View
1
Download
0
Category
Preview:
Citation preview
Motivation Numerical scheme Applications Summary
A numerical algorithmfor general HJB equations :
a jump-constrained BSDE approach
Nicolas Langrene
Univ. Paris Diderot - Sorbonne Paris Cite, LPMA, FiME
Joint work with Idris Kharroubi (Paris Dauphine), Huyen Pham (Paris Diderot)
Workshop on Stochastic Games, Equilibrium, and Applications to
Energy & Commodities Markets - Fields Institute, August 29, 2013
Nicolas Langrene A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 1 /20
Motivation Numerical scheme Applications Summary
Modeling volatilityof an asset price St
Constant : σ > 0
⇓Deterministic : σ (t)
⇓Local : σ (t, St)
⇓Stochastic : dσt = . . .
⇓Uncertain : σ ∈ [σmin, σmax]
Nicolas Langrene A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 2 /20
Motivation Numerical scheme Applications Summary
Uncertain Volatility Model
Example
dSt = σStdWt
σ ∈ [σmin,σmax] uncertain
Super-replication price
Payoff Φ = Φ (T , ST )
P+
0= sup
Q∈QEQ
[Φ (T , ST )]
Q =
�Q ↔ σQ
; σmin ≤ σQ ≤ σmax
�
Nicolas Langrene A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 3 /20
Motivation Numerical scheme Applications Summary
Stochastic control problemwith controlled driver & drift & volatility
Formulation
dXαs = b (Xα
s ,αs) ds + σ (Xαs ,αs) dWs
v (t, x) = supα∈A
Et,x
�� T
tf (Xα
s ,αs) ds + g (XαT )
�
General HJB equation
∂v
∂t+ sup
a∈A
�b (x , a) .Dxv +
1
2tr�σσ�(x , a)D2
x v�+ f (x , a)
�= 0
v (T , x) = g (x) , x ∈ Rd on [0,T )× Rd
Nicolas Langrene A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 4 /20
Motivation Numerical scheme Applications Summary
STEP 1 : Randomization of controls
Poisson random measure µA (dt, da) on R+ × A , ⊥⊥ W
associated to the marked point process I ↔ (τi , ζi )i , valued in A
It = ζi , τi ≤ t < τi+1
Uncontrolled randomized problem
dXs = b (Xs , Is) ds + σ (Xs , Is) dWs
v (t, x , a) = Et,x ,a
�� T
tf (Xs , Is) ds + g (XT )
�
Linear FBSDE
Yt = g (XT ) +
� T
tf (Xs , Is) ds −
� T
tZsdWs −
� T
t
�
AUs (a) µA (ds, da)
Yt ←→ v (t,Xt , It)
Nicolas Langrene A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 5 /20
Motivation Numerical scheme Applications Summary
STEP 2 : Constraint on jumps
Ut (a) = v (t,Xt , a)− v (t,Xt , It−)
Now, how to retrieve HJB ?
⇒ Add the constraint Ut (a) ≤ 0 ∀ (t, a) !
Jump-constrained BSDE
Minimal solution (Y ,Z ,U,K ) of
Yt = g (XT ) +
� T
tf (Xs , Is ,Ys ,Zs) ds −
� T
tZsdWs
+ KT − Kt −� T
t
�
AUs (a) µA (ds, da) , 0 ≤ t ≤ T
subject to Ut (a) ≤ 0 ∀ (t, a)
Nicolas Langrene A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 6 /20
Motivation Numerical scheme Applications Summary
Link with general HJB equations
(X , I ) Markov ⇒ ∃v = v (t, x , a) s.t. Yt = v (t,Xt , It)
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
∂t+sup
a∈A
�b(x ,a).Dxv+
1
2tr�σσ�(x ,a)D2
x v�+f
�x ,a,y ,σ�(x ,a).Dxv
��=0
v (T , x) = g (x) , x ∈ Rd on [0,T )× Rd
Proofs : cf. [Kharroubi, Pham, 2012]
Nicolas Langrene A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 7 /20
Motivation Numerical scheme Applications Summary
Numerical scheme
Ut (a) = v (t,Xt , a)− v (t,Xt , It−) ≤ 0 ∀ (t, a)
⇒ v (t,Xt , It−) ≥ supa∈A v (t,Xt , a)
⇒ Minimal solution v (t,Xt , It−) = supa∈A v (t,Xt , a)
v (t,Xt , It)− v (t,Xt , It−) ←→ Kt − Kt−
Forward-Backward numerical scheme
YN = g (XN)
Zi = Ei
�Yi+1∆W�
i
�/∆i
Yi = Ei [Yi+1 + fi (Xi , Ii ,Yi+1,Zi )∆i ]
Yi = supA∈Ai
Ei ,A [Yi ]
where Ei ,A [.] := E [. |Xi , Ii = A ]
Nicolas Langrene A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 8 /20
Motivation Numerical scheme Applications Summary
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 |Fti ] = arg infV∈L(Fti ,P)
E�(V − U)
2
�
E [U |Fti ] = arg infV∈S
1
M
M�
m=1
(Vm − Um)2
where S ⊂ L (Fti ,P)
Nicolas Langrene A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 9 /20
Motivation Numerical scheme Applications Summary
Empirical regression schemes
First algorithm (“Tsitsiklis - van Roy”)
Y N = g (XN)
Y i = Ei
�Y i+1 + fi (Xi , Ii )∆i
�
Y i = supA∈Ai
Ei,A
�Y i
�
Upward biased (up to Monte Carlo error & regression bias)
Second algorithm (“Longstaff - Schwartz”)
αi = arg supA∈Ai
Ei,A
�Y i
�
X i+1 = b(X i , αi )∆i + σ(X i , αi )∆Wi
v (t0, x0) =1
M
M�
m=1
�N�
i=1
f (X i+1, αi )∆i + g(XN)
�
Downward biased (up to Monte Carlo error)
Nicolas Langrene A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 10 /20
Motivation Numerical scheme Applications Summary
Uncertain correlation model
Model
dS it = σiS
itdW
it , i = 1, 2
�dW 1
t , dW2
t
�= ρdt
−1 ≤ ρmin ≤ ρ ≤ ρmax ≤ 1
Super-replication price
Payoff Φ = Φ�T , S1
T , S2
T
�
P+
0= sup
Q∈QEQ �
Φ�T , S1
T , S2
T
��
Q =
�Q ↔ ρQ ; ρmin ≤ ρQ ≤ ρmax
�
Nicolas Langrene A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 11 /20
Motivation Numerical scheme Applications Summary
Call spread on spread S1 (T )− S2 (T )
Φ = (S1(T)−S2(T)−K1)+ − (S1(T)−S2(T)−K2)
+
S1 (0) S2 (0) σ1 σ2 ρmin ρmax K1 K2 T
50 50 0.4 0.3 −0.8 0.8 −5 5 0.25
Regression basis
φ (t, s1, s2, ρ) = (K2−K1)×S(β0+β1s1+β2s2+β3ρ+β4ρs1+β5ρs2)
S (x) = 1/ (1 + exp (−x))
⇒ Bang-bang optimal control
ρ∗ (t, s1, s2) = argmaxα
φ (t, s1, s2, ρ) = ρmax if β3+β4s1+β5s2 ≥ 0
= ρmin else
Nicolas Langrene A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 12 /20
Motivation Numerical scheme Applications Summary
Results
!20 !15 !10 !5 0 5 10 15 20 250
1
2
3
4
5
6
7
8
9
10
Moneyness ( = S1(0)!S2(0) )
Price of Call Spread on S1(T)!S2(T)
Superhedging!=!max!=0!=!minSubhedging
Nicolas Langrene A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 13 /20
Motivation Numerical scheme Applications Summary
Impact of correlation range
!15 !10 !5 0 5 10 15 20 250
1
2
3
4
5
6
7
8
9
10
Moneyness ( = S1(0)!S2(0) )
Price of Call Spread on S1(T)!S2(T)
CorrelationRanges (+!)
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Nicolas Langrene A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 14 /20
Motivation Numerical scheme Applications Summary
Call Spread (S (T )− K1)+ − (S (T )− K2)
+
S (0) = 100 , K1 = 90 , K2 = 110 , T = 1 , uncertain σ ∈ [0.1, 0.2]
16 17 18 19 20 2110.9
11.0
11.1
11.2
11.3
11.4
11.5
log2(M)
Estimated superreplication price: Algorithm 1
Time Step
1/8
1/16
1/32
1/64
1/128
Nicolas Langrene A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 15 /20
Motivation Numerical scheme Applications Summary
Call Spread (S (T )− K1)+ − (S (T )− K2)
+
S (0) = 100 , K1 = 90 , K2 = 110 , T = 1 , uncertain σ ∈ [0.1, 0.2]
16 17 18 19 20 2110.9
11.0
11.1
11.2
11.3
11.4
11.5
log2(M)
Estimated superreplication price: Algorithm 2
Time Step
1/8
1/16
1/32
1/64
1/128
Nicolas Langrene A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 16 /20
Motivation Numerical scheme Applications Summary
Outperformer Spread (S2(T )−K1S1(T ))+−(S2(T )−K2S1(T ))+
Si (0) = 100 , K1 = 0.9 , K2 = 1.1 , T = 1 , uncertain σi ∈ [0.1, 0.2] , ρ = −0.5
16 17 18 19 20 2110.8
11.0
11.2
11.4
11.6
11.8
12.0
log2(M)
Estimated superreplication price: Algorithm 1
Time Step
1/8
1/16
1/32
1/64
1/128
Nicolas Langrene A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 17 /20
Motivation Numerical scheme Applications Summary
Outperformer Spread (S2(T )−K1S1(T ))+−(S2(T )−K2S1(T ))+
Si (0) = 100 , K1 = 0.9 , K2 = 1.1 , T = 1 , uncertain σi ∈ [0.1, 0.2] , ρ = −0.5
16 17 18 19 20 2110.8
11.0
11.2
11.4
11.6
11.8
12.0
log2(M)
Estimated superreplication price: Algorithm 2
Time Step
1/8
1/16
1/32
1/64
1/128
Nicolas Langrene A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 18 /20
Motivation Numerical scheme Applications Summary
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 Langrene A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 19 /20
? ? ? ?
Thank you !Questions ?
Nicolas Langrene 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-Labordere (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 Langrene A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 21 /20
? ? ? ?
ExampleA linear-quadratic stochastic control problem
Problem
dXαs = (−µ0X
αs + µ1αs) ds + (σ0 + σ1αs) dWs
Xα0 = 0
v (t, x) = supα∈A
E�−λ0
� T
t(αs)
2 ds − λ1 (XαT )
2
�
Set of parameters
µ0 µ1 σ0 σ1 λ0 λ1 T
0.02 0.5 0.2 0.1 20 200 2
Nicolas Langrene A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 22 /20
? ? ? ?
Numerical parameters
n = 52 time steps
M = 106
Monte Carlo simulations
Regression basis
φ (t, x ,α) = β0 + β1x + β2α+ β3xα+ β4x2+ β5α
2
⇒ Linear optimal control
α∗(t, x) = argmax
αφ (t, x ,α) = A (t) x + B (t)
A (t) = −0.5β3/β5
B (t) = −0.5β2/β5
Nicolas Langrene A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 23 /20
? ? ? ?
Estimated optimal controls
0
0.5
1
1.5
!0.5
0
0.5
!3
!2
!1
0
1
2
3
Time t
Optimal Control
!!(t, x)
Diffusion value x
Nicolas Langrene A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 24 /20
? ? ? ?
Control impact (1/2) : no control
0.5 1 1.5 2
!0.6
!0.4
!0.2
0
0.2
0.4
0.6
Time
Uncontrolled diffusion
InterquantileRanges
99%
90%
80%
70%
60%
50%
40%
30%
20%
10%
Nicolas Langrene A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 25 /20
? ? ? ?
Control impact (2/2) : optimal control
0.5 1 1.5 2
!0.6
!0.4
!0.2
0
0.2
0.4
0.6
Time
Optimally controlled diffusion
InterquantileRanges
99%
90%
80%
70%
60%
50%
40%
30%
20%
10%
Nicolas Langrene A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 26 /20
? ? ? ?
Accuracy
0 0.5 1 1.5 2
!4
!3
!2
!1
0
Comparison of control coefficients A(t) and B(t)
Time t
B(t)estimated
theoretical
A(t)estimated
theoretical
Value
Function
v(0,0)=−5.761
v(0,0)=−5.705
Relative
Error :
1%
Nicolas Langrene 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, Lg , Lf ,Cf ,0 > 0 s.t. ∀i = 0, . . . ,N − 1
|g (x)− g (x �)| ≤ Lg |xp − x �p||fi (x ,a,y,z)−fi (x
�,a�,y �,z �)| ≤ Lf (|xp−x �p|+|ap−a�p|+|y−y �|+|z−z �|)|fi (0, 0, 0, 0)| ≤ Cf ,0
Bounded control domain A ⊂ Rd �: ∃A > 0 s.t. ∀a ∈ A, |a| ≤ A
Nicolas Langrene A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 28 /20
? ? ? ?
Theoretical regression (1/2)
Definition
λi (U) := arg infλ∈RB
E�(λ.p (Xi , Ii )− U)
2
�
Pi (U) := λi (U) .p (Xi , Ii )
Associated scheme
YN = g (XN)
λYi = regression coefficients at time ti
Yi = supA∈Ai
λYi .pi (Xi ,A)
Problem : Yi is not itself the projection of some random variable...
Nicolas Langrene A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 29 /20
? ? ? ?
Theoretical regression (2/2)
Alternative definition
λi ,A (U) := arg infλ∈RB
E�(λ.p (Xi ,A)− UA)
2
�
Pi ,A (U) := λi (U) .p (Xi ,A)
Regression error
max
����Yi − Y i
���2
, ∆i
���Zi − Z i
���2�
≤ eC(T−ti )N−1�
k=i
�E�supA∈Ak
��Yk,A − PYk,A (Yk,A)
��2�
+C∆kE�supA∈Ak
��Zk,A − PZk,A (Zk,A)
��2��
But its empirical version cannot be (efficiently) implemented...
Nicolas Langrene A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 30 /20
? ? ? ?
Outperformer (S1(T )−S2(T ))+Si (0) = 100 , T = 1 , uncertain σi ∈ [0.1, 0.2] , ρ = 0
16 17 18 19 20 2110.5
11
11.5
12
log2(M)
Estimated superreplication price: Algorithm 1
Time Step
1/8
1/16
1/32
1/64
1/128
Nicolas Langrene A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 31 /20
? ? ? ?
Outperformer (S1(T )−S2(T ))+Si (0) = 100 , T = 1 , uncertain σi ∈ [0.1, 0.2] , ρ = 0
16 17 18 19 20 2110.5
11
11.5
12
log2(M)
Estimated superreplication price: Algorithm 2
Time Step
1/8
1/16
1/32
1/64
1/128
Nicolas Langrene A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 32 /20
? ? ? ?
Outperformer Spread (S2(T )−K1S1(T ))+−(S2(T )−K2S1(T ))+
Si (0) = 100 , K1 = 0.9 , K2 = 1.1 , T = 1 , uncertain σi ∈ [0.1, 0.2] & ρ ∈ [−0.5, 0.5]
16 17 18 19 20 2111.5
12
12.5
13
13.5
14
log2(M)
Estimated superreplication price: Algorithm 1
Time Step
1/8
1/16
1/32
1/64
1/128
Nicolas Langrene A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 33 /20
? ? ? ?
Outperformer Spread (S2(T )−K1S1(T ))+−(S2(T )−K2S1(T ))+
Si (0) = 100 , K1 = 0.9 , K2 = 1.1 , T = 1 , uncertain σi ∈ [0.1, 0.2] & ρ ∈ [−0.5, 0.5]
16 17 18 19 20 2111.5
12
12.5
13
13.5
14
log2(M)
Estimated superreplication price: Algorithm 2
Time Step
1/8
1/16
1/32
1/64
1/128
Nicolas Langrene A numerical algorithm for general HJB equations : a jump-constrained BSDE approach 34 /20
Recommended