Fast Pricing and Calculation of Sensitivities of OTM European Options Under Lévy Processes

Transcription

1 Fast Pricing and Calculation of Sensitivities of OTM European Options Under Lévy Processes Sergei Levendorskĭi Jiayao Xie Department of Mathematics University of Leicester Toronto, June 24, 2010 Levendorskĭi and Xie (University of Leicester) Fast Pricing of OTM Options 1 / 26

2 Outline 1 Motivation and Main Results 2 Pricing of European options in Lévy Models 3 Integration-Along-Cut (IAC) Method 4 Sensitivities 5 Realization of IAC in KoBoL Model 6 More Numerical Examples Levendorskĭi and Xie (University of Leicester) Fast Pricing of OTM Options 1 / 26

3 Motivation FFT method: first applied to pricing European options by Carr and Madan [1999] sizable computational error for deep OTM options Integration-Along-Cut (IAC) method: Levendorskĭi and Zherder [2001], Boyarchenko and Levendorskĭi [2002] Later, deficiencies of FFT techniques were analyzed by Lord, Fang, Bervoets and Oosterlee [2007] Lord and Kahl [2007] N. Boyarchenko and Levendorskĭi [2007] M. Boyarchenko and Levendorskĭi [2008, 2009] (refined and enhanced (enh-ref) FFT) Carr and Madan [2009] (saddlepoint method) Levendorskĭi and Xie (University of Leicester) Fast Pricing of OTM Options 2 / 26

4 Main Results 1 Accurate and fast numerical realizations for KoBoL (a.k.a. CGMY) and VG model 2 A modification, which makes IAC method applicable to European options far from expiry in VG model 3 An efficient procedure for calculation of option prices at many strikes 4 Calculation of sensitivities using IAC method Levendorskĭi and Xie (University of Leicester) Fast Pricing of OTM Options 3 / 26

5 Study of Relative Efficiency: An Example Table: Prices of European call options in the KoBoL model: IAC vs. FFT, the approximate saddle point method saddlepoint (SP) and enh-ref ifft (Enh-ref). Strike Option price Relative difference between IAC and IAC FFT SP Enh-ref FFT SP Enh-ref European call option parameters: r = 0.03, T = 0.5, S = 100. KoBoL parameters: ν = 0.5, c + = c = 2, λ + = 5, λ = 10, µ Enh-ref ifft algorithm parameters (CPU time: 0.06 seconds): M = 2 13, M 2 = 2, M 3 = 2, = Levendorskĭi and Xie (University of Leicester) Fast Pricing of OTM Options 4 / 26

6 Study of Relative Efficiency: An Example (cont.) The second column is obtained by using Integration-Along-Cut method for a given relative error tolerance The third and fourth column are taken from the tables in P. Carr and D.B. Madan, Saddlepoint Methods for Option Pricing, Journal of Computational Finance, Vol. 13, No. 1 (Fall 2009). Refined and enhanced ifft is a flexible version of ifft developed in M. Boyarchenko and S. Levendorskĭi, Prices and sensitivities of barrier and first-touch digital options in Lévy-driven models, International Journal of Theoretical and Applied Finance, Vol. 12, No. 8, 2009, pp N.B.: For a very long and fine grid, enh-ref ifft method produces results with relative differences less than , but this requires much more CPU time than IAC Levendorskĭi and Xie (University of Leicester) Fast Pricing of OTM Options 5 / 26

7 Speed: An Example in VG Model 0.6 ε = ε = ε = CPU time (in millisecond) Strike price K (S = 1000) Figure: CPU time (in millisecond) of IAC method, for the European call option in VG model: dependence on the relative error tolerance ɛ. European call option parameters: r = 0.01, T = 0.5, S = VG model parameters: c + = c = 0.15, λ + = 9, λ = 8, µ Levendorskĭi and Xie (University of Leicester) Fast Pricing of OTM Options 6 / 26

8 Speed: An Example in KoBoL Model 0.3 ε = ε = ε = CPU time (in millisecond) Strike price K (S = 100) Figure: CPU time (in millisecond) of IAC method, for the European call option in KoBoL model: dependence on the relative error tolerance ɛ. European call option parameters: r = 0.03, T = 0.5, S = 100. KoBoL parameters: ν = 0.5, c + = c = 2, λ + = 5, λ = 10, µ Levendorskĭi and Xie (University of Leicester) Fast Pricing of OTM Options 7 / 26

9 Strong Features of Our Approach Application Variance Gamma (VG) model and strongly Regular Lévy Process of Exponential type (srlpe), which includes Normal Inverse Gaussian (NIG) model, and KoBoL (a.k.a CGMY) model in the finite variation case Levendorskĭi and Xie (University of Leicester) Fast Pricing of OTM Options 8 / 26

10 Strong Features of Our Approach Application Variance Gamma (VG) model and strongly Regular Lévy Process of Exponential type (srlpe), which includes Normal Inverse Gaussian (NIG) model, and KoBoL (a.k.a CGMY) model in the finite variation case Accuracy & Efficiency For one strike, hundreds times faster than FFT For multiple options, IAC method together with the quadratic interpolation, is still faster and more accurate than FFT based approach For sensitivities, a relative advantage of IAC is even greater Levendorskĭi and Xie (University of Leicester) Fast Pricing of OTM Options 8 / 26

11 Strong Features of Our Approach (cont.) Efficient Error Control Real-valued integrand, and For VG, a simple rule for truncation; and asymptotic expansion may be used to calculate the truncated part For KoBoL, the calculations are reduced to summation of oscillating series; even easier Levendorskĭi and Xie (University of Leicester) Fast Pricing of OTM Options 9 / 26

12 Strong Features of Our Approach (cont.) Efficient Error Control Real-valued integrand, and For VG, a simple rule for truncation; and asymptotic expansion may be used to calculate the truncated part For KoBoL, the calculations are reduced to summation of oscillating series; even easier Simplicity Straightforward reduction to cuts Simpson s and enhanced Simpson s (a.k.a product integration) rule Levendorskĭi and Xie (University of Leicester) Fast Pricing of OTM Options 9 / 26

13 Pricing of European Options in Lévy Models Market structure Bond yielding riskless rate of return r Stock S t = e Xt, where X t is a Lévy process under an EMM Q; the characteristic exponent ψ(ξ) of X t admits the analytic continuation into the complex plane with two cuts i(, λ ] and i[λ +, + ) Notation: V (t, x) := V (G; T ; t, x) the price of the European option with maturity T and payoff G(X T ) Levendorskĭi and Xie (University of Leicester) Fast Pricing of OTM Options 10 / 26

14 Lévy processes: General Definitions ψ characteristic exponent of X = (X t ): ] E [e iξxt = e tψ(ξ) Explicit formula (Lévy-Khintchine formula, in 1D) is ψ(ξ) = σ2 2 ξ2 iµξ + (1 e iyξ + iyξ1 y <1 (y))f (dy), R\0 where F (dy), the Lévy density, satisfies min{ y 2, 1}F (dy) <. R\0 Levendorskĭi and Xie (University of Leicester) Fast Pricing of OTM Options 11 / 26

15 Examples a) KoBoL model (a.k.a. CGMY model and extended Koponen s family): ψ(ξ) = iµξ+γ( ν) [c + (( λ ) ν ( λ iξ) ν )+c (λ ν + (λ + +iξ) ν ) ], where ν 1, c ± > 0, λ < 0 < λ +. b) Normal Tempered Stable Lévy processes (NTS): [ (α ψ(ξ) = iµξ + δ 2 (β + iξ) 2) ν/2 (α 2 β 2 ) ν/2], where α > β > 0, δ > 0 and µ R. With ν = 1, NTS is NIG. c) Variance Gamma process (VG): ψ(ξ) = iµξ + c + [ln( λ iξ) ln( λ )] + c [ln(λ + + iξ) ln(λ + )], where c ± > 0, λ < 0 < λ +, µ R. Levendorskĭi and Xie (University of Leicester) Fast Pricing of OTM Options 12 / 26

16 Generalized Black-Scholes Formula Boyarchenko and Levendorskĭi [1998], Carr and Madan [1998] Assume that for ω (λ, λ + ), function G ω (x) := e ωx G(x) L 1 (R). Then [ ] V (t, x) = E e r(t t) G(X T ) X t = x where τ = T t > 0. Equivalently, G(x) = (2π) 1 V (t, x) = (2π) 1 V (t, x) = (2π) 1 Im ξ=ω Im ξ=ω Im ξ=ω e ixξ Ĝ(ξ)dξ, e ixξ τ(r+ψ(ξ)) Ĝ(ξ)dξ, (1) e i(x+µτ)ξ τ(r+ψ0 (ξ)) Ĝ(ξ)dξ. (2) Levendorskĭi and Xie (University of Leicester) Fast Pricing of OTM Options 13 / 26

17 Generalized Black-Scholes Formula Boyarchenko and Levendorskĭi [1998], Carr and Madan [1998] Assume that for ω (λ, λ + ), function G ω (x) := e ωx G(x) L 1 (R). Then [ ] V (t, x) = E e r(t t) G(X T ) X t = x where τ = T t > 0. Equivalently, G(x) = (2π) 1 V (t, x) = (2π) 1 V (t, x) = (2π) 1 Im ξ=ω Im ξ=ω Im ξ=ω e ixξ Ĝ(ξ)dξ, e ixξ τ(r+ψ(ξ)) Ĝ(ξ)dξ, (1) e i(x+µτ)ξ τ(r+ψ0 (ξ)) Ĝ(ξ)dξ. (2) Levendorskĭi and Xie (University of Leicester) Fast Pricing of OTM Options 13 / 26

18 Generalized Black-Scholes Formula (cont.) Let x = ln(s t /K). European call option: G(X T ) = (e X T K) + V (t, x) = Keτr 2π for any ω (λ, 1). Im ξ=ω e i(x+µτ)ξ τψ0 (ξ) (ξ + i)ξ dξ. (3) European put option: G(X T ) = (K e X T ) +, we have the same formula as above, but with ω (0, λ + ). Levendorskĭi and Xie (University of Leicester) Fast Pricing of OTM Options 14 / 26

19 srlpe Definition Let λ < 0 < λ + and ν (0, 2]. We call X a strongly regular Lévy process of exponential type (λ, λ + ) and order ν if the following conditions hold: (i) the characteristic exponent ψ admits the analytic continuation into the complex plane with two cuts i(, λ ] and i[λ +, + ); (ii) for z λ and z λ +, the limits ψ(iz ± 0) exist; (iii) there exists µ R such that the function ψ 0 (ξ) := ψ(ξ) + iµξ is asymptotically positively homogeneous of order ν as ξ in the complex plane with these cuts. In VG model, the logarithmic asymptotics at infinity: srlpe of order 0+. Levendorskĭi and Xie (University of Leicester) Fast Pricing of OTM Options 15 / 26

20 Idea of IAC Method 1 ω λ Example: European OTM Call Option (λ < ω < 1, x + µτ < 0) Assumption: X is an srlpe of order ν (0, 1] Integral: Im ξ=ω φ(ξ)dξ φ(ξ) = e i(x+µτ)ξ τψ0 (ξ) 1 (ξ + i)ξ Levendorskĭi and Xie (University of Leicester) Fast Pricing of OTM Options 16 / 26

21 Idea of IAC Method 1 ω λ Example: European OTM Call Option (λ < ω < 1, x + µτ < 0) Assumption: X is an srlpe of order ν (0, 1] Integral: Im ξ=ω φ(ξ)dξ φ(ξ) = e i(x+µτ)ξ τψ0 (ξ) 1 (ξ + i)ξ Levendorskĭi and Xie (University of Leicester) Fast Pricing of OTM Options 16 / 26

22 Idea of IAC Method 1 ω λ Example: European OTM Call Option (λ < ω < 1, x + µτ < 0) Assumption: X is an srlpe of order ν (0, 1] Integral: Im ξ=ω φ(ξ)dξ φ(ξ) = e i(x+µτ)ξ τψ0 (ξ) 1 (ξ + i)ξ Levendorskĭi and Xie (University of Leicester) Fast Pricing of OTM Options 16 / 26

23 Idea of IAC Method 1 ω λ Example: European OTM Call Option (λ < ω < 1, x + µτ < 0) Assumption: X is an srlpe of order ν (0, 1] Integral: Im ξ=ω φ(ξ)dξ φ(ξ) = e i(x+µτ)ξ τψ0 (ξ) 1 (ξ + i)ξ Levendorskĭi and Xie (University of Leicester) Fast Pricing of OTM Options 16 / 26

24 Idea of IAC Method By Cauchy theorem, φ(ξ)dξ = φ(ξ)dξ L ω + φ(ξ)dξ L R + φ(ξ)dξ L ɛ + φ(ξ)dξ L λ =0 Levendorskĭi and Xie (University of Leicester) Fast Pricing of OTM Options 16 / 26

25 Idea of IAC Method Let Im ξ ω. As R := ξ +, L R φ(ξ)dξ 0 As ɛ 0, L ɛ φ(ξ)dξ 0 Levendorskĭi and Xie (University of Leicester) Fast Pricing of OTM Options 16 / 26

26 Idea of IAC Method = φ(ξ)dξ Im ξ=ω iλ 0 φ(ξ)dξ i 0 i +0 + iλ +0 φ(ξ)dξ Levendorskĭi and Xie (University of Leicester) Fast Pricing of OTM Options 16 / 26

27 IAC Formula for European OTM Call and Put Option OTM Call (x := x µτ > 0) V call (t, x) = Ke rτ π λ zx Ω(τ, z) e dz, (4) z(z + 1) OTM Put (x := x + µτ > 0) V put (t, x) = Ke rτ π + Ω(τ, z) e zx dz, (5) λ + z(z + 1) where Ω(τ, z) = (i/2)(e τψ0 (iz 0) e τψ0 (iz+0) ). (6) Levendorskĭi and Xie (University of Leicester) Fast Pricing of OTM Options 17 / 26

28 Ω(τ, z) in The Case of KoBoL For z < λ : Ω(τ, z) = e τcγ( ν)[( λ ) ν (λ z) ν cos(νπ)+λ ν + (λ+ z)ν ] sin (τcγ( ν)(λ z) ν sin(νπ)) ; For z > λ + : Ω(τ, z) = e τcγ( ν)[( λ ) ν ( λ +z) ν +λ ν + (z λ+)ν cos(νπ)] sin (τcγ( ν)(z λ + ) ν sin(νπ)) ; Levendorskĭi and Xie (University of Leicester) Fast Pricing of OTM Options 18 / 26

29 Sensitivities Denote by S = Ke x, the spot price of the underlying. For a European call option, = Γ = Θ = V / S = e x 2π 2 V / S 2 = e 2x 2πK V / τ = K 2π Im ξ=ω Im ξ=ω Im ξ=ω e i(x+µτ)ξ τ(r+ψ0 (ξ)) iξ 1 dξ; e i(x+µτ)ξ τ(r+ψ0 (ξ)) dξ, e i(x+µτ)ξ τ(r+ψ0 (ξ)) (ξ + i)ξ (r iµξ + ψ 0 (ξ))dξ. Levendorskĭi and Xie (University of Leicester) Fast Pricing of OTM Options 19 / 26

30 Sensitivities (cont.) If X is an srlpe of order ν (0, 1], then = e x τr π Γ = e 2x τr πk Θ = Ke τr π λ e x z λ λ Ω(τ, z) z 1 dz e x z Ω(τ, z)dz, e x z Ω 1(τ, z) z(z + 1) dz where Ω 1 (τ, z) = i 2 (e τψ0 (iz 0) [ r + µz + ψ 0 (iz 0) ] e τψ0 (iz+0) [ r + µz + ψ 0 (iz + 0) ] ) Levendorskĭi and Xie (University of Leicester) Fast Pricing of OTM Options 20 / 26

31 Realization of IAC in KoBoL Model Change the variable y = (λ z) ν in (4): V (t, x) = R ν e x y 1/ν F (y) sin(δ + sin(νπ)y)dy where x = x µτ > 0, δ ± = τc ± Γ( ν), 0 R ν = Ke rτ+x λ +δ +( λ ) ν +δ λ ν +/(νπ), F (y) = e δ+ cos(νπ)y δ (λ + λ +y 1/ν ) ν (λ y 1/ν )(λ + 1 y 1/ν ) y 1/ν 1 Levendorskĭi and Xie (University of Leicester) Fast Pricing of OTM Options 21 / 26

32 Realization of IAC in KoBoL Model (cont.) Set na = nπ/ (δ + sin(νπ)), n = 0, 1,..., and change the variable y = z + na: a V (t, x) = R ν ( 1) n F n (x, z)dz, where f n (z) = n=0 F n (x, z) = e x (z+na) 1/ν f n (z) sin(πz/a) e δ+ cos(νπ)(z+na) δ (λ + λ +(z+na) 1/ν ) ν (λ (z + na) 1/ν ) (λ + 1 (z + na) 1/ν ) (z + na) 1 1/ν 0 Levendorskĭi and Xie (University of Leicester) Fast Pricing of OTM Options 22 / 26

33 Realization of IAC in KoBoL Model (cont.) Lemma Let z (0, a). Then a) if ν (0, 0.5], then the sequence {F n (z)} is monotonically decreasing; b) if ν (0.5, 1), then there exists N such that for n N, the sequence {F n (z)} is monotonically decreasing. Levendorskĭi and Xie (University of Leicester) Fast Pricing of OTM Options 23 / 26

34 Example: Fitting The Smile Relative Error Strike price K (S = 1000) Time to maturity Figure: Relative error of implied volatility surface obtained by enh-ref ifft method. VG parameters: c + = c = 0.15, λ + = 9, λ = 8, µ European call option parameters: r = 0.01, S = IAC (multiple options) algorithm parameters: 1 = , 2 = 0.5, ɛ = , A(x, ɛ) = A( 0.1, ), x = Enh-ref ifft algorithm parameters: = 0.01, M = 2 10, M 2 = 3, M 3 = 3. Relative error: (V enh ref V IAC )/V IAC. Levendorskĭi and Xie (University of Leicester) Fast Pricing of OTM Options 24 / 26

35 Example: Fitting The Smile (cont.) Implied Volatility Strike price, K (S = 1000) Time to maturity Figure: Implied volatility surface. VG parameters: c + = c = 0.15, λ + = 9, λ = 8, µ European call option parameters: r = 0.01, S = IAC (multiple options) algorithm parameters: 1 = , 2 = 0.5, ɛ = , A(x, ɛ) = A( 0.1, ), x = Levendorskĭi and Xie (University of Leicester) Fast Pricing of OTM Options 25 / 26

36 Possible Efficiency Improvement C++ implementation with the x -dependent truncation parameter Non-uniformly spaced grid More Results S.Z. Levendorskĭi and J. Xie, Fast pricing and calculation of sensitivities of out-of-the-money European options under Lévy processes (working paper, April 14, 2010). Available at SSRN: PC Characterization The calculations presented were performed in MATLAB c (R2007a), on a PC with characteristics Intel R Core TM 2 Duo CPU (3.16GHz, 6MB L2 Cache, 1333MHz FSB), under the Genuine Windows R XP Professional operating system. Levendorskĭi and Xie (University of Leicester) Fast Pricing of OTM Options 26 / 26