Anumericalalgorithm for general HJB equations : a jump-constrained BSDE approach Nicolas Langrené Univ. Paris Diderot - Sorbonne Paris Cité, LPMA, FiME Joint work with Idris Kharroubi (Paris Dauphine), Huyên Pham (Paris Diderot) Workshop on Stochastic Games, Equilibrium, and Applications to Energy & Commodities Markets - Fields Institute, August 29, 2013 Nicolas Langrené A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 1/20
Modeling volatility of an asset price S t Constant : σ>0 Deterministic : σ (t) Local : σ (t, S t ) Stochastic : dσ t =... Uncertain : σ [σ min,σ max ] Nicolas Langrené A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 2/20
Uncertain Volatility Model Example ds t = σs t dw t σ [σ min,σ max ] uncertain Super-replication price Payoff Φ=Φ(T, S T ) P + 0 =sup E Q [Φ (T, S T )] Q Q Q = Q σ Q ; σ min σ Q σ max Nicolas Langrené A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 3/20
Stochastic control problem with controlled driver & drift & volatility Formulation dx α s = b (X α s, α s ) ds + σ (X α s, α s ) dw s T v (t, x) =supe t,x f (Xs α, α s ) ds + g (XT α ) α A t General HJB equation v b t +sup (x, a).d x v + 12 tr σσ (x, a) Dx 2 v + f (x, a) =0 a A v (T, x) =g (x), x R d on [0,T ) R d Nicolas Langrené A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 4/20
STEP 1 : Randomization of controls Poisson random measure µ A (dt, da) on R + A, W associated to the marked point process I (τ i,ζ i ) i,valuedina I t = ζ i, τ i t <τ i+1 Uncontrolled randomized problem Linear FBSDE Y t = g (X T )+ dx s = b (X s, I s ) ds + σ (X s, I s ) dw s T v (t, x, a) =E t,x,a f (X s, I s ) ds + g (X T ) T Y t v (t, X t, I t ) t f (X s, I s ) ds t T t Z s dw s T t A U s (a) µ A (ds, da) Nicolas Langrené A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 5/20
STEP 2 : Constraint on jumps U t (a) =v (t, X t, a) v (t, X t, I t ) Now, how to retrieve HJB? Add the constraint U t (a) 0 (t, a)! Jump-constrained BSDE Minimal solution (Y, Z, U, K) of Y t = g (X T )+ T + K T K t t f (X s, I s, Y s, Z s ) ds T subject to U t (a) 0 (t, a) t A T t Z s dw s U s (a) µ A (ds, da), 0 t T Nicolas Langrené A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 6/20
Link with general HJB equations (X, I ) Markov v = v (t, x, a) s.t.y t = v (t, X t, I t ) Key Lemma v = v (t, x, a) does not depend on a! v = v (t, x) Theorem v = v (t, x) is solution of the HJB equation v b(x,a).d t +sup x v + 12 tr σσ (x,a)dx 2 v +f x,a,y,σ (x,a).d x v =0 a A v (T, x) =g (x), x R d on [0,T ) R d Proofs : cf. [Kharroubi, Pham, 2012] Nicolas Langrené A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 7/20
Numerical scheme U t (a) =v (t, X t, a) v (t, X t, I t ) 0 (t, a) v (t, X t, I t ) sup a A v (t, X t, a) Minimal solution v (t, X t, I t )=sup a A v (t, X t, a) v (t, X t, I t ) v (t, X t, I t ) K t K t Forward-Backward numerical scheme Y N = g (X N ) Z i = E i Y i+1 W i / i Y i = E i [Y i+1 + f i (X i, I i, Y i+1, Z i ) i ] Y i = sup A A i E i,a [Y i ] where E i,a [.] :=E [. X i, I i = A] Nicolas Langrené A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 8/20
Towards an implementable scheme How to compute the conditional expectations? (quantification, Malliavin calculus, empirical regression,...) cf. comparative tests in [Bouchard, Warin, 2012] Conditional expectation approximation E [U F ti ]=arg inf V L(F ti,p) E (V U) 2 where S L (F ti, P) 1 Ê [U F ti ]=arg inf V S M M (V m U m ) 2 m=1 Nicolas Langrené A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 9/20
Empirical regression schemes First algorithm ( Tsitsiklis - van Roy ) Ŷ N = g (X N ) Ŷ i = Ê i Ŷ i+1 + f i (X i, I i ) i Ŷ i = sup A A i E i,a Ŷi Upward biased (up to Monte Carlo error & regression bias) Second algorithm ( Longstaff - Schwartz ) ˆα i =arg sup E i,a Ŷi A A i ˆX i+1 = b( ˆX i, ˆα i ) i + σ( ˆX i, ˆα i ) W i ˆv (t 0, x 0 )= 1 M N f ( ˆX i+1, ˆα i ) i + g( ˆX N ) M m=1 i=1 Downward biased (up to Monte Carlo error) Nicolas Langrené A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 10 /20
Uncertain correlation model Model dst i = σ i StdW i t i, i =1, 2 dw 1 t, dwt 2 = ρdt 1 ρ min ρ ρ max 1 Super-replication price Payoff Φ=Φ T, ST 1, ST 2 P 0 + =sup E Q Φ T, ST 1, S 2 T Q Q Q = Q ρ Q ; ρ min ρ Q ρ max Nicolas Langrené A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 11 /20
Call spread on spread S 1 (T ) S 2 (T ) Φ=(S 1 (T) S 2 (T) K 1 ) + (S 1 (T) S 2 (T) K 2 ) + S 1 (0) S 2 (0) σ 1 σ 2 ρ min ρ max K 1 K 2 T 50 50 0.4 0.3 0.8 0.8 5 5 0.25 Regression basis φ (t, s 1, s 2,ρ)=(K 2 K 1 ) S(β 0 +β 1 s 1 +β 2 s 2 +β 3 ρ+β 4 ρs 1 +β 5 ρs 2 ) S (x) =1/ (1 + exp ( x)) Bang-bang optimal control ρ (t, s 1, s 2 )=argmax α φ (t, s 1, s 2,ρ)=ρ max if β 3 +β 4 s 1 +β 5 s 2 0 = ρ min else Nicolas Langrené A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 12 /20
Results Price of Call Spread on S1(T) S2(T) Superhedging ρ=ρ max ρ=0 ρ=ρ min Subhedging 10 9 8 7 6 5 4 3 2 1 20 15 10 5 0 5 10 15 20 25 0 Moneyness ( = S1(0) S2(0) ) Nicolas Langrené A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 13 /20
Impact of correlation range 1.0 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 Correlation Ranges (+ ) Price of Call Spread on S1(T) S2(T) 15 10 5 0 5 10 15 20 25 0 Moneyness ( = S1(0) S2(0) ) Nicolas Langrené A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 14 /20 10 9 8 7 6 5 4 3 2 1
Call Spread (S (T ) K 1 ) + (S (T ) K 2 ) + S (0) = 100, K 1 =90,K 2 =110,T =1,uncertainσ [0.1, 0.2] Estimated superreplication price: Algorithm 1 Time Step 11.5 1/128 11.4 11.3 1/64 11.2 1/32 11.1 1/16 11.0 1/8 10.9 16 17 18 19 20 21 log2(m) Nicolas Langrené A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 15 /20
Call Spread (S (T ) K 1 ) + (S (T ) K 2 ) + S (0) = 100, K 1 =90,K 2 =110,T =1,uncertainσ [0.1, 0.2] Estimated superreplication price: Algorithm 2 Time Step 11.5 1/128 11.4 11.3 1/64 11.2 1/32 11.1 1/16 11.0 1/8 10.9 16 17 18 19 20 21 log2(m) Nicolas Langrené A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 16 /20
Outperformer Spread (S 2 (T ) K 1 S 1 (T )) + (S 2 (T ) K 2 S 1 (T )) + S i (0) = 100, K 1 =0.9,K 2 =1.1,T =1,uncertainσ i [0.1, 0.2], ρ = 0.5 Estimated superreplication price: Algorithm 1 Time Step 12.0 1/128 11.8 11.6 1/64 11.4 1/32 11.2 1/16 11.0 1/8 10.8 16 17 18 19 20 21 log2(m) Nicolas Langrené A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 17 /20
Outperformer Spread (S 2 (T ) K 1 S 1 (T )) + (S 2 (T ) K 2 S 1 (T )) + S i (0) = 100, K 1 =0.9,K 2 =1.1,T =1,uncertainσ i [0.1, 0.2], ρ = 0.5 Estimated superreplication price: Algorithm 2 Time Step 12.0 1/128 11.8 11.6 1/64 11.4 1/32 11.2 1/16 11.0 1/8 10.8 16 17 18 19 20 21 log2(m) Nicolas Langrené A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 18 /20
Summary 1 Probabilistic representation of stochastic control problem with controlled volatility : jump-constrained BSDE 2 Numerical scheme for jump-constrained BSDEs 3 Application to pricing under uncertain volatility 4 Extension : stochastic games, HJB-Isaacs equations Nicolas Langrené A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 19 /20
???? Thank you! Questions? Nicolas Langrené A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 20 /20
???? References J-P. Lemor and E. Gobet and X. Warin (2006) Rate of convergence of an empirical regression method for solving generalized backward stochastic differential equations J. Guyon and P. Henry-Labordère (2011) Uncertain volatility model : a Monte Carlo approach E. Gobet and P. Turkedjiev (2011) Approximation of discrete BSDE using least-squares regression B. Bouchard and X. Warin (2012) Monte Carlo valorisation of American options : facts and new algorithms to improve existing methods I. Kharroubi and H. Pham (2012) Feynman-Kac representation for Hamilton-Jacobi-Bellman IPDE S. Alanko and M. Avellaneda (2013) Reducing variance in the numerical solution of BSDEs Nicolas Langrené A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 21 /20
???? Example A linear-quadratic stochastic control problem Problem dx α s =( µ 0 X α s + µ 1 α s ) ds +(σ 0 + σ 1 α s ) dw s X0 α =0 T v (t, x) =supe λ 0 α A t (α s ) 2 ds λ 1 (X α T ) 2 Set of parameters µ 0 µ 1 σ 0 σ 1 λ 0 λ 1 T 0.02 0.5 0.2 0.1 20 200 2 Nicolas Langrené A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 22 /20
???? Numerical parameters n = 52 M = 10 6 time steps Monte Carlo simulations Regression basis φ (t, x,α)=β 0 + β 1 x + β 2 α + β 3 xα + β 4 x 2 + β 5 α 2 Linear optimal control α (t, x) =argmaxφ (t, x,α)=a (t) x + B (t) α A (t) = 0.5 β 3 /β 5 B (t) = 0.5 β 2 /β 5 Nicolas Langrené A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 23 /20
???? Estimated optimal controls Optimal Control α (t, x) 3 2 1 0 1 2 3 0.5 0 Diffusion value x 0.5 1.5 1 0.5 Time t 0 Nicolas Langrené A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 24 /20
???? Control impact (1/2) : no control Uncontrolled diffusion 10% 20% 30% 40% 50% 60% 70% 80% 90% 99% Interquantile Ranges 0.6 0.4 0.2 0 0.2 0.4 0.6 0.5 1 1.5 2 Time Nicolas Langrené A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 25 /20
???? Control impact (2/2) : optimal control Optimally controlled diffusion 10% 20% 30% 40% 50% 60% 70% 80% 90% 99% Interquantile Ranges 0.6 0.4 0.2 0 0.2 0.4 0.6 0.5 1 1.5 2 Time Nicolas Langrené A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 26 /20
???? Accuracy 0 1 2 3 Comparison of control coefficients A(t) and B(t) B(t) estimated theoretical Value Function ˆv(0,0)= 5.761 v(0,0)= 5.705 Relative Error : 1% 4 A(t) estimated theoretical 0 0.5 1 1.5 2 Time t Nicolas Langrené A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 27 /20
???? Convergence rate 1 Localizations 2 Theoretical regressions 3 Empirical regressions Assumptions p 1, L g, L f, C f,0 > 0s.t. i =0,...,N 1 g (x) g (x ) L g x p x p f i (x,a,y,z) f i (x,a,y,z ) L f ( x p x p + a p a p + y y + z z ) f i (0, 0, 0, 0) C f,0 Bounded control domain A R d : Ā > 0s.t. a A, a Ā Nicolas Langrené A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 28 /20
???? Theoretical regression (1/2) Definition ˆλ i (U) :=arg inf E (λ.p (X i, I i ) U) 2 λ R B P i (U) :=ˆλ i (U).p (X i, I i ) Associated scheme Ŷ N = g (X N ) ˆλ Y i = regression coefficients at time t i Ŷ i = sup A A i ˆλ Y i.p i (X i, A) Problem : Ŷ i is not itself the projection of some random variable... Nicolas Langrené A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 29 /20
???? Theoretical regression (2/2) Alternative definition ˆλ i,a (U) :=arg inf E (λ.p (X i, A) U A ) 2 λ R B P i,a (U) :=ˆλ i (U).p (X i, A) Regression error Yi 2 Zi 2 max Ŷ i, i Ẑ i N 1 e C(T t i ) k=i E sup Y k,a Pk,A Y (Y k,a ) 2 A A k +C k E sup Zk,A Pk,A Z (Z k,a ) 2 A A k But its empirical version cannot be (efficiently) implemented... Nicolas Langrené A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 30 /20
???? Outperformer (S 1 (T ) S 2 (T )) + S i (0) = 100, T =1,uncertainσ i [0.1, 0.2], ρ =0 Estimated superreplication price: Algorithm 1 Time Step 12 1/128 1/64 11.5 1/32 11 1/16 10.5 16 17 18 19 20 21 log2(m) Nicolas Langrené A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 31 /20 1/8
???? Outperformer (S 1 (T ) S 2 (T )) + S i (0) = 100, T =1,uncertainσ i [0.1, 0.2], ρ =0 Estimated superreplication price: Algorithm 2 Time Step 12 1/128 1/64 11.5 1/32 11 1/16 10.5 16 17 18 19 20 21 log2(m) Nicolas Langrené A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 32 /20 1/8
???? Outperformer Spread (S 2 (T ) K 1 S 1 (T )) + (S 2 (T ) K 2 S 1 (T )) + S i (0) = 100, K 1 =0.9,K 2 =1.1,T =1,uncertainσ i [0.1, 0.2] & ρ [ 0.5, 0.5] Estimated superreplication price: Algorithm 1 Time Step 14 1/128 13.5 1/64 13 1/32 12.5 1/16 12 11.5 16 17 18 19 20 21 log2(m) Nicolas Langrené A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 33 /20 1/8
???? Outperformer Spread (S 2 (T ) K 1 S 1 (T )) + (S 2 (T ) K 2 S 1 (T )) + S i (0) = 100, K 1 =0.9,K 2 =1.1,T =1,uncertainσ i [0.1, 0.2] & ρ [ 0.5, 0.5] Estimated superreplication price: Algorithm 2 Time Step 14 1/128 13.5 1/64 13 1/32 12.5 1/16 12 11.5 16 17 18 19 20 21 log2(m) Nicolas Langrené A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 34 /20 1/8