Stochastic Grid Bundling Method for Backward Stochastic Di ff - - PowerPoint PPT Presentation

Stochastic Grid Bundling Method for Backward Stochastic Differential Equations

Ki Wai Chau

Centrum Wiskunde & Informatica

17th Winter school on Mathematical Finance 22 January 2018

(A joint work with Kees Oosterlee (CWI & TU Delft))

K.W. Chau (CWI) SGBM for BSDEs 22 January 2018 1 / 20

Backward Stochastic Differential Equations

Settings:

A filtered complete probability space (Ω, F, F, P) W := (Wt)0≤t≤T is a d-dimensional Brownian motion adapted to F

Forward Backward Stochastic Differential Equation ⇢ dXt = µ(t, Xt)dt + σ(t, Xt)dWt, X0 = x0, dYt = f (t, Xt, Yt, Zt)dt + ZtdWt, YT = Φ(XT),

µ : Ω ⇥ [0, T] ⇥ Rq ! Rq and σ : Ω ⇥ [0, T] ⇥ Rq ! Rq×d f : Ω ⇥ [0, T] ⇥ Rq ⇥ R ⇥ Rd ! R Φ : Ω ⇥ Rq ! R Solution: (Yt, Zt) which satisfies the equation, adapts to F and satisfies some regularity requirements.

K.W. Chau (CWI) SGBM for BSDEs 22 January 2018 2 / 20

Discretization

For a given time grid π = {0 = t0 < . . . < tN = T}, we define the backward time discretizations (Y π, Z π) based on the theta-scheme from [Zhao et al., 2012]: Y π

tN =Φ(X π tN),

Z π

tN = rΦ(X π tN)σ(tN, X π tN)

Z π

tk = θ1 2 (1 θ2)Etk

h Z π

tk+1

i + 1 ∆k θ1

2 Etk

h Y π

tk+1∆W T k

i + θ1

2 (1 θ2)Etk

h fk+1(Y π

tk+1, Z π tk+1)∆W T k

i , k = N 1, . . . , 0 Y π

tk =Etk

h Y π

tk+1

i + ∆kθ1fk(Y π

tk, Z π tk)

+ ∆k(1 θ1)Etk h fk+1(Y π

tk+1, Z π tk+1)

i , k = N 1, . . . , 0, where fk(y, z) := f (tk, X π

tk, y, z), 0  θ1  1 and 0 < θ2  1.

K.W. Chau (CWI) SGBM for BSDEs 22 January 2018 3 / 20

Discretization (cont.)

Note that:

the globally Lipschitz driver assumption is in force; we use a Markovian approximation X π

tk, tk 2 π:

X π

tk+1 = X π tk + b(tk, X π tk )∆k + σ(tk, X π tk )∆Wk;

due to the Markovian setting, there exist functions y (θ1,θ2)

k

(x) and z(θ1,θ2)

k

(x) such that Y π

tk = y (θ1,θ2) k

(X π

tk), Z π tk = z(θ1,θ2) k

(X π

tk).

Question: How to approximate Ex

tk

h y(θ1,θ2)

k+1

(X π

tk+1)

i , Ex

tk

h y(θ1,θ2)

k+1

(X π

tk+1)∆W T k

i , and other similar quantities along the time grid?

K.W. Chau (CWI) SGBM for BSDEs 22 January 2018 4 / 20

Stochastic Grid Bundling Method

Non-nested Monte Carlo scheme

It starts with the simulation of M independent samples of (X π

tk)0≤k≤N,

denoted by (X π,m

tk

)1≤m≤M,0≤k≤N. The simulation is only performed once. The terminal values for each path are: y (θ1,θ2),R,I

N

(X π,m

tN

) = Φ(X π,m

tN

), z(θ1,θ2),R

N

(X π,m

tN

) = rΦ(X π,m

tN

)σ(tN, X π,m

tN

), m = 1, . . . , M.

K.W. Chau (CWI) SGBM for BSDEs 22 January 2018 5 / 20

Recurring steps in time (I)

Non-nested Monte Carlo scheme Regress-later

The least-squares regression technique for functions is performed on the random variable X π

tk+1

Then we use the (analytical) expectation of the resulting approximation in our algorithm. This will remove the ”statistical” error at the regression step. To ensure the stability of our algorithm, the regression coefficients must be bounded above. It means that an error notion should be given by the program when the Euclidean norm of any regression coefficient vector is greater than a predetermined constant L.

K.W. Chau (CWI) SGBM for BSDEs 22 January 2018 6 / 20

Regress now and Regress later

Regress now B @ η1(X π,1

tk )

ηQ(X π,1

tk )

... η1(X π,#B

tk

) ηQ(X π,#B

tk

) 1 C A B @ α1 . . . αQ 1 C A = B @ g(X π,1

tk+1)

. . . g(X π,#B

tk+1 )

1 C A E[g(X π

tk+1)|X π tk = x] ⇡ Q

X

l=1

αlηl(x) Regress later B @ η1(X π,1

tk+1)

ηQ(X π,1

tk+1)

... η1(X π,#B

tk+1 )

ηQ(X π,#B

tk+1 )

1 C A B @ α1 . . . αQ 1 C A = B @ g(X π,1

tk+1)

. . . g(X π,#B

tk+1 )

1 C A E[g(X π

tk+1)|X π tk = x] ⇡ Q

X

l=1

αlE[ηl(X π

tk+1)|X π tk = x]

K.W. Chau (CWI) SGBM for BSDEs 22 January 2018 7 / 20

Recurring steps in time (II)

Non-nested Monte Carlo scheme Regress-later Localization (Bundling)

At each time period, all paths are bundled into Btk(1), . . . , Btk(B) (almost) equal-size, non-overlapping partitions based on the result of (X π,m

tk

). We perform the approximation separately at each bundle.

K.W. Chau (CWI) SGBM for BSDEs 22 January 2018 8 / 20

Bundling

K.W. Chau (CWI) SGBM for BSDEs 22 January 2018 9 / 20

Formulation

Specifically, the bundle regression parameters αk+1(b), βk+1(b), γk+1(b) are defined as αk+1(b) = arg min

α2RQ

PM

m=1(p(X π,m tk+1)α y(θ1,θ2),R,I k+1

(X π,m

tk+1))21Btk (b)(X π,m tk

) PM

m=1 1Btk (b)(X π.m tk

) βi,k+1(b) = arg min

β2RQ

PM

m=1(p(X π,m tk+1)β z(θ1,θ2),R i,k+1

(X π,m

tk+1))21Btk (b)(X π,m tk

) PM

m=1 1Btk (b)(X π.m tk

) γk+1(b) = arg min

γ2RQ

PM

m=1(p(X π,m tk+1)γ fk+1(y(θ1,θ2),R,I k+1

, z(θ1,θ2),R

k+1

))21Btk (b)(X π,m

tk

) PM

m=1 1Btk (b)(X π.m tk

)

K.W. Chau (CWI) SGBM for BSDEs 22 January 2018 10 / 20

Formulation (cont.)

The approximate functions within the bundle at time k are defined by : z(θ1,θ2),R

r,k

(b, x) = θ1

2 (1 θ2)Ex tk

h p(X π

tk+1)

i βk+1(b) + θ1

2 Ex tk

∆Wr,k ∆k p(X π

tk+1)

• (αk+1(b) + (1 θ2)∆kγk+1(b)),

y(θ1,θ2),R,0

k

(b, x) = Ex

tk

h p(X π

tk+1)

i αk+1(b), y(θ1,θ2),R,i

k

(b, x) = ∆kθ1fk(yπ,R,i1

k

(x), zπ,R

k

(x)) + hk(x), hk(b, x) = Ex

tk

h p(X π

tk+1)

i (αk+1(b) + ∆k(1 θ1)γk+1(b)), i = 1, . . . , I, with y(θ1,θ2),R,I

k

(x) =

B

X

b=1

1x2Btk (b)y(θ1,θ2),R,I

k

(b, x) and similarly for z.

K.W. Chau (CWI) SGBM for BSDEs 22 January 2018 11 / 20

Refined Regression

Theorem 1

Assume that for a real function v that is bounded in a compact set and R v2(x)ν(dx)  1, then ˆ ES ZZ |v(y) ˜ v(x, y)|2ν(dx, dy)

•  ϑ(L0)

ˆ E[1S] ˆ E "X

B2B

Z

B

Z ν(dx, dy)(log(PM

m=1 1B(X m)) + 1)Q

PM

m=1 1B(X m)

# + 8 ˆ E[1S] ˆ E "X

B2B

Z

B

Z ν(dx, dy)( inf

φ2H sup x2B

E ⇥ |v(Y ) φ(Y )|2|X = x ⇤ ^ L0) # +ˆ ES ZZ |v(y) ˜ v(x, y)|2(1 1A(y))ν(dx, dy)

• K.W. Chau (CWI)

SGBM for BSDEs 22 January 2018 12 / 20

Example 1

We consider the BSDE: 8 < : dXt = dWt, dYt = (YtZt Zt + 2.5Yt sin(t + Xt) cos(t + Xt) 2 sin(t + Xt))dt + ZtdWt, with the initial and terminal conditions x0 = 0 and YT = sin(XT + T). The exact solution is given by (Yt, Zt) = (sin(Xt + t), cos(Xt + t)). The terminal time is set to be T = 1 and (Y0, Z0) = (0, 1). We run the examples with the basis functions η(x) = (1, x, x2) and bundle based on the value of x.

K.W. Chau (CWI) SGBM for BSDEs 22 January 2018 13 / 20

Test Case Example θ1 θ2 I M N B L D 1 0.5 0.5 4 22J 2J 2J 100 E 1 0.5 0.5 4 22J 2J 2J 10000 F 1 0.5 0.5 4 22J 2J 2J

• |Y0 y(θ1,θ2),R

(x0)| J 2 3 4 5 D NA 9.2870 ⇥ 102 1.0114 ⇥ 101 8.1415 ⇥ 102 E 29.2228 7.8601 ⇥ 101 3.9639 ⇥ 101 5.2388 ⇥ 102 F 2.2154 ⇥ 1015 1.9059 ⇥ 1056 3.4731 ⇥ 101 5.8511 ⇥ 102 J 6 7 8 D 3.9920 ⇥ 103 1.5486 ⇥ 102 NA E 1.1931 ⇥ 102 1.2395 ⇥ 102 1.4347 ⇥ 103 F 2.0485 ⇥ 103 6.8277 ⇥ 103 2.6705 ⇥ 103

K.W. Chau (CWI) SGBM for BSDEs 22 January 2018 14 / 20

Example 2: European option

We consider a market where the assets satisfy: dSi,t = µiSi,tdt + σiSi,tdBi,t, 1  i  q with Bt being a correlated q-dimension Wiener process with dBi,tdBj,t = ρijdt. The parameters ρij form a symmetric matrix ρ, ρ = B B B @ 1 ρ12 ρ13 · · · ρ1q ρ21 1 ρ23 · · · ρ2q . . . . . . . . . . . . ρq1 ρq2 ρq3 · · · 1 1 C C C A , and we assume it is invertible. By performing a Cholesky decomposition

• n ρ such that LLT = ρ, we relate Bt to standard Brownian motion

Bt = LWt.

K.W. Chau (CWI) SGBM for BSDEs 22 January 2018 15 / 20

Example 2: European option (cont.)

For a European option with terminal payoff g(St), a replicating portfolio Yt, containing ωi,t of asset Si,t and Zt = (ω1,tσ1S1,t, . . . , ωq,tσq, Sq,t)L solve the BSDE, ( dYt =

• rYt ZtL1 µr

σ

• dt + ZtdWt;

YT = g(ST), where µr

σ

• =

⇣

µ1r σ1 , · · · , µqr σq

⌘T . In this numerical test, we use the 5-dimensional example from [Reisinger and Wittum, 2007].

K.W. Chau (CWI) SGBM for BSDEs 22 January 2018 16 / 20

Example 2: European option (cont.)

We would consider a European weighted basket put option for 1 year in

• ur test, therefore, the payoff function g is given by

g(s) = 1

5

X

i=1

wisi !+ , where (w1, w2, w3, w4, w5) = (38.1, 6.5, 5.7, 27.0, 22.7). The reference price is given as 0.175866. We use equal-partitioning and sorting the paths according to P5

i=1 wiX m tp,i.

The regression basis is pk(x) = ⇣P5

i=1 wixi

⌘k1 for k = 1, . . . , K.

K.W. Chau (CWI) SGBM for BSDEs 22 January 2018 17 / 20

Test Case Example θ1 θ2 I M N B L K AA 2 0.5 0.5 4 212 10 22J

• 3

AB 2 1

• 211

10 22J

• 2

|Y0 y(θ1,θ2),R (x0)| J 1 2 AA 2.0321 ⇥ 103 2.2567 ⇥ 103 1.9883 ⇥ 103 AB 2.9314 ⇥ 103 1.8934 ⇥ 103 2.2151 ⇥ 104

K.W. Chau (CWI) SGBM for BSDEs 22 January 2018 18 / 20

References

Reisinger, C. and Wittum, G. (2007). Efficient hierarchical approximation of highdimensional option pricing problems. SIAM Journal on Scientific Computing, 29(1):440–458. Zhao, W., Li, Y., and Zhang, G. (2012). A generalized θ-scheme for solving backward stochastic differential equations. Discrete and Continuous Dynamical Systems - Series B, 17(5):1585–1603.

K.W. Chau (CWI) SGBM for BSDEs 22 January 2018 19 / 20

Thank You

K.W. Chau (CWI) SGBM for BSDEs 22 January 2018 20 / 20