Markovian RIBSDE Approximation Results Numerics Pricing Game Options with Call Protection: Doubly Reflected Intermittent BSDEs and their Approximation J.-F. Chassagneux , S. Crépey , A. Rahal Équipe Analyse et Probabilité Université d’Évry Val d’Essonne 91025 Évry Cedex, France Workshop “Stochastic Control and Finance”, Roscoff, March 23 2010 The research of the authors benefited from the support of Ito33 and of the ‘Chaire Risque de crédit’, Fédération Bancaire Française Chassagneux, Crépey, Rahal RIBSDEs
Markovian RIBSDE Approximation Results Numerics Convertible bond with underlying stock S Coupons from time 0 onwards Terminal payoff at 휁 = 휏 ∧ 휃 1 휁 = 휏< T ℓ ( 휏, S 휏 ) + 1 휗<휏 h ( 휗, S 휗 ) + 1 휁 = T g ( S T ) [ 0 , T ] -valued bond holder put time 휏 and bond issuer call time 휃 Cancelable American claim, or game option Call protections preventing the issuer from calling the bond on certain random time intervals Typically monitored at discrete monitoring times In a possibly very path-dependent way Chassagneux, Crépey, Rahal RIBSDEs
Markovian RIBSDE Approximation Results Numerics Agenda Mathematical issues Doubly reflected backward stochastic differential equations with an intermittent upper barrier, only active on random time intervals (RIBSDE) Related variational inequality approach (VI) Highly-dimensional pricing problems (path dependence) Deterministic pricing schemes ruled out by the curse of dimensionality Simulation methods → Contributions A convergence rate for a discrete time approximation scheme by simulation to an RIBSDE VI approach Practical value of this approach on the benchmark problem of pricing by simulation highly path-dependent convertible bonds A demonstration of the real abilities of simulation/regression numerical schemes in high dimension (up to d = 30 in this work) Chassagneux, Crépey, Rahal RIBSDEs
Markovian RIBSDE Diffusion Set-Up with Marker Process Approximation Results Markovian RIBSDE Numerics Connection with Finance Solution of the RIBSDE Outline Markovian RIBSDE 1 Diffusion Set-Up with Marker Process Markovian RIBSDE Connection with Finance Solution of the RIBSDE Approximation Results 2 BSDE Approach Variational Inequality Approach Numerics 3 Benchmark Model No Call Protection Call Protection Reducible Case General Case Chassagneux, Crépey, Rahal RIBSDEs
Markovian RIBSDE Diffusion Set-Up with Marker Process Approximation Results Markovian RIBSDE Numerics Connection with Finance Solution of the RIBSDE Diffusion Set-Up with Marker Process Diffusion with Lipschitz coefficients in ℝ q dX t = b ( t , X t ) dt + 휎 ( t , X t ) dW t Call protection monitoring times 픗 = { 0 = T 0 < . . . < T N = T } Marker process H keeping track of the path-dependence, in view of ‘markovianizing’ the model ℝ q × 풦 -valued factor process 풳 = ( X , H ) (finite set 풦 ) u = u ( t , x , k ) = u k ( t , x ) 풦 -valued pure jump marker process H supposed to be constant except for deterministic jumps at the T I s H T I = 휅 I ( X T I , H T I − ) Jump functions 휅 k I continuous in x outside ∂ 풪 (constant on 풪 and on c 풪 ) for an open, ‘regular’ domain 풪 ⊆ ℝ q Chassagneux, Crépey, Rahal RIBSDEs
Markovian RIBSDE Diffusion Set-Up with Marker Process Approximation Results Markovian RIBSDE Numerics Connection with Finance Solution of the RIBSDE Call Protection Subset K of 풦 Call forbidden/possible whenever H t ∈ K / / ∈ K 픗 -valued stopping times given as successive times of exit from and entrance to K , so 휗 0 = 0 and then 휗 2 l + 1 = inf { t > 휗 2 l ; H t / ∈ K } ∧ T , 휗 2 l + 2 = inf { t > 휗 2 l + 1 ; H t ∈ K } ∧ T Call forbidden/possible on the ‘even’/‘odd’ intervals [ 휗 l , 휗 l + 1 ) H t ∈ K / / ∈ K Starting from H 0 = k / ∈ K (‘Call at the beginning’) 0 = 휗 0 = 휗 1 < 휗 2 ≤ . . . ≤ 휗 N + 1 = T Call possible on the first non-void time interval [ 휗 1 = 0 = 휗 0 , 휗 2 > 0 ) Starting from H 0 = k ∈ K (‘No Call at the beginning’) 0 = 휗 0 < 휗 1 ≤ . . . ≤ 휗 N + 1 = T Call forbidden on the first non-void time interval [ 휗 0 = 0 , 휗 1 > 0 ) Chassagneux, Crépey, Rahal RIBSDEs
Markovian RIBSDE Diffusion Set-Up with Marker Process Approximation Results Markovian RIBSDE Numerics Connection with Finance Solution of the RIBSDE Markovian RIBSDE Reflected BSDE ( 풮 ) with data f ( t , X t , y , z ) , 휉 = g ( X T ) , ℓ ( t , X t ) , h ( t , X t ) , 휗 ‘Standard Lipschitz and L 2 -integrability assumptions’ (if not for 휗 ) Mokobodski condition Existence of a square-integrable quasimartingale Q between L and U Doubly reflected BSDE with lower barrier L t = ℓ ( t , X t ) and intermittent (the ‘I’ in RIBSDE) upper barrier given by, for t ∈ [ 0 , T ] [ N / 2 ] [( N + 1 ) / 2 ] ∑ ∑ U t = 1 [ 휗 2 l ,휗 2 l + 1 ) ∞ + 1 [ 휗 2 l − 1 ,휗 2 l ) h ( t , X t ) l = 0 l = 1 ‘Nominal’ upper obstacle h ( t , X t ) only active on the ‘odd’ random time intervals [ 휗 2 l − 1 , 휗 2 l ) Call protection on the ‘even’ random time intervals [ 휗 2 l , 휗 2 l + 1 ) Chassagneux, Crépey, Rahal RIBSDEs
Markovian RIBSDE Diffusion Set-Up with Marker Process Approximation Results Markovian RIBSDE Numerics Connection with Finance Solution of the RIBSDE Risk-neutral pricing problems in finance Driver coefficient function f typically given as f = f ( t , x , y ) = c ( t , x ) − 휇 ( t , x ) y Dividend and interest-rate related functions c and 휇 Single-name credit risk (counterparty risk) Recovery-adjusted dividend-yields c Credit-spread adjusted interest-rates 휇 Pre-default factor process X Affine in y , does not depend on z Historical rather than RN modeling → ‘ z -dependent’ f Market imperfections → nonlinear f Terminal cost functions typically given by ℓ ( t , x ) = ¯ P ∨ S , h ( t , x ) = ¯ C ∨ S , g ( x ) = ¯ N ∨ S P ≤ ¯ ¯ N ≤ ¯ C Constants S = x 1 first component of x Mokobodski condition satisfied with Q = S provided S is a square-integrable Itô process Chassagneux, Crépey, Rahal RIBSDEs
Markovian RIBSDE Diffusion Set-Up with Marker Process Approximation Results Markovian RIBSDE Numerics Connection with Finance Solution of the RIBSDE Highly path dependent call protection Example (‘ l out of d ’) Given a constant trigger level ¯ S and constants l ≤ d ≤ N , call possible iff S has been ≥ ¯ S on at least l of the last d monitoring times 풦 = { 0 , 1 } d , 휅 k I ( x ) = ( 1 S ≥ ¯ S , k 1 , . . . , k d − 1 ) H t vector of the indicator functions of the events S T I ≥ ¯ S at the last d monitoring dates preceding time t ∈ K with ∣ k ∣ = ∑ Call possible iff ∣ H t ∣ ≥ l ⇔ H t / 1 ≤ p ≤ d k p and K = { k ∈ 풦 ; ∣ k ∣ < l } Chassagneux, Crépey, Rahal RIBSDEs
Markovian RIBSDE Diffusion Set-Up with Marker Process Approximation Results Markovian RIBSDE Numerics Connection with Finance Solution of the RIBSDE Solution of the RIBSDE Definition A solution 풴 to ( 풮 ) is a triple 풴 = ( Y , Z , A ) such that: (i) Y ∈ 풮 2 , Z ∈ ℋ 2 q , A ∈ 풜 2 ∫ T ∫ T Z s dW s t ∈ [ 0 , T ] (ii) Y t = 휉 + f ( s , X s , Y s , Z s ) ds + A T − A t − t t (iii) L t ≤ Y t on [ 0 , T ] , Y t ≤ U t on [ 0 , T ] ∫ T ∫ T ( Y t − L t ) dA + ( U t − − Y t − ) dA − and t = 0 t = 0 0 (iv) A + is continuous, and { ( 휔, t ) ; Δ Y ∕ = 0 } = { ( 휔, t ) ; Δ A − ∕ = 0 } ⊆ ∪ [ N / 2 ] l = 0 [ [ 휗 2 l ] ] Δ Y = Δ A − on ∪ [ N / 2 ] l = 0 [ [ 휗 2 l ] ] q and 풜 2 ‘usual L 2 spaces’ 풮 2 , ℋ 2 A ± Jordan component of A Convention that 0 × ±∞ = 0 in (iii) Obvious extension to a random terminal time 휃 Chassagneux, Crépey, Rahal RIBSDEs
Markovian RIBSDE Diffusion Set-Up with Marker Process Approximation Results Markovian RIBSDE Numerics Connection with Finance Solution of the RIBSDE For l decreasing from N to 0, let us define 풴 l = ( Y l , Z l , A l ) on [ 휗 l , 휗 l + 1 ] as the solution, with A l continuous, to the stopped RBSDE (for l even) or R2BSDE (for l odd) with data (with Y N + 1 휗 N + 1 ≡ g ( X T ) ) { f ( t , X t , y , z ) , Y l + 1 ( l even) 휗 l + 1 , ℓ ( t , X t ) f ( t , X t , y , z ) , min ( Y l + 1 ( l odd) 휗 l + 1 , h ( 휗 l + 1 , X 휗 l + 1 )) , ℓ ( t , X st ) , h ( t , X t ) Let us define 풴 = ( Y , Z , A ) on [ 0 , T ] by, for every l = 0 , . . . , N : ( Y , Z ) = ( Y l , Z l ) on [ 휗 l , 휗 l + 1 ) , and also at 휗 N + 1 = T in case l = N . So in particular { Y 0 0 , k ∈ K Y 0 = Y 1 0 , k / ∈ K where k is the initial condition of the marker process H . dA = dA l on ( 휗 l , 휗 l + 1 ) , ( ) + = Δ Y 휗 l (= 0 for l odd ) Y l Δ A 휗 l = Δ A − 휗 l = 휗 l − h ( 휗 l , X 휗 l ) and Δ A T = Δ Y T = 0. Proposition 풴 = ( Y , Z , A ) is the unique solution to ( 풮 ) Chassagneux, Crépey, Rahal RIBSDEs
Markovian RIBSDE Diffusion Set-Up with Marker Process Approximation Results Markovian RIBSDE Numerics Connection with Finance Solution of the RIBSDE Verification principle Risk-neutral pricing problems in finance Financial interpretation of a solution 풴 to ( 풮 ) Y 0 ‘NFLVR’ Arbitrage price at time 0 for the game option with payoff functions c , l , h , g and call protection 휗 Bilateral super-hedging price and infimal issuer super-hedging price up to a local martingale cost process Z Hedging strategy Chassagneux, Crépey, Rahal RIBSDEs
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