σ t is the risk premium process.

6 Linear pricing theory Pricing and hedging European contingent claims We consider a European contingent claim with maturity T > 0 and payoff ξ which is F T -measurable. The problem is to find a self-financing strategy (V t, ϕ t ) which reaches ξ at time T. Classical approach (without BSDE theory) : (Karatzas-Shreve...) Discounting and change of probability measure. integrability conditions under Q (unique martingale measure) BSDE approach No discounting and no change of probability measure. integrability conditions (for ξ and ϕ) under P (reference probability)

7 Linear pricing theory Classical approach Let Q be the unique martingale probability measure defined by dq dp F T T = exp( θ s dw s 1 T θs 2 ds) From Girsanov theorem, under Q, Wt := W t + t 0 θ sds is a F- Brownian motion and by Itô, S := e t 0 rsds S t is a martingale under Q. d S = Sσ t d W t. We consider the discounted wealth Ṽ := e t 0 rsds V t. We get dṽ = ϕσ t d W t, where ϕ := e t 0 rsds ϕ t is the discounted portfolio.

8 The pricing/hedging problem is to construct a couple (V t, ϕ t ) (value of portfolio, portfolio) such that { dv t = (r t V t + θ t ϕ t σ t )dt + ϕ t σ t dw t ; 0 t T. V T = ξ a.s. that is to find (Ṽ t, ϕ t ) (2) { d Ṽ t = ϕσ t d W t ; 0 t T. Ṽ T = ξ a.s. where ξ = e T 0 ξ. In order to have uniqueness we need integrability conditions. Here the integrability conditions are under Q. It is required that E Q (ξ 2 ) < + (or equivalently, E Q ( ξ 2 ) < + since r t is bounded), and we look for ϕ in H 2 Q (set of predictable processes, square integrable under Q, that is E Q ( T equivalently, E Q ( T 0 ϕ 2 s ds < ), since σ t, σt 1 0 ϕ2 s ds < ) (or bounded).

9 Theorem If ξ L 2 Q (F T ),!(Ṽ, ϕ) H 2 Q H2 Q satisfying (2). Proof. Take Ṽ t = E Q ( ξ F t ). By the theorem of representation of martingales under Q with respect to Brownian motion W, t!(ψ t ) H 2 Q, s.t. Ṽ t = Ṽ0 + Ψ s d W s. 0 Set ϕ t = Ψ t σt 1. We obtain dṽ t = ϕσ t d W t ; Ṽ T = ξ. We get ϕ t = e t 0 rsds ϕ t and V t = e t 0 rsds Ṽ t, i.e.,under P: V t = E P [e T t r sds e T t θ sdw s T t θ 2 s ds ξ F t ] = E P [Γ t,t ξ F t ] ; 0 t T a.s. where the process Γ, called deflator or change of numeraire, is given by dγ t,s = Γ t,s [ r s ds θ s dw s ] ; Γ t,t = 1, Γ t,s = exp( s t θ u dw u + s t i.e. { r u 1 2 θ2 u})du.

10 Linear pricing theory BSDE approach The problem can be formulated in terms of a BSDE problem. Set (3) Z t = ϕ t σ t. Suppose E P [ξ 2 ] < (in L 2 P (F T )). We want to find a couple of square integrable (with respect to P) predictable processes (V, Z) such that { dv t = (r t V t + θ t Z t )dt + Z t dw t ; 0 t T. (4) V T = ξ a.s. Equation (4) is an example of a (linear) BSDE in the pair (V t, Z t ) of unknown processes. If we can solve this equation, then the replicating portfolio ϕ t is given by Z t σt 1 and V t is the hedging price of the claim at time t. Note that, in contrast to ordinary SDEs, Equation (4) has two unknown processes and the terminal value V T of V is given, not the initial value.

11 Both classical and BSDE approaches are possible for the linear case, but in the case of imperfections (nonlinear wealth), the classical approach can no more be used.

12 II. Imperfect financial markets An example leading to nonlinear wealth Different borrowing and lending interest rates, denoted respectively by R t and r t, adapted and bounded with R t r t. Amount borrowed at t: (V t ϕ t ) (at interest rate R t ) Amount lended at t: (V t ϕ t ) + (invested at bond rate r t ) We have: V t ϕ t = (V t ϕ t ) + (V t ϕ t ) Self- financing property gives: dv t = (V t ϕ t ) + r t dt (V t ϕ t ) R t dt + ϕ t ds 1 t S 1 t Using (V t ϕ t ) + = (V t ϕ t ) + (V t ϕ t ), we get, by easy computations dv t = r t V t dt (R t r t )(V t ϕ t ) dt + ϕ t σ t θ t dt + ϕ t σ t dw t.

13 Consider a European option with payoff ξ L 2 (F T ). Pricing and hedging this claim by constructing a replicating portfolio lead to the following problem : Find (V, ϕ) H 2 H 2 such that { dv t = (r t V t (R t r t )(V t ϕ t ) )dt + ϕ t σ t θ t dt + ϕ t σ t dw t ; V T = ξ. This is an example of nonlinear BSDE (coefficient of dt is nonlinear in V and φ). Does a solution exists?

14 Nonlinear BSDEs (Ω, F, P) W = (W t ) t 0 : d-dim Brownian motion; W t = (W 1 t,..., W d t ) F t = σ(w s, 0 s t) augmented with the P-null sets. T > 0 : terminal time H 2 (R) = {(φ t ) 0 t T, real-valued predictable processes s.t. E( T 0 φ2 t dt) < }. We denote φ H 2 = E( T 0 φ2 t dt). H 2 (R d ) = {(φ t ) 0 t T, predictable processes s.t. E( T 0 φ t 2 dt) < }.

15 Definition (Standard coefficients) ξ L 2 (F T ) (the terminal condition) g (the driver) : Ω [0, T ] R R d g(ω, t, y, z) such that (i) g is P B(R) B(R d ) - measurable (ii) E T 0 g(t, 0, 0)2 dt < (iii) g is uniformly Lipschitz w.r.t. y, z, i.e. C > 0 s.t. dt dp- a.s., y, y R, z, z R d, g(ω, t, y, z) g(ω, t, y, z ) C( y y + z z ).

16 Theorem (Existence and uniqueness for BSDEs) There exists a unique solution (Y, Z) H 2 (R) H 2 (R d ) of the BSDE associated to standard coefficients (ξ, g), that is satisfying: { dy t = g(t, Y t, Z t )dt Zt dw t 0 t T (5) Y T = ξ. This result can be extended when the terminal time T is replaced by a stopping time S T 0,T.

17 Proof Suppose g does not depend on y and z, i.e. g(ω, t, y, z) = g(t, ω) H 2 (R) (6) dy t = g(t)dt Z t dw t ; Y T = ξ. Uniqueness. Integrating (6) between t and T, we get Y t = ξ + Taking E( F t ) we get T (7) Y t = E[ξ + t g(s)ds T t T t Z s dw s g(s)ds F t ]. Existence. Consider the candidate Y given by (7). Then Y t + t 0 g(s)ds = E[ξ + T 0 g(s)ds F t]. It is a martingale. By the martingale representation thm, there exists a unique Z in H 2 s. t. Y t + t 0 g(s)ds = Y 0 + t 0 Z s dw s. and we get dy t + g(t)dt = Zt dw t and Y T = ξ. (Y, Z) is thus a solution in H 2 (R) H 2 (R d ) of the BSDE.

18 Moreover we have i.e. Y S 2. E[ sup Yt 2 ] < + 0 t T Hint: Use Burkholder-Davis-Gundy inequalities.

19 General standard driver g(t, ω, y, z) Proof of existence and uniqueness of the solution of the BSDE: We define a mapping Φ as follows. Given (U, V ) H 2 (R) H 2 (R d ), let Φ(U, V ) = (Y, Z) where (Y, Z) is the solution of the BSDE associated with driver g(s) = g(ω, s, U s (ω), V s (ω)) and terminal condition ξ (which is well defined.) We can show that Φ is a contraction (for the norm φ 2 β := E[ T 0 eβs φ 2 s ds] with β well chosen large enough) and thus admits an unique fixed point, which corresponds to the solution of BSDE (5). Detailed proof on the blackboard (see also Pham s book).

20 Linear BSDEs Theorem Let α, β be bounded predictable processes, ξ L 2 P (F T ) and φ predictable with E P [ T 0 φ2 t dt] <. Then there exists a unique solution (Y, Z) of square integrable predictable processes of the linear BSDE dy t = [φ t + α t Y t + β t Z t ]dt Z t dw t ; 0 t T ; Y T = ξ given by T Y t = E[Γ t,t ξ + Γ t,s φ s ds F t ] ; 0 t T a.s. t where the process Γ, usually called adjoint process, is given by dγ t,s = Γ t,s [α s ds + β s dw s ] ; Γ t,t = 1, i.e. s Γ t,s = exp( β u dw u + t s t {α u 1 2 β2 u})du.

21 Sketch of Proof: - Apply Itô formula to Y s Γ t,s - we obtain Y s Γ t,s + s t Γ t,r φ r dr is a martingale. Hence, T Y t = E[Γ t,t ξ + Γ t,s φ s ds F t ] ; 0 t T a.s. t Remark: Note that if ξ 0 and φ 0, then Y t 0.

22 Application to the replication portfolio problem in the linear case: { dy t = ( r t Y t θ t Z t )dt Z t dw t ; 0 t T. Y T = ξ a.s. In this case, we have φ t = 0, α t = r t, β t = θ t. This leads to the classical quasi explicit expression for the wealth Y t (value of replicating portfolio): Y t = E[Γ t,t ξ F t ], where Γ t,s is the deflator process, started at t, such that Γ t,s = exp{ s t r u du} exp{ s t θ u dw u 1 2 s t θ 2 udu}. This defines a linear price system Y : L 2 P H2 P ; ξ Y (ξ), which is increasing and corresponds to the classical free-arbitrage price system. Y (ξ) is called the hedging price of ξ. Note that in particular if ξ 0 a.s. then the price Y t 0.

23 Linear BSDEs were first introduced by Bismut 1973 as the adjoint equation associated with the stochastic Pontryagin maximum principle for stochastic control.

24 Theorem (Comparison theorem for BSDEs) Let (g 1, ξ 1 ), (g 2, ξ 2 ) be standard coefficients and (Y 1, Z 1 ), (Y 2, Z 2 ) be the solutions of the associated BSDEs. Assume ξ 1 ξ 2 g 1 (t, Yt 2, Zt 2 ) g 2 (t, Yt 2, Zt 2 ) Then Yt 1 Yt 2, 0 t T a.s.

25 Proof: Set Ȳ = Y 1 Y 2, Z = Z 1 Z 2 and take d = 1 to simplify. By difference of the 2 BSDEs, { dȳ t = {g 1 (t, Yt 1, Zt 1 ) g 2 (t, Yt 2, Zt 2 )}dt Z t dw t Ȳ T = ξ 1 ξ 2. Now g 1 (t, Yt 1, Zt 1 ) g 2 (t, Yt 2, Zt 2 ) = {g 1 (t, Yt 1, Zt 1 ) g 1 (t, Yt 2, Zt 1 )} + {g 1 (t, Yt 2, Zt 1 ) g 1 (t, Yt 2, Zt 2 )} + {g 1 (t, Yt 2, Zt 2 ) g 2 (t, Yt 2, Zt 2 )} = y g 1 (t)(yt 1 Yt 2 ) + z g 1 (t)(zt 1 Zt 2 ) + φ t where y g 1 (t) := g 1 (t, Yt 1, Zt 1 ) g 1 (t, Yt 2, Zt 1 ) Yt 1 Yt 2 z g 1 (t) := g 1 (t, Yt 2, Zt 1 ) g 1 (t, Yt 2, Zt 2 ) Zt 1 Zt if Yt 1 Yt 2 0; 0 otherwise if Z 1 t Z 2 t 0; 0 otherwise

26 So { dȳ t = { y g 1 (t)ȳ t + z g 1 (t) Z t + φ t }dt Z t dw t Ȳ T = ξ 1 ξ 2 This is a linear BSDE. Note that y g 1 (t) and z g 1 (t) are bounded since g is Lipschitz. Since φ t 0 and ξ 1 ξ 2 0 we get that Ȳt 0, 0 t T a.s.

27 Theorem (Strict Comparison theorem for BSDEs) Let (g 1, ξ 1 ), (g 2, ξ 2 ) be standard coefficients and (Y 1, Z 1 ), (Y 2, Z 2 ) be the solutions of the associated BSDEs. Assume ξ 1 ξ 2 a.s. g 1 (s, Y 2 s, Z 2 s ) g 2 (s, Y 2 s, Z 2 s ) a.s 0 s T Yt 1 = Yt 2 on A F t (for fixed t in [0, T ]) Then ξ 1 = ξ 2 on A and g 1 (s, Ys 2, Zs 2 ) = g 2 (s, Ys 2, Zs 2 ) a.s on A [t, T ]

28 Corollary (Sufficient condition of positivity) If ξ 0 a.s. and g(t, 0, 0) 0 a.s., then Y t 0, 0 t T a.s. Proof. We apply the comparison theorem to (g 1 = g, ξ 1 = ξ), (g 2 = 0, ξ 2 = 0). Solution associated to (g 1, ξ 1 ) : (Y 1, Z 1 ) = (Y, Z). Solution associated to (0, 0) : (Y 2, Z 2 ) = (0, 0).

29 Application to the example of higher interest rate for borrowing Consider an option with payoff ξ L 2 (F T ) in this market. Recall that the pricing/hedginp problem of European option with payoff ξ leads to the BSDE: find (V, ϕ) such that dv t = (r t V t (R t r t )(V t ϕ t ) )dt +ϕ t σ t θdt +ϕ t σ t dw t ; V T = ξ. or equivalently, setting Z t = ϕ t σ t, dv t = g(t, V t, Z t )dt Z t dw t, V T = ξ. with g(t, y, z) = r t y + (R t r t )(y z σ ) + θ t z By the thm of existence and uniqueness, there exists a unique solution (V, Z) of this BSDE in H 2 H 2 with nonlinear driver g. Note that g(t, 0, 0) = 0 in this example. By the corollary, we obtain that if ξ 0 a.s, then V t 0, 0 t T a.s.

30 Other examples of imperfections leading to nonlinear dynamics: dv t = g(t, V t, ϕ t σ t )dt ϕ t σ t dw t Large investor with trading strategy ϕ t affecting the market prices: r t (ω) = r(t, ω, ϕ t ) and similarly for σ, θ. g(t, V t, ϕ t σ t ) = r(t, ϕ t )V t ϕ t ( θ σ)(t, ϕ t ). Taxes on the profits on risky investments. g(t, V t, ϕ t σ t ) = (r t V t + ϕ t θ t σ t ) + ρϕ + t. Here, ρ ]0, 1[ represents an instantaneous tax coefficient on the gains made from the investments in the risky assets.

31 Imperfect financial markets More generally, when there are market imperfections captured by the nonlinearity of the wealth process dv t = g(t, V t, σ t ϕ t )dt σ t ϕ t dw t ; with driver g(t, y, z) uniformly Lipschitz with respect to y and z, the theory of BSDEs implies that given a claim ξ L 2 (F T ), there exists a unique solution (Y, ϕ) with Y and ϕ in H 2 (R) (i.e. predictable and square integrable) such that dy t = g(t, Y t, σ t ϕ t )dt σ t ϕ t dw t ; Y T = ξ. Y t is the price of the option at time t and ϕ t is the hedging portfolio.

32 Nonlinear pricing system E g The operator E g : (ξ, T ) Y (ξ, T ), where (Y (ξ, T ), Z(ξ, T )) is solution of the associated BSDE with coefficients (ξ, g) and terminal time T was first introduced in El-Karoui-Quenez 96 and called nonlinear pricing system, and later called g-conditional expectation in Peng This notion can be extended to the case where T is replaced by a stopping time τ T 0 and ξ by a random variable η L 2 (F τ ). Definition The g-conditional expectation E g is defined for all stopping time τ T 0 and all η L 2 (F τ ) by E g t,τ (η) := Y t (η, τ); 0 t T, where (Y (η, τ), Z(η, τ)) is the solution of the BSDE associated with driver g, terminal time τ and terminal condition η.

33 In the case of a perfect market, g(t, y, z) = r t y θ t z (linear) and the pricing system is linear, i.e. α, β R, E g t,t (αξ1 + βξ 2 ) = αe g t,t (ξ1 ) + βe g t,t (ξ2 )

34 Properties of the nonlinear pricing system (or g-evaluation) The price system is increasing: if ξ 1 ξ 2 a.s. then E t,t (ξ 1 ) E t,t (ξ 2 ) a.s. (By comparison thm) if g(t, 0, 0) = 0 a.s. on [0, T ] Ω, then E t,t (0) = 0 t. (i.e. the price of the null option is 0) (By uniqueness of the solution: (0, 0) is solution and it is the only one) Moreover the price system is positive, i.e. if ξ 0 a.s. then E t,t (ξ) 0. (By comparison thm) If g(t, 0, 0) = 0 a.s. then the pricing system satifies the non arbitrage property, i.e. if for t, A F t given, if if ξ 1 ξ 2 a.s. on A and Et,T (ξ 1 ) = E t,t (ξ 2 ) a.s. on A F t Then ξ 1 = ξ 2 a.s. on A Proof: Apply strict comparison thm with g 1 = g 2 = g.

35 if g is convex w.r.t. y, z, then the price system is convex, i.e. α [0, 1], ξ 1, ξ 2 L 2 (F T ), E t,t (αξ 1 + (1 α)ξ 2 ) αe t,t (ξ 1 ) + (1 α)e t,t (ξ 2 ) (By comparison thm)

36 The price system is time-consistent, i.e. if S T, then E t,t (ξ) = E t,s (E S,T (ξ)) 0 t S This corresponds to the flow property of BSDEs (also holds if S is a stopping time T ). In other words, s [0, T ], ξ L 2 (F s ), the g-evaluation E g t,s(ξ) is an E g -martingale. A continuous adapted process X t in S 2 is said to be an E-martingale if E σ,τ (X τ ) = X σ a.s. on σ τ, for all σ, τ T 0.

37 A brief introduction to risk measures Static framework We assume that uncertainty is described by a measurable space (Ω, F) of possible scenarios, and that risky positions belong to a linear space X. Let A X : Subset of acceptable positions. We require that A contains all constants and that X A whenever X Y for some Y A. We define the monetary risk measure by ρ(x ) : inf{m R X + m A} The functional ρ satisfies: 1. ρ(x + m) = ρ(x ) m for all X X and all constants m. (Translation invariance) 2. If X 1 X 2 then ρ(x 1 ) ρ(x 2 ). (Monotonicity) Interpretation: ρ(x ) is the amount to be added to X to make it acceptable, since by translation invariance, ρ(x + ρ(x )) = 0.

38 Example: Value at Risk (VaR) Suppose that a probability measure is given on the set Ω of all scenarios. A position is acceptable for the VaR if the probability of loss P[X < 0] is below some level β: VaR(X ) := inf{m R P[X + m < 0] β} It is thus a β-quantile of the distribution of X under P. VaR is widely used in practice but it is not convex; however convexity is a good property for a risk measure since it means that diversification should not increase the risk.

39 This has motivated an axiomatic approach for a general theory of monetary risk measures called coherent (A is a cone), Arzner et al. 99 then extended to the convex case (A convex) For all λ [0, 1] and all X 1, X 2 X. ρ(λx 1 + (1 λ)x 2 ) λρ(x 1 ) + (1 λ)ρ(x 2 ) (Föllmer & Schied 02, Frittelli & Rosazza-Gianin 02)

40 Example: the entropic risk measure ρ(x ) = 1 γ ln E P[exp( γx )] = sup {E Q [ X ] 1 H(Q P)} Q Q γ where H(Q P) is the relative entropy of Q with respect to P defined by: { E Q [ln dq H(Q P) = dp ] if Q admits a density w.r.t. P + otherwise

41 Dual Representation: Convex dualiy theory implies that convex risk measures are typically of the form: ρ(x ) = sup {E Q [ X ] α(q)} Q Q where Q is a set of probability measures and α is a penalty fonction which can take the value +. The penalty term vanishes in the special case of coherent risk measures. Interpretation in terms of model uncertainty : Worst case

42 Dynamic risk measures and relations to BSDEs Extension to the dynamic case. Additional axioms are required to get time-consistency ; Natural link with BSDEs. In the Brownian case, cf e.g. P. Barrieu - N. El Karoui 2004). in the case with jumps, see M.C. Quenez, A.S. SPA 13

43 Dynamic risk measures associated to BSDEs Let g be a standard Lipschitz driver. For each T > 0 and ξ L 2 (F T ), set ρ t (ξ, T ) := E g t,t (ξ), 0 t T, If T represents a given maturity and ξ a financial position at time T, then ρ t (ξ, T ) is interpreted as the risk measure of ξ at time t. The functional ρ : (ξ, T ) ρ (ξ, T ) defines then a dynamic risk measure induced by the BSDE with driver g.

44 ρ t (, T ) is monotonous (nonincreasing with respect to financial position ξ:) if ξ 1 ξ 2 then ρ t (ξ 1, T ) ρ t (ξ 2, T ) a.s. ρ satisfies the nonarbitrage property: If ξ 1 ξ 2 and if ρ t (ξ 1, T ) = ρ t (ξ 2, T ) a.s. on an event A F t, then ξ 1 = ξ 2 a.s. on A. If g is concave w.r.t. y, z, then ρ t (, T ) is convex. If g only depends on z (not on y), then ρ is invariant by translation (monetary risk measure), i.e. ρ t (ξ + ξ, T ) = ρ t (ξ, T ) ξ, ξ L 2 (F T ), ξ L 2 (F t ). Example: the dynamic entropic risk measure ρ t (ξ, T ) := 1 γ ln E[exp( γξ) F t] is associated to the BSDE with driver g(t, y, z) := 1 2 γz2.

45 Recursive utility (Duffie & Epstein (1992), Epstein & Zin (1989), Kreps & Porteus (1978)). Let g(t, y, c) be an F t -adapted process. Assume that c g(t, y, c) is concave for all t, y. The recursive utility process of a given consumption process c( ) 0 is defined as the solution Y of the equation [ T ] (8) Y t = E g(s, Y s, c s ))ds F t t ; 0 t T. This equation is equivalent to the following BSDE in (Y, Z): { dy t = g(t, Y t, c t )dt Z t db t (9) Y T = 0.

46 The Markovian case - Links of BSDEs with PDEs Let (t, x) [0, T ] R p. Let (Ss t,x, t s T ) the solution of the forward SDE: { ds s = b(s, S s )ds + σ(s, S s )dw s, t s T S t = x where b : [0, T ] R p R p and σ : [0, T ] R p R p d are Borelian Lipschitz w.r.t. x. Let (Ys t,x, Zs t,x ) be the solution of the BSDE { dy s = g(t, Ss t,x, Y s, Z s )dt Zs dw s t s T (10) Y T = Ψ(S t,x T ), where Ψ : R p R, g is Lipschitz w.r.t. (y, z) with constant C (uniformly wrt (t, s)), and b, σ, g, Ψ have linear growth. Y t,x t is a deterministic function of (t, x), denoted by u(t, x).

47 t,x t,x Theorem (Generalized Feyman-Kac representation) Let v a function of class C 1,2 such that v(t, x) + σ(t, x) x v(t, x) C(1 + x ), solution of the PDE { t v(t, x) + Lv(t, x) + g(t, x, v(t, x), σ(t, x) x v(t, x)) = 0 v(t, x) = Ψ(x) where x v denotes the gradient of v and L t,x is the infinitesimal generator of S s : L t,x := i b i (t, x) xi + i,j a i,j (t, x) 2 x i x j with a ij = 1 2 (σσ ) ij. Then v(t, x) = Y t,x t

48 Moreover Z t,x s Y t,x s = σ(s, S t,x s = v(s, Ss t,x ) s t ) x v(s, S t,x s ) s t. Proof. Apply Itô to v(s, S t,x s ). We get : dv(s, Ss t,x ) = { t v(s, S s ) + Lv(s, S s )}ds + x v(s, S s ) σ(s, S s )dw s Since v satisfies the PDE, the term in ds above is equal to g(t, S s, v(s, S s ), σ(s, S s ) x v(s, S s )). Moreover v(t, S T ) = Ψ(S T ) (terminal condition). So (v(s, Ss t,x ), σ (s, Ss t,x ) x v(s, Ss t,x )) is solution of the BSDE (10). By uniqueness it is equal a.s. to (Y s, Z s ). reference: Pardoux-Peng: BSDEs and quasi-linear PDEs 1992

49 This is a verification theorem. We have also a theorem which states that the function u(t, x) := Yt t,x is a viscosity solution of the PDE. (El Karoui-Peng-Quenez 97).

50 III. Nonlinear pricing of American Contingent claims Recall that an American option gives the right to the owner to exercise the option at any time before maturity. Consider the imperfect market M g, when under nonlinear constraints, the strategy (wealth V, portfolio ϕ) satisfies: dv t = g(t, V t, σ t ϕ t )dt σ t ϕ t dw t. where g is nonlinear (Lipschitz).

51 Consider an American option associated with horizon T > 0 and a payoff given by a continuous process (ξ t, 0 t T ). At time 0, it consists in the selection of a stopping time ν T leading to the payment of the payoff ξ ν from the seller to the buyer. We have seen that for an European option with payoff ξ and maturity T, its initial hedging price is given by E g 0,T (ξ). A natural price for the American option is the quantity (11) sup E g 0,ν (ξ ν), ν T which we call the g-value of the American option (first introduced in NEK-Quenez 96 and called fair value ).

52 Main results: 1. The g-value sup ν T E g 0,ν (ξ ν) of the American option with payoff ξ coincides with the solution of an associated reflected BSDE with barrier ξ. 2. It is equal to the superhedging price u 0 of the American option, (i.e. the minimal initial capital which enables the seller to invest in a portfolio which will cover his liability to pay to the buyer no matter the exercise time of the buyer). Mathematical tools: theory of (nonlinear) reflected BSDEs.

53 Reflected BSDEs Reflected BSDEs have been introduced by El Karoui et al. (1997). The solution of such a BSDE is forced to stay above a given process, called obstacle (or barrier). An increasing process is introduced to push the solution upwards in a minimal way so that it remains above the obstacle.

54 Reflected BSDEs Given: a standard driver g(t, y, z) an obstacle (ξ t ),0 t T, continuous on [0, T [, adapted, belonging to S 2, i.e. E(sup t ξ t 2 ) < and satisfying lim t T,t<T ξ t ξ T. A solution of the reflected BSDE associated with driver g and obstacle ξ. consists in a triplet (Y, Z, K) in H 2 (R) H 2 (R n ) S 2 (R) such that (12) dy t = g(t, Y t, Z t )dt + dk t Z t dw t ; Y T = ξ T Y t ξ t, 0 t T a.s., K is a nondecreasing continuous predictable process with K 0 = 0 and T 0 (Y t ξ t )dk t = 0 ( i.e. t K t increases only on{t, Y t = ξ t }) The process K is the minimal push which allows the solution to stay above the obstacle.

55 Reflected BSDEs The special case of a driver process Suppose g does not depend on y, z, k, that is g(ω, t, y, z) = g(ω, t), where g is a process in H 2. Theorem (Link with classical optimal stopping problems) If (Y, Z, K) satisfies the reflected BSDE associated with driver g = g(ω, t) and obstacle ξ, then Y t = ess sup τ T t E[ξ τ + τ t g(s)ds F t ] a.s. and the supremum is attained at D t = inf{s t; Y s = ξ s }. Proof. on blackboard

56 Theorem (Existence and Uniqueness) The reflected BSDE associated with driver g = g(ω, t) and obstacle ξ admits a unique solution (Y, Z, K). Proof. Uniqueness. It comes from the previous characterization theorem. Existence. The proof is based on Doob-Meyer decomposition for super martingales, the theorem of martingale representation and the optimal stopping theory. (Proof on the blackboard).

57 Reflected BSDEs The general case of a Lipschitz driver process Theorem (Existence and Uniqueness) Let g(t, y, z) and ξ. be a couple of standard parameters. Then, the associated reflected BSDE admits a unique solution (Y, Z, K).

58 Proof. Define a mapping Φ as follows. Given (U, V ) H 2 (R) H 2 (R d ), let (Y, Z) = Φ(U, V ) where (Y, Z, K), with t t K t = Y t + Y 0 g(s, U s, V s ) ds + Zs dw s, 0 t T, 0 0 is the solution of the Reflected BSDE associated with driver g(s) = g(s, U s, V s ) and obstacle ξ (which is well defined from previous proposition) Prove that Φ is a contraction in the Banach space H 2 (R) H 2 (R d ) equipped with the norm β (with β well chosen) and thus admits an unique fixed point, which corresponds to the solution of RBSDE (12). Detailed proof on the blackboard.

59 Theorem (Characterization) Let (Y, Z, K) be the solution of the reflected BSDE. Then and the supremum is attained at Y t = ess sup τ T t,t E g t,τ (ξ τ ) D t := inf{s t; Y s = ξ s }, that is Y t = E t,dt (ξ Dt ).

60 Proof. Let D t := inf{s t; Y s = ξ s }. On [t, D t ], the solution Y t > ξ t, so (K t ) remains constant and (Y, Z) the solution of the BSDE with driver g, terminal time D t and terminal condition ξ Dt (= Y Dt by continuity of Y and ξ). In other terms, Y t = E t,dt (ξ Dt ). It remains to prove that for each τ T t,t, we have: Y t E t,τ (ξ τ ). Set X t := E t,τ (ξ τ ); It satisfies the BSDE dx s = g(s, X s, π s)ds π s dw s, t s τ ; X τ = ξ τ Note that on [t, τ], the pair (Y, Z) satisfies dy s = g(s, Y s, Z s )ds + dk s Z s dw s ; Y τ = Y τ. In other terms, (Y, Z) is a solution of the BSDE associated with terminal time τ, terminal condition Y τ and driver g(s,, ) + dks ds. But we have : g(s, y, z)ds + dk s g(s, y, z) and Y τ ξ τ. The comparison thm for BSDEs implies Y t E t,τ (ξ τ ).

61 Theorem (Comparison theorem for reflected BSDEs) Let (f 1, ξ 1 ), (f 2, ξ 2 ) be standard coefficients and (Y 1, Z 1, K 1 ), (Y 2, Z 2, K 2 ) be the solutions of the associated reflected BSDEs. Assume ξ 1 t ξ 2 t, 0 T, a.s. f 1 (t, y, z f 2 (t, y, z) Then Y 1 Y 2, P-a.s. t, y, z Proof. By the comparison thm for BSDEs, for all stopping time τ t, we have: Et,τ 1 (ξτ 1 ) Et,τ 2 (ξτ 2 ). Taking the supremum over τ, we get: Y 1 t Y 2 t.

62 Theorem (Optimality criterium) We have the following equivalence: (i) τ is t-optimal, i.e. Y t = ess sup τ T t,t E g t,τ (ξ τ ) = E t,τ (ξ τ ) (ii) dk s = 0, t s τ and Y τ = ξ τ. In other terms (Y s, t s τ ) is solution of the BSDE associated with the terminal condition ξ τ. Proof. (ii) (i): clear (i) (ii) application of the strict comparison thm (detailed proof on the blackboard)

63 Application to American option pricing Under nonlinear constraints, the wealth is supposed to satisfy: dv t = g(t, V t, σ t π t )dt σ t π t dw t. If the exercise time τ T t,t is fixed, then the American option is a European option with maturity τ and payoff ξ τ. The price of this European option is given by E t,τ (ξ τ ). Definition The g-value of the American option at time t is defined by the F t -measurable random variable: Y t = ess sup Et,τ g (ξ τ ) a.s. τ T t,t Using the previous theory we have that Y is the solution of the reflected BSDE assciated to driver g and obstacle ξ. More precisely,

64 Proposition: Z in H(R d ), K continuous, increasing process, such that dy t = g(t, Y t, Z t )dt + dk t Zt dw t Y T = ξ T Y t ξ t 0 t T a.s. T 0 (Y s ξ s )dk s = 0 Moreover, D t := inf{s t; Y s = ξ s } is optimal i.e. Y t = E t,dt (ξ Dt ).

65 Theorem The g-value Y 0 := sup ν T E g 0,ν (ξ ν) of the American option with payoff ξ satisfies Y 0 = u 0 where u 0 is the superhedging price of the American option defined as the lowest price which allows the seller to be superhedged, i.e. where u 0 = inf H H = {x R, ϕ H 2 s.t. V x,ϕ t ξ t a.s. 0 t T } Proof: 1.Y 0 u 0. Let x H. Let us show that Y 0 x. For all τ T 0, we have: ξ τ Vτ x,ϕ. So by comparison thm, E 0,τ (ξ τ ) E 0,τ (Vτ x,ϕ ) = V x,ϕ 0 = x. Taking supremum over τ, we get Y 0 = sup τ T0 E 0,τ (ξ τ ) x.

66 2. u 0 Y 0 Consider V Y 0,ϕ t the wealth associated to portfolio ϕ t = (σt ) 1 Z t and initial value Y 0. It satisfies: and we have V t = Y 0 Y t = Y 0 t 0 t 0 g(s, Y s, Z s )ds + t 0 Z s dw s t g(s, Y s, Z s )ds + Zs dw s K t. 0 We can prove by using a comparison argument for forward differential equations, that Now V Y 0,ϕ t Y t. Y t ξ t, So V Y 0,ϕ t ξ t, for all 0 t T. So Y 0 belongs to H.

67 Relation with PDEs (Variational inequalities) We now focus on the Markovian case. Let b and σ be continuous mappings, globally Lipschitz. For each (t, x) [0, T ] R, let {Xs t,x, t s T } the solution of s s Xs t,x = x + b(xr t,x )dr + σ(xr t,x )dw r t t We consider the Reflected BSDE associated driver f (s, Xs t,x,.) and obstacle ξ t,x of the following form: { ξ t,x s ξ t,x T := h(s, Xs t,x ), s < T t,x := g(xt ) where g C(R) has at most polynomial growth at infinity, h : [0, T ] R R is jointly continuous in t and x and there exist p N and C > 0, such that h(t, x) C(1 + x p ), t, x and h(t, x) g(x), x.

68 For each (t, x) [0, T ] R, there exists a unique triple (Y t,x, Z t,x, K t,x ) S 2 H 2 S 2 of adapted processes, which solves the Reflected BSDE: (13) T = g(x t,x ) + Y t,x s T T s f (r, X t,x r, Y t,x r, Zr t,x )dr + K t,x T K s t,x Zr t,x dw r s Ys t,x ξs t,x, 0 s T a.s., K t,x is a nondecreasing, continuous predictable process with T t (Y t,x s K t,x t = 0 and such that ξs t,x )dks t,x = 0 a.s.

69 Theorem The function u, defined by u(t, x) := Y t,x t, t [0, T ], x R is a viscosity solution (i.e. both a viscosity sub- and supersolution) of the following PDE with obstacle (14) min(u(t, x) h(t, x), u u (t, x) Lu(t, x) f (t, x, u(t, x), (σ )(t, x)) = 0, (t, x) [0, T ) t x u(t, x) = g(x), x R where Lφ(x) := 1 2 σ2 (x) 2 φ (x) + b(x) φ x 2 x (x).

70 Definition A continuous function u is said to be a viscosity subsolution of (14) if u(t, x) g(x), x R, and if for any point (t 0, x 0 ) (0, T ) R and for any φ C 1,2 ([0, T ] R) such that φ(t 0, x 0 ) = u(t 0, x 0 ) and φ u attains its minimum at (t 0, x 0 ), we have min(u(t 0, x 0 ) h(t 0, x 0 ), φ t (t 0, x 0 ) Lφ(t 0, x 0 ) f (t 0, x 0, u(t 0, x 0 ), (σ φ x )(t, x))(t 0, x 0 )) 0. In other words, if u(t 0, x 0 ) > h(t 0, x 0 ), φ t (t 0, x 0 ) Lφ(t 0, x 0 ) f (t 0, x 0, u(t 0, x 0 ), (σ φ x )(t, x))(t 0, x 0 )) 0. A continuous function u is said to be a viscosity supersolution of (14) if u(t, x) g(x), x R, and if for any point (t 0, x 0 ) (0, T ) R and for any φ C 1,2 ([0, T ] R) such that φ(t 0, x 0 ) = u(t 0, x 0 ) and φ u attains its maximum at (t 0, x 0 ), we have min(u(t 0, x 0 ) h(t 0, x 0 ), t φ(t 0, x 0 ) Lφ(t 0, x 0 ) f (t 0, x 0, u(t 0, x 0 ), (σ φ x )(t 0, x 0 )) 0.

71 Under some additional hypothesis on f, there exists a unique solution of the obstacle problem (27) in the class of continuous functions with polynomial growth.

72 IV. Nonlinear pricing of Game options Definition Game options are derivative contracts that can be terminated by both counterparties at any time before maturity T (introduced by Kifer 2000). extend the setup of American options by allowing the seller to cancel the contract. More precisely If the buyer exercises at time τ, he gets ξ τ from the seller, If the seller cancels at σ before τ, then he pays ζ σ to the buyer. We assume ζ t ξ t 0 for all t and this difference is interpreted as a penalty for the seller for cancellation of the contract.

73 Game options Superhedging The game option consists for the seller to select a cancellation time σ T and for the buyer to choose an exercise time τ T, so that the seller pays to the buyer at time τ σ the amount I (τ, σ) = ξ τ 1 τ σ + ζ σ 1 σ<τ. Definition (Superhedging price) It is the minimal initial wealth which enables the seller to choose a cancellation time σ and a portfolio which cover his liability to pay to the buyer up to σ no matter the exercise time chosen by the buyer (Kifer). Definition: For each initial wealth x, a super-hedge against the game option is a pair (σ, ϕ) of a stopping time σ and a portfolio ϕ s.t. Vt σ x,ϕ I (t, σ), 0 t T a.s. (or equivalently Vt x,ϕ ξ t, 0 t σ a.s. and Vσ x,ϕ ζ σ a.s.).

74 Let A(x) be the set of all super-hedges associated with initial wealth x. The superhedging price u 0 is then defined as the infimum of initial wealths such that there exists a hedge against the game option, that is u 0 = inf{x R, (σ, ϕ) A(x)}.

75 In the case of perfect markets, Kifer shows in the CRR discrete-time model and in the Black-Scholes model that u 0 = value function of a Dynkin game: u 0 = sup inf E Q[ ξ τ 1 τ σ + ζ σ 1 τ>σ ] = inf sup E Q [ ξ τ 1 τ σ + ζ σ 1 τ>σ ], τ σ σ τ where ξ t and ζ t are the discounted values of ξ t and ζ t. Here, E Q denotes the expectation under the unique martingale probability measure Q of the market model.

76 Nonlinear pricing of Game options Consider the imperfect market M g, when under nonlinear constraints, the wealth satisfies: dv t = g(t, V t, σ t ϕ t )dt σ t ϕ t dw t. where g is nonlinear (Lipschitz). Suppose that the seller has chosen his cancellation time σ. Then, the game option reduces to an American option with payoff I (., σ), whose superhedging initial price is given by sup τ T E g 0,τ σ [I (τ, σ)]. It is thus natural to define the g-value of the game option as inf sup E g 0,τ σ (I (τ, σ)). σ T τ T

77 Questions :? inf sup E g 0,τ σ (I (τ, σ)) = sup inf σ T τ T τ T σ T Eg 0,τ σ (I (τ, σ)) In other words, does this generalized Dynkin game has a value?? Links with superhedging price of game options The classical approach based on a change of probability measure and an actualization procedure (see Hamadène) cannot be adapted to the nonlinear case. Mathematical tools: theory of (nonlinear) doubly reflected BSDEs and links with generalized Dynkin games. Dumitrescu-Quenez-Sulem 2014

78 Doubly reflected BSDEs We define Doubly reflected BSDEs (DRBSDEs), for which the solution is constrained to stay between two given processes called barriers ξ ζ. Two nondecreasing processes A and A are introduced in order to push the solution Y above ξ and below ζ in a minimal way. This minimality property of A and A is ensured by the Skorohod conditions.

79 Definition (Doubly reflected BSDEs) Let T > 0 be a fixed terminal time and g be a Lipschitz driver. Let ξ and ζ be two adapted continuous on [t,t) processes with ζ T = ξ T a.s., ξ S 2, ζ S 2, ξ t ζ t, t [0, T ] a.s. A process (Y, Z, A, A ) in S 2 H 2 A 2 A 2 is a solution of the doubly reflected BSDE associated with driver g and barriers ξ, ζ if dy t = g(t, Y t, Z t )dt + da t da t Z t dw t ; 0 t T Y T = ξ T, ξ t Y t ζ t, da t da t T 0 t T a.s., 0 (Y t ξ t )da t = 0 and T 0 (ζ t Y t )da t = 0 a.s. (Skorohod conditio A 2 = {nondecreasing continuous adapted processes A with A 0 = 0 and E(A 2 T ) < }

80 Classical Dynkin games Links with linear doubly reflected BSDEs We start by the case of the driver g = 0 and ξ T = 0 a.s. Then the associated doubly reflected BSDE is related to a classical Dynkin game problem with payoff: I (τ, σ) = ξ τ 1 {τ σ} + ζ σ 1 {σ<τ}.

81 Definition (Classical Dynkin game) For any S T 0, 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[I (τ, σ) F S ] V (S) := ess sup τ T S ess inf σ T S E[I (τ, σ) F S ]. We clearly have the inequality V (S) V (S) a.s. We say that the Dynkin game is fair (or there exists a value function) at time S if V (S) = V (S) a.s.

82 Definition (S-saddle point) Let S T 0. A pair (τ, σ ) TS 2 all (τ, σ) TS 2, we have is called an S-saddle point if for E[I (τ, σ ) F S ] E[I (τ, σ ) F S ] E[I (τ, σ) F S ] a.s.

83 When it exists, the value function of the Dynkin game can be written as the difference of 2 supermartingales J and J solutions of a coupled optimal stopping problems: V (S) = V (S) = J S J S, where for all θ T 0, J θ = ess sup τ T θ E [ J τ + ξ τ F θ ] and J θ = ess sup σ T θ E [J σ ζ σ F θ ] a.s. J and J are finite (and in S 2 ) iff Mokobodzki s condition holds, that is if there exist two nonnegative continuous supermartingales H and H in S 2 such that: (15) ξ t H t H t ζ t 0 t T a.s. (which is for example satisfied when ξ and ζ are semimartingales)

84 is an S-saddle point. Theorem Under this condition, the doubly reflected BSDE dy t = da t da t Z t dw t ; Y T = 0 ξ t Y t ζ t, 0 t T a.s., da t da t T 0 (Y t ξ t )da t = 0 and T 0 (ζ t Y t )da t = 0 admits a unique solution (Y, Z, A, A ) in S 2 H 2 (A 2 ) 2. S T 0, Y S is the common value function of the Dynkin game associated with the gain I (τ, σ) = ξ τ 1 {τ σ} + ζ σ 1 {σ<τ}., i.e. (16) Y S = V (S) = V (S) a.s. Moreover, S T 0, the pair of stopping times (τ s, σ s ) defined by (17) σ S := inf{t S, Y t = ζ t }; τ S := inf{t S, Y t = ξ t }

85 Sketch of proof : Suppose that J, J S 2. Define (18) Y t := J t J t; 0 t T. Let us show that there exist (Z, A, A ) H 2 A 2 A 2 such that (Y, Z, A, A ) is a solution of DRBSDE (15) associated with driver process g = 0. By assumption, J and J are square integrable supermartingales. The process Y is thus well defined. We have J T = J T a.s. Hence, Y T = ξ T a.s. By the Doob-Meyer decomposition, there exist two square integrable martingales M and M and two processes A and A A 2 such that: (19) dj t = dm t da t ; dj t = dm t da t. Set M t := M t M t. By (19), (18), we derive dy t = dm t dα t, with α := A A. By the martingale representation theorem, there exist Z H 2 such that dm t = Z t dw t. Hence, dy t = dα t Z t dw t.

86 By the optimal stopping theory, the process A increases only when the value function J is equal to the corresponding reward J + ξ. Now, {J t = J t + ξ} = {Y t = ξ t }. Hence, T 0 (Y t ξ t )da t = 0 a.s. Similarly the process A satisfies T 0 (Y t ζ t )da t = 0 a.s. We have Y σ S = ζ σ S and Y τ S = ξ τ S a.s. On [S, τs [, we have Y t > ξ t a.s., so A is constant on [S, τs ]. Similarly, A is constant on [S, σs ] a.s. The process (Y t + t 0 g(s)ds, S t τ S σ S ) is thus a martingale. Hence, Y S = E[I S (τs, σ S ) F S] a.s. Similarly we can show that τ, σ T S, E[I S (τ, σs ) F S] Y S and Y S E[I S (τs, σ) F S] a.s., So (τs, σ S ) is an S-saddle point.

87 This result can easily be extended to the case when g = (g t ) in H 2 is a driver process and ξ T 0, by doing the change of variables: T T ξ t g := ξ t E[ξ T + g(s)ds F t ], ζ t g := ζ t E[ζ T + g(s)ds F t ], t t The associated payoff of the Dynkin game is then I S (τ, σ) = σ τ S g(u)du + ξ τ 1 {τ σ} + ζ σ 1 {σ<τ}.

88 Generalized Dynkin games and links with nonlinear DRBSDEs From now on, we are given a standard Lipschitz driver g and we suppose that Mokobodzki s condition is satisfied. Theorem (Existence and uniqueness for DRBSDEs) Then, DRBSDE (15) admits a unique solution (Y, Z, A, A ) S 2 H 2 (A 2 ) 2. Proof. based on a contraction argument (similar to the case of BSDEs and Reflected BSDEs)

89 Generalized Dynkin games We introduce a generalized Dynkin game expressed in terms of E g -expectations. Let g(t, y, z) be a given a Lipschitz driver. For each pair (τ, σ) of stopping times valued in [0, T ], the criterium is given by ( ) ξτ 1 {τ σ} + ζ σ 1 {σ<τ} E g 0,τ σ with ξ, ζ two continuous adapted processes satisfying ξ ζ. When the driver g does not depend on the solution, that is, when it is given by a process (g(t)), the criterium coincides with ( τ σ ) E g s ds + ξ τ 1 {τ σ} + ζ σ 1 {σ<τ}. 0

90 For each stopping time S T 0, the upper and lower value functions at time S are defined respectively by (20) V (S) := ess inf σ T S ess sup τ T S E S,τ σ (I (τ, σ)); (21) V (S) := ess sup τ T S ess inf σ T S E S,τ σ (I (τ, σ)). with I (τ, σ) := ξ τ 1 τ σ + ζ σ 1 σ<τ. We clearly have the inequality V (S) V (S) a.s. By definition, we say that the game is fair (or there exists a value function) at time S if V (S) = V (S) a.s.

91 Definition (S-saddle) Let S T 0. A pair (τ, σ ) TS 2 is called an S-saddle point for the generalized Dynkin game if for each (τ, σ) TS 2 we have E g S,τ σ (I (τ, σ )) E g S,τ σ (I (τ, σ )) E g S,τ σ (I (τ, σ)) a.s. Definition Let Y be an RCLL process in S 2. The process Y is said to be a E g -supermartingale (resp E g -submartingale), if E g σ,τ (Y τ ) Y σ (resp. E g σ,τ (Y τ ) Y σ ) a.s. on σ τ, for all σ, τ T 0.

92 We first provide a sufficient condition for the existence of an S-saddle point and for the characterization of the common value function as the solution of the DRBSDE. Lemma Let (Y, Z, A, A ) be the solution of the DRBSDE. Let S T 0. Let (ˆτ, ˆσ) T S. Suppose that (Y t, S t ˆτ) is a E-submartingale and that (Y t, S t ˆσ) is a E-supermartingale with Yˆτ = ξˆτ and Yˆσ = ζˆσ a.s. The pair (ˆτ, ˆσ) is then an S-saddle point for the generalized Dynkin game (20)-(21) and Y S = V (S) = V (S) a.s.

93 Proof. Since the process (Y t, S t ˆτ ˆσ) is a E-martingale and since Yˆτ = ξˆτ and Yˆσ = ζˆσ a.s., we have Y S = E g S,ˆτ ˆσ (Yˆτ ˆσ) = E g S,ˆτ ˆσ (ξˆτ 1ˆτ ˆσ +ζˆσ 1ˆσ<ˆτ ) = E g S,ˆτ ˆσ (I (ˆτ, ˆσ)) a.s. Let τ T S. We want to show that for each τ T S (22) Y S E S,τ ˆσ (I (τ, ˆσ)) a.s. Since the process (Y t, S t τ ˆσ) is a E-supermartingale, we get (23) Y S E S,τ ˆσ (Y τ ˆσ ) a.s. Since Y ξ and Yˆσ = ζˆσ a.s., we also have Y τ ˆσ = Y τ 1 τ ˆσ + Yˆσ 1ˆσ<τ ξ τ 1 τ ˆσ + ζˆσ 1ˆσ<τ = I (τ, ˆσ) a.s. By inequality (23) and the monotonicity property of E, we derive inequality (22). Similarly, one can show that for each σ T S, we have: Y S E S,ˆτ σ (I (ˆτ, σ)) a.s. The pair (ˆτ, ˆσ) is thus an S-saddle point and Y S = V (S) = V (S) a.s.

94 Theorem (Existence of S-saddle points) Let (Y, Z, A, A ) be the solution of the DRBSDE (15). For each S T 0, let τ S := inf{t S, Y t = ξ t }; σ S := inf{t S, Y t = ζ t }. Then, for each S T 0, the pair of stopping times (τs, σ S ) is a S-saddle point for the generalized Dynkin game and Y S = V (S) = V (S) a.s. Moreover, Y σ S = ζ σ S, Y τ S = ξ τ S, A τ S = A S and A σs = A S a.s.

95 Proof. Let S T 0. Since Y and ξ are continuous processes, we have Y σ S = ζ σ S and Y τ S = ξ τ S a.s. By definition of τs, Y t > ξ t (ω) for each t [S, τs [. Hence, since Y is solution of the DRBSDE, the process A is constant on [S, τs ] a.s. because A is continuous. Hence, A τs = A S a.s. Similarly, A σs = A S a.s. By previous Lemma, (τs, σ S ) is an S-saddle point and Y S = V (S) = V (S) a.s.

96 Application to nonlinear pricing of game options We consider now a game option associated with continuous payoffs ξ and ζ in S 2 with ζ T = ξ T ; ξ t ζ t, 0 t T a.s. satisfying Mokobodski s condition (which is for example satisfied when ξ and ζ are semimartingales) The game option consists for the seller to select a cancellation time σ T and for the buyer an exercise time τ T, so that the seller pays to the buyer at time τ σ the payoff I (τ, σ) := ξ τ 1 τ σ + ζ σ 1 σ<τ. A natural price for the seller of the game option, is the g-value given by (24) Y (0) := inf sup E g 0,τ σ [I (τ, σ)] σ T τ T

97 Under the assumptions above, this generalized Dynkin game is fair, i.e inf sup E g 0,τ σ [I (τ, σ)] = sup inf σ T τ T τ T σ T Eg 0,τ σ [I (τ, σ)] and its value Y (0) equals Y 0, where (Y, Z, A, A ) is the unique solution in S 2 L 2 (W ) A 2 A 2 of the Doubly Reflected BSDE associated to driver g and obstacles ξ, ζ.

98 Definition: Superhedging price u 0 of the game option := minimal initial wealth which enables the seller to choose a cancellation time σ and a portfolio which will cover his liability to pay to the buyer up to σ no matter what exercise time the buyer chooses (Kifer). Definition: For each initial wealth x, a super-hedge against the game option is a pair (σ, ϕ) of a stopping time σ and a portfolio ϕ s.t. Vt σ x,ϕ I (t, σ), 0 t T a.s. (or equivalently Vt x,ϕ ξ t, 0 t σ a.s. and Vσ x,ϕ ζ σ a.s.). Let A(x) be the set of all super-hedges associated with initial wealth x. The superhedging price is then defined as the infimum of initial wealths such that there exists a hedge against the game option, that is u 0 = inf{x R, (σ, ϕ) A(x)}.

99 Theorem superhedging price u 0 = g-value of the game option, that is u 0 = inf sup E g 0,τ σ [I (τ, σ)] = sup inf σ T τ T τ T σ T Eg 0,τ σ [I (τ, σ)]. Let (Y, Z, A, A ) be the solution of the DRBSDE associated with driver g and barriers ξ, ζ. We have Let u 0 = Y 0. σ := inf{t 0, Y t = ζ t } and ϕ t = (σ t ) 1 Z t. The pair (σ, ϕ ) belongs to A(u 0 ), i.e. the cancellation time σ and the strategy ϕ allow the seller of the game option to be super-hedged. The proof is based on links between generalized Dynkin games and DRBSDEs

100 Buyer point of view Definition: (Kifer) A stopping time τ is a rational exercise time for the game option if V Y 0,ϕ t ζ t, 0 t τ a.s. and V Y 0,ϕ τ = ξ τ a.s. Lemma: The stopping time (25) τ := inf{t 0 : Y t = ξ t } is a rational exercise time for the buyer of the game option.

101 The pair (τ, σ ) is a saddle point for the generalized Dynkin game, that is for each (τ, σ) T 2 we have E g 0,τ σ [I (τ, σ )] Y 0 = E g 0,τ σ [I (τ, σ )] E g 0,τ σ [I (τ, σ)]. In this case, τ is optimal for the optimal stopping problem sup τ T E g [I (τ, σ )] (as in the case of an American option) and σ is optimal for inf σ T E g [I (τ, σ)].

102 Relations with variational inequalities (VI) We consider now the Markovian case and study the links between generalized Dynkin games (or equivalently DRBSDEs) and obstacle problems. Let b : R R, σ : R R be continuous, globally Lipschitz. (t, x) [0, T ] R, let (Xs t,x, t s T ) solution of s s Xs t,x = x + b(xr t,x )dr + t t and set X t,x s = x for s t. σ(xr t,x )dw r

103 We consider the DRBSDE associated with obstacles ξ t,x, ζ t,x of the following form: ξ t,x s := h 1 (s, Xs t,x ), ζs t,x := h 2 (s, Xs t,x ), s < T, ξ t,x T = ζt,x T t,x := g(xt ) with g C(R), h 1, h 2 : [0, T ] continuous with respect to t and Lipschitz continuous with respect to x, uniformly in t and g, h 1, h 2 have at most polynomial growth with respect to x. Moreover, we assume that the obstacles ξs t,x and ζs t,x satisfy Mokobodski s condition, which holds e.g; if h 1 and h 2 are C 1,2. Let f : [0, T ] R 3 R be continuous in t uniformly with respect to x, y, z, uniformly Lipschitz with respect to x, y, z uniformly in t, such that f (t, x, 0, 0, ) at most polynomial growth with respect to x. The driver is defined by f (s, Xs t,x (ω), y, z).

104 For each (t, x) [0, T ] R, there exists an unique solution (Y t,x, Z t,x, A t,x, A t,x ) of the associated DRBSDE. We define: (26) u(t, x) := Y t,x t, t [0, T ], x R. which is a deterministic quantity. u is continuous in (t, x) and has at most polynomial growth at infinity.

105 A solution of the obstacle problem is a function u : [0, T ] R R which satisfies the equality u(t, x) = g(x) and (27) where h 1 (t, x) u(t, x) h 2 (t, x) if u(t, x) < h 2 (t, x) then Hu 0 if h 1 (t, x) < u(t, x) then Hu 0 Lφ(x) := 1 2 σ2 (x) 2 φ (x) + b(x) φ x 2 x (x), Hφ(t, x) := φ φ (t, x) Lφ(t, x) f (t, x, φ(t, x), (σ )(t, x)). t x

106 Definition A continuous function u is said to be a viscosity subsolution (resp. supersolution) of (27) if u(t, x) g(x), (resp ) x R, for all (t, x), h 1 (t, x) u(t, x) h 2 (t, x) for all φ C 1,2 ([0, T ] R) such that φ(t 0, x 0 ) = u(t 0, x 0 ) and φ u attains its minimum (resp. maximum) at (t 0, x 0 ), if u(t 0, x 0 ) > h 1 (t 0, x 0 ) (resp. <), then (Hφ)(t 0, x 0 ) 0 (resp.)

107 Theorem The function u defined by (26) is a viscosity solution (i.e. both a viscosity sub- and supersolution) of the obstacle problem (27). This result provides a probabilistic interpretation of semi linear PDEs with two barriers in terms of game problems.

108 References Classical Dynkin Games in continuous time: Alario-Nazaret, Lepeltier and Marchal, B. (1982). Links with Doubly RBSDEs when the driver g does not depend on y, z: Cvitanic and Karatzas (1996), Hamadène (2002) and Lepeltier (2000) (Hyp: Brownian+ regularity). Generalized Dynkin games and DRBSDEs, Dumitrescu-Quenez-A.S EJP Game options in an imperfect market with default, Dumitrescu-Quenez-A.S

109 Market model with default Setup Let (Ω, G, P) be a probability space, equipped with two stochastic processes: a one-dimensional standard Brownian motion W and a jump process N defined by N t := 1 ϑ t for any t 0, where ϑ is a random variable which represents a default time. We assume that this default can appear at any time that is P(ϑ t) > 0 for any t 0. We denote by G = (G t, t 0)) the complete natural filtration generated by W and N. We suppose that W is a G-Brownian motion.

110 Let Λ be the predictable compensator of the nondecreasing process N (i.e. the unique RCLL increasing predictable process null at 0 such that N t Λ is a martingale). Note that Λ t ϑ is then the predictable compensator of N t ϑ = N t. By uniqueness of the predictable compensator, Λ t ϑ = Λ t. We assume that Λ is absolutely continuous w.r.t. Lebesgue s measure, so that there exists a nonnegative process λ, called the intensity process, such that Λ t = t 0 λ sds. Since Λ t ϑ = Λ t, λ vanishes after ϑ. We denote by M the compensated martingale which satisfies t M t = N t λ s ds. 0

111 Let T > 0. We introduce the following sets : S 2 is the set of G-adapted RCLL processes ϕ such that E[sup 0 t T ϕ t 2 ] < +. A 2 is the set of real-valued non decreasing RCLL predictable processes A with A 0 = 0 and E(A 2 T ) <. H 2 is the set [ of G-predictable processes such that ] Z 2 T := E 0 Z t 2 dt <. is the set [ of G-predictable processes such that ] U 2 λ := E T 0 U t 2 λ t dt <. H 2 λ

112 We recall the martingale representation theorem (see e.g. Jeanblanc-Yor-Chesnay 09): Lemma Any G-local martingale m has the representation t t (28) m t = m 0 + z s dw s + l s dm s, t [0, T ] a.s., 0 0 where z and l are predictable such that the two above stochastic integrals are well defined. If m is a square integrable martingale, then z H 2 and l H 2 λ.