Notes on the BENCHOP implementations for the COS method

Transcription

1 Notes on the BENCHOP implementtions for the COS method M. J. uijter C. W. Oosterlee Mrch 29, 2015 Abstrct This text describes the COS method nd its implementtion for the BENCHOP-project. 1 Fourier cosine expnsion formul COS formul We explin the COS method to pproximte the Europen option vlue ux, t 0 = e r t E [ux T, T X t0 = x], 1 with t = T t 0. Here X t is the stte process, which cn be ny monotone function of the underlying sset price S t, for exmple, the scled log-sset price, X t = lns t /K, where K is the options strike price. We ssume continuous trnsitionl density, which is denoted by py x. In other words, B py xdy = PX T B X t0 = x, Borel subsets B. We omit the dependence on t for nottionl convenience. We rewrite ux, t 0 = e r t uy, T py xdy. 2 The numericl method is bsed on Fourier cosine series expnsions of the option vlue t time level T nd the density function, s we will show below. The resulting eqution is clled the COS formul, due to the use of Fourier cosine series expnsions. In the derivtion of the COS formul, we distinguish three different pproximtion steps. Step 1: For the problems we work on, the integrnd decys to zero s y ±. Becuse of tht, we cn truncte the infinite integrtion rnge of the Centrum Wiskunde & Informtic, Amsterdm, the Netherlnds, emil: mrjonruijter@gmil.com. Centrum Wiskunde & Informtic, Amsterdm, the Netherlnds, emil: c.w.oosterlee@cwi.nl, nd Delft University of Technology, Delft, the Netherlnds. 1

2 expecttion to some intervl [, b] without losing significnt ccurcy. This gives the pproximtion u 1 x, t 0 ; [, b] = e r t uy, T py xdy. 3 Step 2: Next, we consider the Fourier cosine series expnsions of the density function nd the option vlue t time T on [, b]: py x = nd uy, T = P k x cos U k T cos, 4 with series coefficients {P k nd {U k given by P k x = 2 nd U k T = 2 py x cos uy, T cos, 5 dy 6 dy, 7 respectively. in 1 indictes tht the first term in the summtion is weighted by one-hlf. eplcing the density function by its Fourier cosine series, interchnging summtion nd integrtion, using the definition of coefficients U k, nd truncting the series summtion, we obtin the next pproximtion u 2 x, t 0 ; [, b], N = 2 e r t P k xu k T. 8 Step 3: The coefficients P k x cn now be pproximted s follows P k x 2 = 2 {φ py x cos x e i dy := Φ k x. 9 {. denotes tking the rel prt of the input rgument. φ. x is the conditionl chrcteristic function of X T, given X t0 = x. The density function of stochstic process is usully not known, but often its chrcteristic function is known see [FO08]. For Lévy processes the chrcteristic function cn be represented by the Lévy-Khintchine formul nd there holds φω x = φω 0e iωx := ϕ levy ωe iωx. 10 2

3 Inserting the bove equtions into 8 gives us the COS formul for pproximtion of ux, t 0 : ûx, t 0 := u 3 x, t 0 ; [, b], N = 2 e r t Φ k xu k T = e r t {ϕ levy e i U k T. 11 Since the terms U k T re independent of x, we cn clculte the option vlue for mny vlues of x simultneously. 1.1 Fourier cosine coefficients cll nd put pyoff function We switch to the scled log-sset price process, X t := lns t /K. The pyoff functions of cll nd put options then red: gy = Ke y 1 + nd gy = K1 e y +, 12 respectively, where z + := mxz, 0 nd K denotes the strike price. The Fourier cosine coefficients of the option vlue t time T, we use uy, T = gy re known nlyticlly: U k T = 2 gy cos U cll k T = 2 K χ k0, b,, b ψ k 0, b,, b, dy, 13 U put k T = 2 K ψ k, 0,, b χ k, 0,, b, 0 b. 14 The functions χ k nd ψ k re given by: χ k z 1, z 2,, b = z2 z 1 nd ψ k z 1, z 2,, b = nd dmit the following nlytic solutions [ 1 χ k z 1, z 2,, b = 2 cos ψ k z 1, z 2,, b = 1+ + sin { [ sin z 2 z 2 e y cos z2 z2 z 1 cos dy e z2 cos e z 2 sin ] sin z 1 z 1 dy 15 z1 e z1 ] e z 1, 16, for k 0, z 2 z 1, for k =

4 2 Method prmeters The uthors of [FO08] provide the following rule-of-thumb for the computtionl domin for Europen options [ [, b] = ξ 1 L ξ 2 + ξ 4, ξ 1 + L ξ 2 + ] ξ 4, L [6, 10], 18 where ξ 1, ξ 2,... re the cumulnts of the underlying stochstic process. For the cumulnts of the Merton jump diffusion model nd Heston model, we refer to [FO08]. For some problems we further optimized the width of intervl [, b], such tht lower number of Fourier cosine coefficients, i.e. N, is needed to obtin the required ccurcy. In Tble 1 our choices for the computtionl domin re presented, which is either prescribed by vlue L or the intervl itself. Also the number of Fourier coefficients is reported. Tble 1: Method prmeters [, b] nd N. Problem 1 stndrd u Γ V [, b] L = 8 L = 8 L = 8 L = 8 N Problem 1 stndrd Americn Up-nd-out [ [, b] ln 50 K, ln 160 ] [ K ln 60 K, ln 140 ] K N Problem 1 chllenging u Γ V [ [, b] ln 60 K, ln 170 ] [ K ln 60 K, ln 170 ] [ K ln 60 K, ln 170 ] [ K ln 60 K, ln 170 ] K N Problem 1 chllenging Americn Up-nd-out [ [, b] ln 60 K, ln 160 ] [ K ln 160 K, ln 128 ] K N Problem 2 Europen 2 Americn 3 smooth [, b] L = 8 L = 8 [50, 360] N Problem [, b] L = 8 L = 6 L = 8 N The Blck-Scholes-Merton model for one underlying sset The sset price is modeled by geometric Brownin motion ds t = rs t dt + σs t dw t. 19 4

5 We switch to the scled log-sset price process, X t := lns t /K. We then del with the Brownin motion The corresponding chrcteristic function reds dx t = r 1 2 σ2 dt + σdw t. 20 ϕ levy ω = exp iωr 1 2 σ2 t 1 2 ω2 σ 2 t Europen option nd Greeks The COS formul to pproximte the Europen options is given by eqution 11. The Greeks cn then be pproximted by the following formuls: S ûx, t 0 = e r t i {ϕ levy e i U k T 1 S, 22 2 S ûx, t 2 0 = e r t i {ϕ levy e i i 2 U k T 1 S, 2 23 σ ûx, t 0 = e r t i {ϕ levy e iω ω 2 σ t U k T Bermudn nd Americn put A Bermudn-style option cn be exercised t fixed set of M erly-exercise dtes prior to the expirtion time T, t 0 < t 1 <... t m <... < t M = T, with timestep t := t m+1 t m. The uthors in [FO09] developed recursive lgorithm, bsed on the COS method, for pricing Bermudn options bckwrds in time vi Bellmn s principle of optimlity. The problem is solved bckwrds in time, with ux, t M = gx, cx, t m 1 = e r t E [ ux tm, t m X tm 1 = x ], ux, t m 1 = mx[gx, cx, t m 1 ], 2 m M, ux 0, t 0 = cx 0, t Function cx, t m 1 is clled the continution vlue nd is pproximted by the COS formul ĉx, t m 1 := e r t {ϕ levy e i The Fourier coefficients of the vlue function in 26 re given by U k t m = 2 uy, t m cos U k t m, 26 dy. 27 5

6 The recursive lgorithm to recover the coefficients U k t m mkes use of n FFT lgorithm for the fst computtion of mtrix-vector multiplictions see [FO09]. Incresing the number of erly-exercise dtes to infinity resembles n Americn option. We will use 4-point ichrdson-extrpoltion scheme on the Bermudn option vlues with smll M to pproximte Americn option vlues. Let ûx 0, t 0 ; M denote the Bermudn option vlue with M time steps. We clculte the extrpolted vlue, û x 0, t 0 ; M, by the following 4-point ichrdson-extrpoltion scheme with k 0 = 1, k 1 = 2, k 2 = 3 [ û x 0, t 0 ; M := ûx 0, t 0 ; 8M 56ûx 0, t 0 ; 4M ] + 14ûx 0, t 0 ; 2M ûx 0, t 0 ; M. 28 For the stndrd prmeters we compute û x 0, t 0 ; 4 nd for the chllenging prmeters û x 0, t 0 ; Brrier cll up-nd-out Similr s the Bermudn-style option we solve discrete brrier cll up-nd-out bckwrds in time with h = lnb/k ux, t M = gx, cx, t m 1 = e r t E [ ux tm, t m X tm 1 = x ] {, 0 x h, ux, t m 1 = cx, t m 1 x < h,, 2 m M, 29 ux 0, t 0 = cx 0, t 0. U cllup&out k T = 2 K χ k0, h,, b ψ k 0, h,, b, 30 Incresing the number of erly-exercise dtes to infinity resembles the continuous brrier option. We will use the following 4-point ichrdson-extrpoltion scheme with k 0 = 1/2, k 1 = 1, k 2 = 3/2 on the discrete brrier option vlues with M time steps, ûx 0, t 0 ; M, to pproximte the continuous brrier cll up-nd-out, [ û x 0, t 0 ; M := ûx 2 0, t 0 ; 8M ûx 0, t 0 ; 4M + 3 ] 2 + 2ûx 0, t 0 ; 2M ûx 0, t 0 ; M. 31 For the stndrd prmeters we compute û x 0, t 0 ; 16 nd for the chllenging prmeters ûx 0, t 0 ; 1. 6

7 4 Problem 2: The Blck-Scholes-Merton model with discrete dividends We cn use the following COS formul to compute the option vlue t time τ: ûx, τ + = e rt τ {ϕ levy ; t = T τ e i U k T. 32 To determine the option vlue t time t 0 we use the following COS formul ûx, t 0 = e rτ with Fourier cosine coefficients U k τ = 2 {ϕ levy ; t = τ e uy, τ cos i U k τ 33 dy 34 There holds uy, τ = uy + ln1 D, τ +. We use discrete Fourier cosine trnsforms DCT to pproximte the Fourier cosine coefficients U k τ. For this, we tke N grid-points nd define n equidistnt y-grid y n := + n N nd y := N. 35 We determine the vlue of function uy, τ = uy + ln1 D, τ + on the N grid-points. The midpoint-rule integrtion gives us U k τ = = n=0 n=0 n=0 2 uy n, τ cos y n uy n, τ cos 2n+1 2 2N N y uy n + ln1 D, τ + cos 2n+1 2 2N N. 36 The ppering DCT Type II cn be clculted efficiently by, for exmple, the function dct of MATLAB. 5 Problem 3: The Blck-Scholes-Merton model with locl voltility The sset price is modeled by locl voltility model ds t = µs t, tdt + σs t, tdw t, 37 7

8 with µs, t = rs nd σs, t = σs, ts. We pproximte the process by n Order 2.0 simplified wek Tylor scheme see [O14], i.e., We define time-grid t 0, t 1,..., t m,..., t M = T, with fixed timesteps t := t m+1 t m. For nottionl convenience we write S m = S tm nd ω m+1 := ω tm+1 ω tm. The pproximted process is denoted by Sm = St m. We strt with S0 = S 0 nd following forwrd scheme is used to determine the vlues Sm+1, for m = 0,..., M 1, with S m+1 = S m + ms m, t m t + ςs m, t m ω m+1 + κs m, t m ω m ms, t = µs, t 1 2 σs, t σ SS, t µt S, t + µs, t µ S S, t µ SSS, t σ 2 S, t t, ςs, t = σs, t 40 µs S, t σs, t + σ t S, t + µs, t σ S S, t σ SSS, t σ 2 S, t t The chrcteristic function of Sm+1, given Sm = S, in eqution 38 is given by [ φ S m+1 ω Sm = S = E exp iωsm+1 ] S m = S 1 = exp iωs + iωms, t m t 2 ω2 ς 2 S,t m t 1 2iωκS,t m t 1 2iωκS, t m t 1/2. 41 The option pricing problem is solved bckwrds in time, with M = 17, { us, tm = gs, [ ] us, t m 1 = e r t E ust m, t m St 42 m 1 = S, 1 m M. We use the COS formul [ ] us, t m 1 = e r t E ust m, t m St m 1 = S { := e r t φ S m+1 S i m = Se Uk t m 43 nd the Fourier cosine coefficients U k t m re pproximted by using DCT s explined in Section 4. 6 Problem 4: The Heston model for one underlying sset The sset price is modeled by the Heston model ds t = rs t dt + σ V t dw 1 t, 44 dv t = κθ V t dt + σ V t dw 2 t, 45 8

9 where W t = Wt 1, Wt 2 is 2D correlted Wiener process with correltion dwt i dw j t = ρ ij dt. We switch to the scled log-sset price process, X t := lns t /K. The chrcteristic function reds ϕ levy ω; V t0 = exp iωr t + Vt 0 1 e D t σ κ iρσω D 2 1 Ge D t κθ 1 Ge exp σ tκ iρσω D 2 ln D t 2 1 G, 46 D = κ iρσω 2 + ω 2 + iωσ 2, 47 G = κ iρσω D κ iρσω+d Problem 5: The Merton jump diffusion model for one underlying sset The sset price is modeled by the Merton jump diffusion model ds t = r λξs t dt + σs t dw t + e J 1S t dq t. 49 Here ξ := E[e J 1] nd q t is Poisson process with intensity rte λ. The jumps J re normlly distributed with men γ nd stndrd devition δ. We switch to the scled log-sset price process, X t := lns t /K, dx t = r λξ 1 2 σ2 ds + σdw t + Jdq t. 50 The corresponding chrcteristic function reds ϕ levy ω = exp iωr λξ 1 2 σ2 t 1 2 ω2 σ 2 t e λ texpiγω 1 2 ω2 δ Problem 6: The Blck-Scholes-Merton model for two underlying ssets The sset prices evolve ccording to the following dynmics: ds i t = rs i tdt + σ i S i tdw i t, i = 1, 2, 52 where W t = Wt 1, Wt 2 is 2D correlted Wiener process with correltion dwt i dw j t = ρ ij dt. We switch to the log-processes Xt i := ln St: i dx i t = r 1 2 σ2 i dt + σ i dw i t. 53 The log-sset prices t time T, given the vlues t time t 0, re bivrite normlly distributed, X T N X 0 + µ t, Σ, 54 with µ i = r 1 2 σ2 i nd covrince mtrix Σ ij = σ i σ j ρ ij t. The chrcteristic function reds s φω x = e ix ω ϕ levy ω, with ϕ levy ω = expiµ tω 1 2 ω Σω. 55 9

10 The 2D-COS formul for pproximtion of ux, t 0 reds see [O12] ûx, t 0 = e r t N 1 1 N 2 1 k 1 =0 k 2 =0 1 2 [ {ϕ levy { + ϕ k1 π levy b 1 1, k 2π b 2 2 exp k 1π b 1 1, + k2π b 2 2 exp ik 1 π x1 1 b ik 2 π x2 2 b 2 2 ] ik 1 π x 1 1 b 1 1 ik 2 π x 2 2 b 2 2 U k1,k 2 T. The Fourier cosine coefficients of the pyoff function re given by U k1,k 2 T = 2 b b e y 1 e y 2 + cos k 1 π y 1 1 b 1 1 cos k 2 π y 2 2 b 2 2 dy 1 dy 2, 57 for which n nlytic solution is vilble nd cn be found using, for instnce, Mple 14. eferences [FO08] F. Fng nd C. W. Oosterlee. A novel pricing method for Europen options bsed on Fourier-cosine series expnsions. SIAM Journl on Scientific Computing, 312: , [FO09] F. Fng nd C. W. Oosterlee. Pricing erly-exercise nd discrete brrier options by Fourier-cosine series expnsions. Numerische Mthemtik, 1141:27 62, [O12] M. J. uijter nd C. W. Oosterlee. Two-dimensionl Fourier cosine series expnsion method for pricing finncil options. SIAM Journl on Scientific Computing, 345:B642 B671, [O14] M. J. uijter nd C. W. Oosterlee. Numericl Fourier method nd second-order Tylor scheme for bckwrd SDEs in finnce. Working pper vilble t SSN: submitted for publiction,