On the higher-order weak approximationof SDEs

Transcription

1 On the higher-order weak approximationof SDEs Syoiti Ninomiya Tokyo Institute of Technology The 3rd Workshop on numerical methods for solving the filtering problem and high order methods for solving parabolic UK(2012/9/24 28)

2 Outline Abstract Algorithm for barrier options Semi-closed form solutions to SDEs

3 Background: Objective: A new higher-order weak approximation scheme based on [Kusuoka 01] and [Lyons and Victoir 04] Construction of Concrete higher-order weak approximation algorithms that are: Versatile, (applicable to a broad class of SDEs) Easy to use, (Blackbox algorithm)

4 Current status for European Option type problems: Two kinds of schemes of order 2 (say Alg 1 [Victoir &N 08] and Alg 2 [Ninomiya &N 09]) Both work in practice. General extrapolation method for Alg 1 [Oshima, Teichmann and Veluscek 09] Enables arbitraly order weak approximation This talk is on: Higher order algorithms for barrier option pricing problem. New SCF to Asian option under Heston model.

5 References 1/3: Kusuoka, S.: Approximation of Expectation of Diffusion Process and Mathematical Finance. In: T. Sunada (ed.) Advanced Studies in Pure Mathematics, Proceedings of Final Taniguchi Symposium, Nara 1998, vol. 31, pp (2001) Kusuoka, S.: Approximation of Expectation of Diffusion Processes based on Lie Algebra and Malliavin Calculs. Advances in Mathematical Economics 6, (2004) Lyons, T., Victoir, N.: Cubature on Wiener Space. Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences 460, (2004)

6 References 2/3: N. and Victoir, N.: Weak Approximation of Stochastic Differential Equations and Application to Derivative Pricing. Applied Mathematical Finance 15, (2008) Ninomiya, M. and N.: A new weak approximation scheme of stochastic differential equations by using the Runge Kutta method Finance and Stochastics 13, (2009) Oshima, K., Teichmann, J. and Veluscek, D.: A new extrapolation method for weak approximation schemes with applications Preprint: arxiv: v1[math.pr] (2009)

7 References 3/3: Kusuoka, S.: Approximation of Expectation of Diffusion Processes with Dirichlet boundary International Workshop on Mathematical Finance Topics on Leading-edge Numerical Procedures and Models (16 18 Feb 2010, Tokyo)

8 The Problem : Numerical calculation of E[f(X(T, x))], where X(t, x) = x + d j=0 t 0 V j C b (RN ;R N ) V j (X(s, x)) db j (s) B(t) = (B 0 (t), B 1 (t),...,b d (t)), B 0 (t) = t, (B 1 (t),...,b d (t)) : d-dim Std. BM, (1) db j (s) : Stratonovich integral.

9 Two approaches: PDE method where L = V 0 +(1/2) d i=1 V 2 i. u (t, x) = Lu, u(0, x) = f(x). t Probabilistic method Simulation Step 1. Discretize X(t, x) and obtain X n (t, x). n = {partitions of [0, T]} Euler Maruyama, Higher order (Milstein, Kusuoka,..) Step 2. Integrate f(x n (T, x)) over D(n)-dimensional domain [0, 1) D(n) by MC, QMC, etc. D(n) depends on the discretization scheme We consider only the probabilistic method here.

10 Simulation MC and QMC Step. 2 of the simulation is the numerical integration: E[f (X n (T, x))] = F(a 1,..., a D(n) ) da 1 da D(n) [0,1) D(n) Calc. RHS by using MC or QMC.

11 MC and QMC W : r. v./(ω,f, P), M Z >0 MC(W, M) := 1 M W k, where{w i } M i=1 M : iid s. t. W 1 W QMC(W, M) := 1 M k=1 M k=1 W(ω k ), {ω i } i=1 : deterministic sequence (LDS) MC(W, M) : r. v. QMC(W, M) R.

12 Two types of approximation errors in simulation: 1. Discritization error E[f(X(T, x))] E[f (X n (T, x))] 2. Integration error MC(f (X n (T, x)),m)(ω) E[f (X n (T, x))] or QMC(f (X n (T, x)),m) E[f (X n (T, x))]

13 Two important remarks(1/2): Integration error and MC By CLT, ( MC ( f (X n (T, x)),m ) N When we proceed simulation E[f (X n (T, x))], Var[f (X n (T, x))] M Var[f(X(T, x))] Var[f(X n (T, x))] Remark 1 As long as we use MC, the number of sample points M needed to attain the given accuracy is independent of n and discretizing algorithm (Euler Maruyama or Kusuoka etc.). ).

14 Two important remarks(2/2) Integration error and QMC There exist sequences which satisfy C f,n > 0 M Z >0 QMC(f (X n (T, x)),m) E[f (X n (T, x))] (log M) D(n) Cf,D(n). M Remark 2 In contrast to the MC case, the number of sample points M needed by the QMC to attain the given accuracy depends heavily on the dimension of integration D(n). Smaller the dimension, smaller number of samples are needed. n d Euler Maruyama, D(n) = n (d + 1) N. & Victoir, 2n d Ninomiya, & N.

15 Order 1: Euler Maruyama scheme X (EM),n (0, x) = x, ( ) ( ) k + 1 kn X (EM),n n, x = X (EM),n, x + T n d i=0 Ṽ i ( X (EM),n ( k n, x )) Z i k+1, where, j Z k j = T/n, if k = 0, iid. r. v. N(0, 1), if k {1,..., d}, Ṽ i k (y) = V i 0 (y)+ 1 2 j V j V i (y) j if k = 0, if k {1,..., d}. V i k

16 Approx. Error of Euler Maruyama scheme [( E f X(T, x) )] [ ( E f X (EM),n (T, x) )] = ( O n 1) = O ( t) when f : bdd.& measurable and{v i } d : Unif. Hörmander Cond. [Bally & Talay 96][Kohatsu-Higa 00] i=0 Euler Maruyama scheme is an order 1 scheme.

17 Intuitive explanation of the scheme (1/3) ( ) P X t f (x) := E[f (X(t, x))], f C b (RN ) L := V d V 2 2 j. j=1

18 Intuitive explanation of the scheme (2/3) Applying Ito formula repeatedly, we obtain E[f (X(t, x))] = ( P X t f ) (x) = f(x)+.. = f(x)+ = t 0 t 0 ( ) P X s1 Lf (x) ds 1 { s1 ( (Lf)(x)+ ) } P X s2 L 2 f (x) ds 2 ds 1 0 n ( ) (tl) i f (x)+ 1 i! n! i=0 t 0 (t s) n( P X s L n+1 f ) (x) ds.

19 Intuitive explanation of the scheme(3/3) n ( ) (tl) i Observation: f (x) gives a nth order approx. of E[f(X(t, x))]. i! i=0 Slogan: Construct a random variable Ξ s. t. E[Ξ] = n ( ) (tl) i f (x). i! i=0

20 Free Lie Algebra Non-commutative algebra (1) A :={v 0,...,v d } : alphabet A := ( k=1 A k) {1} : free monoid on A For w = w 1 w k A, (w i A) w := k, w := w +card({1 i w ; w i = v 0 }), A m :={w A w = m}, A m :={w A w m}. Concatenation product: For u = u 1 u k, v = v 1 v l A, uv := u 1 u k v 1 v l.

21 Non-commutative algebra (2) R A := a w w a w R :R-algebra of formal series with basis A, w A R A := a w w R A k N s.t. a w = 0 if w k w A : freer-algebra with basis A, (P, w) :=a w, for P = a w w R A, w A (PQ, w) := (P, u)(q, v), for P, Q R A, w A. uv=w u,v A

22 Non-commutative algebra (3) Projection and Truncation: j m (P) := (P, w)w w m P k := (P, w)w w =k Homogeneous component: R A m := { P R A (P, w) = 0 if w m }

23 Free Lie algebra: [P, Q] := PQ QP for P, Q R A, { } J A := K R A subr-module A K, [x, y] K x, y K, L R (A) := :R-coefficients free Lie algebra on A K J A K L R ((A)) := { P R A s.t. P k L R (A), k N }.

24 Logarithm and exponential: For P R A s.t. (P, 1) = 0 For Q R A s.t. (Q, 1) = 1 exp(p) := 1+ log(q) := k=1 P k k!. ( 1) k 1 (Q 1) k. k k=1 log(exp(p)) = P and exp(log(q)) = Q.

25 Hausdorff product: For z 1, z 2 L R ((A)), z 1 z 2 := log(exp(z 1 ) exp(z 2 )). (z 1 z 2 ) z 3 = log(exp(z 1 ) exp(z 2 ) exp(z 3 )) = z 1 (z 2 z 3 ) =: z 1 z 2 z 3 z 2, z 1 L R = z 2 z 1 L R ((A)) the Baker Campbell Hausdorff Dynkin formula.

26 L R (A) andr A Element ofl R Vector field generated by V 0,...,V d Elements ofr A Differential operator generated by V 0,...,V d

27 Resurgence and rescaling Resurgence operator Φ: Φ(1) := id, Φ(v i1 v in ) := V i1 V in Rescaling operator: For s> 0, Ψ s :R A R A is defined as: Ψ s P m = s m/2 P m where P m R A m. m=0 m=0 Example: ( Φ (Ψ s v )) 2 [v 1, v 2 ] = sv 0 + s 2 [V 1, V 2 ]

28 Notation: exp(v)(x) For a smooth vector field V, (i.e. V C b (RN ;R N )) exp(v)(x) := y(1) where y(t) is a solution to the ODE: y(0) = x, dy(t) dt = V(y(t))

29 Order m Integration scheme: IS(m) g IS(m) def g : C b (RN ;R N ) ( R N R N), C m > 0 s.t. W C b (RN ;R N ) sup g(w)(x) exp(w)(x) Cm ( W C m+1) m+1 x R N IS(m) Set of order m ODE solver We need to integrate, for example: sv 0 + s 2 (V 1 + +V d ) Not necessarily s-linear.

30 R A,L R ((A))-valued random variables Topology:R A R Direct product topology R A -valued andl R ((A))-valued probability theory can be considered. (Ito formula, etc.)

31 Approximation theorem (1/2) m 1, M 2, Z 1,...,Z M :L R ((A))-valued random variables s. t. Z i = j m Z i for i = 1,...,M, E[ j m Z i 2 ]< for i = 1,...,M, M E exp a Φ(Ψ s (Z j )) C m+1 < for any a> 0. j=1

32 Approximation theorem (2/2) = p [1, ), g 1,..., g M IS(m), C m,m > 0 s. t. sup g1 (Φ(Ψ s (Z 1 ))) g M (Φ(Ψ s (Z M )))(x) x R N exp(φ(ψ s (j m (Z M Z 1 ))))(x) L p C m,m s (m+1)/2 for s (0, 1] where C m,m depends only on m and M. f g(x) := f (g(x))

33 Cor. to the Approximation theorem Q (s) gives (m 1)/2-order weak approximation. Let Q (s) for s (0, 1] be ( Q(s) f ) (x) := E[f (g(φ(ψ s (Z 1 ))) g(φ(ψ s (Z M )))(x))], where f C b (RN ;R) and g IS(m) then C> 0, P s (f) := E[f(X(t, x))] P s f Q (s) f Cs (m+1)/2 grad(f) (Remark: When{V 0,..., V d } finitely generated.)

34 Algorithm 1 (Victoir N. (2004)) (Λ i, Z i ) i {1,,n} : 2n indep. r. v., i P(Λ i =±1) = 1 2, Z i N(0, I d ). {X (Alg.1),n } k k=0,...,n : a family of r. v. defined as: X (Alg.1),n 0 := x, X (Alg.1),n := (k+1)/n ( V exp 0 2n ( V exp 0 2n ) exp ( Z 1 k V 1 n ) ) exp ( Z d k V d n ) exp exp ( Z d k V d ( Z 1 k V 1 ) ( ) V n exp 0 (Alg.1),n 2n X Λ k k = +1, ) ( ) V n exp 0 (Alg.1),n X Λ k = 1. 2n k

35 Extrapolations of Algorithm 1 To an arbitrary order [Oshima, Teichmann and Veluscek 09] To the 6th order [Fujiwara 06]

36 General framework of Algorithm 2 (1) Remind the Slogan Find the Ξ written as: where logξ = Z 1 Z 2 Z M d Z j = c j v 0 + S i j v i, j {1,...,M} i=1 c j Rs.t. c 1 + +c M = 1 E [ S i j Si j ] = Rjj δ ii S i j N(0, R jj ) (i {1,...,d}, j {1,...,M})

37 General framework of Algorithm 2 (2) The slogan is equivalent to: E[j m (exp(z 1 ) exp(z M ))] = j m exp v d i=1 v 2 i (2) The problem: Find real numbers{c j } M j=1,{ R ij that satisfy (2). }1 i j M #unknown vars = 1 M(M + 3) 2

38 Main tools (Theorem 1): For the LHS of (2) If n w (i) is odd for some i {1,...,d}, then C(w) = 0. If n w (i) is even for every i {1,...,d}, then C(w) = k=(k1,...,k M ) K l (M) d p=1 1 k 1! k M! M (c j ) Nw (0,j, k) j=1 {d ij} 1 i j M e(n w (p,1, k),...,n w (p,m, k)) M 2 M i=1 d j=1 ii ( N w ( p, j, k )! ) 1 i j M(d ij!) 1 i j M R d ij ij (3)

39 Main tools (Theorem 2): For the RHS of (2) Let A 0 ={v 0, v 1 v 1, v 2 v 2,...,v d v d } A. Then exp v d v 2 i 2 = i=1 w=w 1 w l w 1,...,w l A w l l! w, that is, 1 if w A 0 C(w) = 2 w l l! 0 otherwise. (4)

40 Algebraic Relations Algebraic relations between{c j } M j=1,{ } R ij 1 i j M Using following 3 results, those relations are obtained.

41 An example of Algorithm 2 [Ninomiya N. (2009)] When (m, M) = (5, 2), c 1 = 2(2u 1), c 2 = 1± 2 R 22 = 1+u± 2(2u 1), R 12 = u R 11 = u Becomes 1-dimensional ideal. for some u 1/2. 2(2u 1) 2 2(2u 1) 2

42 Partial results (1): The case m 6: (m, M) = (7, 2): ideal becomes trivial (i.e. =C[c 1, c 2, R 11, R 12, R 13 ]) (m, M, d) = (7, 3, 1): ideal becomes trivial.

43 Partial results (2): The case (m, M, d) = (7, 3, 2) and v 0 free: The algebraic relation: 14R R 13 13, 4R R R 11 15, 36R R , 72R 22 R R R R 22 17, 288R 23 R 33 8R R 23 46R , 12R 23 22R R , 8R R 22 R R R 22 27, 22R R R R 22 57, 144R R R R Unfortunately, the solution is: R 33 = 5± R

44 Partial results (3): The case (m, M, d) = (7, 4, 3) and v 0 free: Over 1-month symbolical calculation cannot find the answer...

45 We have some more results (Theorem 3): A = i=0 A i, : A A A : Shuffle product. (A, ) becomes an algebra (shuffle algebra). C( ) is ring homomorphism. Cor. C : (A, ) R[c j, R ij,(1 i j M)] C((0) n 0 (11) (ll)) = C(0) n 0 C(11) l

46 An ( example ) of Algorithm 2 [Ninomiya N. (2005)] j Z, i,k i {1,...,d},j {1,2} k {0,...,n 1} ( ) ( )( ) Z 1 i,k 1/2 1/ 2 η Z 2 = i,k 1/2 1/ 1 i,k 2 η 2, whereη j i.i.d. N(0, 1). i,k i,k {X (Alg.2),n } k k=0,...,n : a family of r. v. defined as: X (Alg.2),n 0 := x, X (Alg.2),n := (k+1)/n 1 exp 2n V 0 + d i=1 Z 1 i,k V i n exp 1 2n V 0 + d i=1 Z M i,k n V i X(Alg.2),n k/n

47 Theorem Both X (Alg.1),n and X (Alg.2),n are of order 2. Recent result by Kusuoka If Q (s) is constructed by X (Alg.1),n or X (Alg.2),n then C> 0 s.t. (P s f)(x) ( Q (s) f ) (x) = Cs (m+1)/2 + O ( s (m+3)/2)

48 D(n) D(n) : dimension of integration domain n d Euler Maruyama, D(n) = n (d + 1) Algorithm 1 n 2 d Algorithm 2

49 The Runge Kutta method How to calc. exp(z i )(x)? Lucky case: We can get exact form of exp(z i )(x). Often the case with Algorithm 1. Otherwise: Forced to proceed with nuerical approximation. Good news: Theorem [Ninomiya N.] Classical order m Runge Kutta methods belongs tois(m)

50 MC or QMC with Algorithms 1 and 2 W: a set of ODEs -valued r. v. by Algorithm 1 or 2 MC or QMC: Draw a set of ODEs W(ω) from W and obtain (something like) exp(w(ω))(x) numerically Iterate the step above and calculate the average: 1 M M exp(w(ω i ))(x) i=1

51 Advantages of the approache Free from symbolical calculation. Calc. in group is easy. Numerical ODE solver works (by the 2nd th m) Calc. in algebra is difficult. Huge symbolical calc. Simultaneous distributions of multiple integrations of BMs are not knowm. (except for the 2nd order.) Universal (applicable to non-commutative{v i } d i=0 ) Naïve Ito-Taylor expansion with the Runge Kutta suffers from the non-commutativity.

52 : Pricing of Asian option under the Heston model: Y 1 (t, x) = x 1 + Y 2 (t, x) = x 2 + W 1, W 2 =ρt t Asian Option: Y 3 (t, x) = t 0 t 0 t 0 µy 1 (s, x) ds + t α(θ Y 2 (s, x)) ds + 0 Y 1 (s, x) Y 2 (s, x) dw 1 (s), t 0 β Y 2 (s, x) dw 2 (s), ( ) Y3 (T, x) Y 1 (s, x) ds, Payoff = max K, 0. T

53 Discretization Error Discretization Error and Num. of Partitions Euler-Maruyama Euler-Maruyama + Romberg Ninomiya-Victoir Ninomiya-Victoir + Romberg New Method New Method + Romberg O(1/n 2 ) O(1/n 3 ) 5e-05 5e-06 1e-04 Error 1e-05 1e-06 1e Num. of Partitions

54 Convergfence Error from quasi-monte Carlo and Monte Carlo 2σ(for MC) and Absolute difference(for QMC) e-04 1e-05 QMC : New Method + Romberg, 1/ t=2 QMC : Ninomiya-Victoir + Romberg, 1/ t=4 QMC : New Method, 1/ t=10 QMC : Ninomiya-Victoir, 1/ t=12 QMC :Euler-Maruyama + Romberg, 1/ t=16 MC : Euler-Maruyama 5e-05 5e-06 1e e+06 1e+07 Num. of Sample points

55 Overall performance comparison: #Partition, #Dim, #Sample, and CPU time required for 10 4 accuracy. Method #Partition #Dim #Sample CPU time (sec) E-M + MC E-M + Extrpltn + QMC N-V + QMC N-V + Extrpltn + QMC KNN + QMC KNN + Extrpltn + QMC

56 Algorithm for barrier options Semi-closed form solutions to SDEs (1/2): Existence of the asymptotic expansion of the errors of Algorithms 1 and 2 [Kusuoka] Extrapolation of Algorithm 1 To order 6 [Fujiwara 06] To arbitrary order [Oshima, Teichmann and Veluscek 09] SPDE case [Teichmann] Levy driven case [Tanaka and Kohatsu-Higa 09]

57 Algorithm for barrier options Semi-closed form solutions to SDEs (2/2): Semi-closed form solutions to generalized SABR models by Algorithm 1 [Bayer, Friz and Loeffen 10] Algorithms for Barrier-option pricing case [Kusuoka 10][Kusuoka, Ninomiya and N. 12] Semi-closed form solutions to Heston models by Algorithms 1 and 2.

58 Algorithm for barrier options Semi-closed form solutions to SDEs Barrier option problem : Calculate E [ f (X(t, x)), min X(s, t)>0 s [0,t] ] (5) numerically, where X(t, x) is a diffusion starting from x.

59 Algorithm for barrier options Semi-closed form solutions to SDEs The 1 dimensional cas (i.e. X : [0, 1] R Ω R) Time horizon = 1. t t X(t, x) = x + V 0 (X(s, x)) ds + db(s) 0 0 [ ] (6) (P t f)(x) = E f (X(t, x)), min X(s, x)>0 s [0,t]

60 Algorithm for barrier options Semi-closed form solutions to SDEs killing function k k : (0, 1] [0, ) R [0, 1] measurable func. Naive killing: 0, if y> 0, k(s, x, y) = 1, otherwise Standard killing: exp( 2xy/s) if y> 0, k(s, x, y) = 1, otherwise Improved standard killing: exp(2xy/s)κ(s, x, y (2) ) if y> 0, k(s, x, y) = 1, otherwise

61 Algorithm for barrier options Semi-closed form solutions to SDEs Approximating operator Q (s) ( ) Q(s) f (x) := [ ( ( E f F X N (n/n) ; x, s))( ( 1 k s, x, ( F X N (n/n); x, s)))] where F is the discretization scheme (ex. Euler-Maruyama, Alg-1, Alg-2).

62 Algorithm for barrier options Semi-closed form solutions to SDEs Naive killing does not work 1 V 0 (x) = 1+x Discr Err:E[f(X(1)), X(t)>0, t<1], dx(t,x)=(1+x(t,x))dt+db(t), f:digital call, naive killing, g=1, M=100M 1 N-N N-V Euler-Maruyama 1/x /sqrt(2) 0.25x -1/ discretization error e-05 1e e+06 # of partitions

63 Algorithm for barrier options Semi-closed form solutions to SDEs Naive killing does not work 2 V 0 (x) = cos(x)+3/2 Discr Err:E[f(X(1)), X(t)>0, t<1], dx(t,x)=(cos(x(t,x))+2/3)dt+db(t), f:euro call, naive killing, g=1, M=100M 1 N-N N-V Euler-Maruyama 1/x /sqrt(2) 0.25x -1/ discretization error e-05 1e e+06 # of partitions

64 Algorithm for barrier options Semi-closed form solutions to SDEs Standard killing V 0 (x) = 1+x Discr Err:E[f(X(1)), X(t)>0, t<1], dx(t,x)=(1+x(t,x))dt+db(t), f:digit. call, gamma=1.0, Standard Killing 1 N-V + kappa N-V N-N E-M 0.1 N-N, naive killing E-M, naive killing 1/x 1/x 2 1/x discretization error e-05 1e e+06 # of partitions

65 Algorithm for barrier options Semi-closed form solutions to SDEs Standard killing V 0 (x) = cos(x)+3/2 Discr Err:E[f(X(1)),X(t)>0, t<1], dx(t,x)=(cos(x(t,x))+3/2)dt+db(t), f:euro call, g= N-V + kappa g=1.0 N-V + g=1.0 E-M g=1.0 N-N g=1.0 1/x 1 1/x 2 1/x 3 discretization error # of partitions

66 Algorithm for barrier options Semi-closed form solutions to SDEs Standard killing V 0 (x) = cos(x)+3/2 Discr Err:E[f(X(1)),X(t)>0, t<1], dx(t,x)=(cos(x(t,x))+3/2)dt+db(t), f:euro call 10 N-V + kappa g=1.0 N-V + g=1.0 N-V + kappa g=1.5 N-V + g=1.5 E-M g=1.0 1 N-N g=1.0 1/x 1/x 2 1/x 3 discretization error # of partitions

67 Algorithm for barrier options Semi-closed form solutions to SDEs Observation Q (s) seems versatile and promising. Standard killing works. But, cannot see the Improved effect. Again,γparadox arises. Straightforward extension of the formula to multi-dimensional cases is possible

68 Algorithm for barrier options Semi-closed form solutions to SDEs Semi-closed form solution (SCF) When all ODEs that arise in Alg. 1 have closed form solutions, the algorithm is called semi-closed form solution (SCF in the following). Bayer-Friz-Loeffen 10 [BFL10] SCFs to SABR and generalized SABR models. Transform the problem by adding constant drift: B Q (t) =γt + B P (t) γ R SABR: One of the two important SV models in finance.

69 Algorithm for barrier options Semi-closed form solutions to SDEs [Kubo N. (2012)] SCFs to the Asian option problem under Heston model by Alg 2. Heston model The other important SV model in finance. The squared volatility process is of CIR type. Alg 2 ODEs are more complex than Alg 1 case. ODEs in Alg 1 exp ( tξ i V i ) x ODEs in Alg 2 exp ( t 2 V 0 + d i=1 tξi V i ) x

70 Algorithm for barrier options Semi-closed form solutions to SDEs Asian Option under Heston model: Numerical Calc. of E [ max ( X 3 (T)/T K, 0 )] where X(t) = (X 1 (t), X 2 (t), X 3 (t)), X(0) = x R 3 dx 1 (t) =µx 1 (t) dt + X 1 (t) X 2 (t) db 1 (t) dx 2 (t) =α(θ X 2 (t)) dt +β X 2 (t) ( ρ db 1 (t)+σ db 2 (t) ) dx 3 (t) = X 1 (t) dt ρ 2 +σ 2 = 1,ρ ( 1, 1) and the Feller condition: 2αθ β 2 > 0, which makes X 2 (t) strictly positive, is satisfied.

71 Algorithm for barrier options Semi-closed form solutions to SDEs Derivation of the SCF (1/3) Consider the following equivalent measure Q: g 1 (t, X) = X 2 (t) G 2 X 2 (t) g 2 (t, X) = 1 σ H X 2 (t)+ (I/β)+ρG X 2 (t) g(t, X) = (g 1 (t, X), g 2 (t, X)), B Q (t) = g τ (s, X) ds + B(t) 0 t 2 L(t) = exp g τ j (s, X) dbj (s) 1 t g τ (s, X) 2 ds 2 Q(A) = E[1 A (L(T)] 0 j=1 where G =βρ/4 µ, H =ρ/2 α/β, I =αθ β 2 /4 andτis the 1st hitting time of g to M. t 0

72 Algorithm for barrier options Semi-closed form solutions to SDEs Derivation of the SCF(2/3) Proceed the calculation considering as if g τ = g, the problem is transformed as follows: V Q 0 (y) =.t (0, 0, y 1, C 1 y 2 + C 2 + C 3 /y 2 ) V 1 (y) = t( y 1 y2,βρ y 2, 0, g 1 (t, y) ) V 2 (y) = t( 0,βσ y 2, 0, g 2 (t, y) ) and 2 dx(t) = V Q (X(t)) dt + V j (X(t)) db j (t) Q j=1 where C 1 = 1/8+H 2 /(2σ 2 ), C 2 = G/2 α/4+h(i/β+ρg)/σ 2, C 3 = (G 2 I/2+(I/β+ρG) 2 /σ 2 )/2 and X 4 (t) := log L(t).

73 Algorithm for barrier options Semi-closed form solutions to SDEs Derivation of the SCF(3/3) At last we get: ( ( s exp t 2 V Q 0 + sz 1 V 1 + )) sz 2 V 2 x ( ( x 1 exp Kt 2 sz1 + )) x 2 2 t (Kt + x 2 ) 2 = x 3 + sx 1 2 e sz 1 (x 2 /2K) t+ x2 /K ( K x2 exp sz1 u2) du /K 2 x 4 + A(t, x 2, 1/ x 2, 1/(Kt + ( ) Kt x 2 ))+log x2 + 1 where K = β x 2 2 (ρz 1 +σz 2 ) and A(u, v, w, z) is a polynomial of u, v, w and z.

74 Algorithm for barrier options Semi-closed form solutions to SDEs On the new SCFs to Heston model Two new SCFs to Asian derivatives under Heston model Another simulation method for CIR processes Need error estimation w.r.t M Work well in practical examples

75 Algorithm for barrier options Semi-closed form solutions to SDEs Why it works well? (in spite of the cheating) Rough calculation of P(X 2 (t) 0): Whenθ = 0.09,α = 2.0, β = 0.1,µ = 0.05,ρ= 1/2, x 2 = 0.09 and T = 1, Var(X 2 (1)) = and x 2 /Var(X 2 (1)) = 6.0. Then P(X 2 (T) 0) = 1 2π 6.0 e u2 /2 du

76 Both approaches are necessary because: PDE approach works only when L is elliptic, dimesion is small. Simulation is the last resort but (people believes) very time consuming. This presentation focuses on Simulation.

77 Bibliography I Christian Bayer, Peter Friz, and Ronnie Loeffen. Semi-Closed Form Cubature and Applications to Financial Diffusion Models. ArXiv e-prints, September 2010.