A Highly Efficient Shannon Wavelet Inverse Fourier Technique for Pricing European Options

Transcription

1 A Highly Efficient Shannon Wavelet Inverse Fourier Technique for Pricing European Options Luis Ortiz-Gracia Centre de Recerca Matemàtica (joint work with Cornelis W. Oosterlee, CWI) Models and Numerics in Financial Mathematics, Lorentz Center Leiden, May 28, 2015

2 Outline 1 Option pricing and wavelets 2 3 Numerical Experiments 4

3 Outline 1 Option pricing and wavelets Luis Ortiz-Gracia (CRM) SWIFT 3 / 33

4 Our point of departure is the risk-neutral option valuation formula, v(x, t) = e r(t t) E Q (v(y, T ) x) = e r(t t) v(y, T )f (y x) dy. (1) Whereas f is typically not known, the characteristic function of the log-asset price is often available. We represent the option values as functions of the scaled log-asset prices, and denote these prices by, x = ln(s t /K) and y = ln(s T /K), with S t the underlying price at time t and K the strike price. The pay-off v(y, T ) for European options in log-asset space is then given by, { v(y, T ) = [α K (e y 1)] + 1, for a call,, with α = (2) 1, for a put. R Luis Ortiz-Gracia (CRM) SWIFT 4 / 33

5 The strategy to follow to determine the price of the option consists of approximating the density function f in (1) by means of a finite combination of Shannon scaling functions and recovering the coefficients of the approximation from its Fourier transform. Multi-resolution analysis: Consider the space, L 2 (R) = {f : + and a family of closed nested subspaces, f (x) 2 dx < },... V 2 V 1 V 0 V 1 V 2..., in L 2 (R) where, V j = {0}, V j = L 2 (R), j Z j Z f (x) V j f (2x) V j+1. Luis Ortiz-Gracia (CRM) SWIFT 5 / 33

6 If these conditions are met, then a function φ V 0 exists (called scaling or father function) such that {φ j,k } k Z forms an orthonormal basis of V j, where, φ j,k (x) = 2 j/2 φ(2 j x k). For any f L 2 (R) a projection map of L 2 (R) onto V m, P m : L 2 (R) V m, is defined by means of: P m f (x) = k Z c m,kφ m,k (x), where c m,k = + f (x)φ m,k(x) dx are the scaling coefficients. Considering higher m values (i.e. when more terms are used), the truncated series representation of the function f improves. As opposed to Fourier series, a key fact regarding the use of wavelets is that wavelets can be moved (by means of the k value), stretched or compressed (by means of the j value) to accurately represent the local properties of a function. Luis Ortiz-Gracia (CRM) SWIFT 6 / 33

7 In our financial mathematics context, we here consider Shannon wavelets, φ m,k (x) = 2 m/2 sin(π(2m x k)) π(2 m, k Z. (3) x k) It is clear that for m = k = 0, we have the basic scaling function or father wavelet, φ(x) = sinc(x), where sinc(x) = sin(πx) πx. Shannon wavelets represent the real part of the so-called harmonic wavelets. They have a slow decay in the time domain but a very sharp compact support in the frequency (Fourier) domain, ˆφ m,k (w) = e i k 2 m w ( w ) 2 rect m/2 2 m+1, (4) π where rect is the rectangle function, defined as, 1, if x < 1/2, rect(x) = 1/2, if x = 1/2, 0, if x > 1/2. Luis Ortiz-Gracia (CRM) SWIFT 7 / 33

8 1 phi 1.2 phi_hat x w Figure : Left plot: Shannon scaling function φ(x). Right plot: ˆφ(w). Luis Ortiz-Gracia (CRM) SWIFT 8 / 33

9 Outline Option pricing and wavelets Coefficients computation Pay-off coefficients Multiple strikes valuation 1 Option pricing and wavelets Luis Ortiz-Gracia (CRM) SWIFT 9 / 33

10 Coefficients computation Pay-off coefficients Multiple strikes valuation Let us consider a probability density function f in L 2 (R) associated to a certain continuous random variable X, and its Fourier transform, ˆf (w) = e iwx f (x) dx. (5) R Following the wavelets theory, function f can be approximated at a level of resolution m, i.e., f (x) P m f (x) = k Z c m,k φ m,k (x), (6) The infinite series in (6) is well-approximated by a finite summation without loss of considerable density mass, P m f (x) f m (x) := k 2 for certain accurately chosen values k 1 and k 2. k=k 1 c m,k φ m,k (x), (7) Luis Ortiz-Gracia (CRM) SWIFT 10 / 33

11 Coefficients computation Pay-off coefficients Multiple strikes valuation We start by considering, c m,k = f, φ m,k = f (x)φ m,k (x) dx = 2 m/2 R R f (x)φ(2 m x k) dx. (8) Using the classical Vieta formula, the cardinal sine can be expressed as an infinite product, i.e., sinc(t) = + j=1 ( πt ) cos 2 j. (9) If we truncate the infinite product to a finite product with J factors, then, thanks to the cosine product-to-sum identity, we have J ( πt ) cos 2 j = 1 2 J 1 ( ) 2j 1 2 J 1 cos 2 J πt. (10) j=1 By (9) and (10) the sinc function can thus be approximated as follows, sinc(t) sinc (t) := 1 2 J 1 ( ) 2j 1 2 J 1 cos 2 J πt. (11) j=1 j=1 Luis Ortiz-Gracia (CRM) SWIFT 11 / 33

12 Coefficients computation Pay-off coefficients Multiple strikes valuation If we replace function φ in (8) by its approximation (11) then, c m,k c m,k := 2m/2 2 J 1 2 J 1 j=1 R ( ) 2j 1 f (x) cos 2 J π(2 m x k) dx. (12) ) Taking into account that R (ˆf (w) = f (x) cos(wx) dx and observing R that, 2j 1 kπ(2j 1) ikπ ˆf (w)e 2 J i(wx = e ) 2 J f (x) dx, R we end up with the following expression for computing the density coefficients, c m,k cm,k = 2m/2 2 J 1 [ 2 J 1 R ˆf j=1 ( (2j 1)π2 m 2 J ) ] e ikπ(2j 1) 2 J. (13) Luis Ortiz-Gracia (CRM) SWIFT 12 / 33

13 Coefficients computation Pay-off coefficients Multiple strikes valuation Summary: Questions: v(x, t) = e r(t t) f (y x) k 2 c m,k cm,k = 2m/2 2 J 1 [ 2 J 1 R ˆf j=1 How do we determine k 1 and k 2? How do we determine J? R v(y, T )f (y x) dy, k=k 1 c m,k (x)φ m,k (y), ( (2j 1)π2 m 2 J ) ] e ikπ(2j 1) 2 J. Luis Ortiz-Gracia (CRM) SWIFT 13 / 33

14 Coefficients computation Pay-off coefficients Multiple strikes valuation To address the first question: We consider a first truncation [a, b] as in Fang and Oosterlee (2008) to get k 1 and k 2 (not necessary but facilitates the application of an FFT) and we control the suitability of the interval,...plus... f ( ) h 2 m P m f ( ) h 2 m = 2 m 2 c m,k δ k,h = 2 m 2 cm,h, (14) k Z ( A = 1 c m,k1 + ) c 2 m/2 m,k + c m,k2, (15) 2 2 k 1<k<k 2 that is, we control the density values and the density mass as a byproduct. Luis Ortiz-Gracia (CRM) SWIFT 14 / 33

15 Coefficients computation Pay-off coefficients Multiple strikes valuation To address the second question: We use the following lemma, which gives us an estimate of the error when approximating the cardinal sine function. This lemma is the basis of further theoretical error estimations. The proof of this lemma and the details of those estimations can be found in Ortiz-Gracia and Oosterlee (2015). Lemma 1 Define the absolute error E V (t) := sinc(t) sinc (t). Then, E V (t) (πc) 2 2 2(J+1) (πc) 2, for t [ c, c], where c R, c > 0 and J log 2 (πc). Luis Ortiz-Gracia (CRM) SWIFT 15 / 33

16 Coefficients computation Pay-off coefficients Multiple strikes valuation We assess the error estimation in Lemma 1 (that is, the theoretical error) by plotting the empirical and the theoretical error. For this purpose, we fix the interval of approximation [ 5, 5], c = 5. We consider J log 2 (πc) 4 and define the empirical error for each J-value as, ε emp = max t j = j j=0,...,1000 sinc(t j ) sinc (t j ). As expected, the thick line corresponding to the empirical error is below the dashed line belonging to the theoretical error. 0.1 Empirical error Theoretical error absolute error e-05 1e-06 1e-07 1e-08 1e J Luis Ortiz-Gracia (CRM) SWIFT 16 / 33

17 Coefficients computation Pay-off coefficients Multiple strikes valuation Alternative: coefficients via Parseval s identity Remark 1 By Parseval s identity, f, φ m,k = 1 2π ˆf, ˆφm,k (where both ˆf and ˆφ m,k are known). After some algebraic manipulation and thanks to the compact support of ˆφ m,k we end up with, c m,k = 2 m/ R (ˆf (2 m+1 πt) e i2πkt) dt, It is worth remarking that an FFT algorithm can be applied in this case as well to speed up the computations. Luis Ortiz-Gracia (CRM) SWIFT 17 / 33

18 Coefficients computation Pay-off coefficients Multiple strikes valuation We consider the approximation to the density function f, f (y x) f m (y x) = k 2 k=k 1 c m,k (x)φ m,k (y). (16) We truncate the infinite integration range to a finite domain I m = [ k1 2, k2 m 2 ], m v(x, t) e r(t t) I m v(y, T )f (y x) dy. If we replace f by its approximation f m, we find, v(x, t) e r(t t) v(y, T )f m (y x) dy = e I m r(t t) with the pay-off coefficients, Vm,k α := v(y, T )φ m,k (y) dy. I k k 2 k=k 1 c m,k (x) V α m,k, (17) Luis Ortiz-Gracia (CRM) SWIFT 18 / 33

19 Coefficients computation Pay-off coefficients Multiple strikes valuation Proposition 1 With the same notation as before, let us define k 1 := max(k 1, 0). The pay-off coefficients for a European call option are computed, as follows, { K2 m/2 [ ( ) ( )] 2 J 1 Vm,k 1 V 1, m,k := 2 J 1 j=1 I k 1 1 2, k2 m 2 I k 1 m 2 2, k2 m 2, if k m 2 > 0, 0, if k 2 0, where I 1 and I 2 are closed formulae. Proof, error estimation and details on how to choose the parameter J rely on Lemma 1 and can be found in Ortiz-Gracia and Oosterlee (2015). FFT can be applied to compute the pay-off coefficients. Luis Ortiz-Gracia (CRM) SWIFT 19 / 33

20 Coefficients computation Pay-off coefficients Multiple strikes valuation For Lévy jump models and for Heston stochastic volatility models, options for many strike prices can be computed simultaneously in a highly efficient way. We distinguish vectors by boldfaced letters here. For Lévy processes (and Heston model), whose characteristic functions can be represented by, ˆf (w; x) = ˆf Levy (w) e iwx, (18) the SWIFT option pricing formula simplifies to, r(t t) 2 m K k 2 2 v(x, t) = e e ikπ j 1 2 j For n K strikes: 2 j+ j 2 R k=k 1 j=0 ˆf Levy ( (2j + 1)π2 m 2 j ) e i(2j+1)π2m ) 2 j x e 2πikj Compute the density coefficients n K times, since x depends on the strike. Compute the pay-off coefficients once, since the values depend on the strike. Efficient but improvable! V α, m,k 2 j (19) do not α, V m,k. Luis Ortiz-Gracia (CRM) SWIFT 20 / 33

21 Coefficients computation Pay-off coefficients Multiple strikes valuation The overall CPU time could be further reduced by reformulating the approximation problem as follows. Observe that, for Lévy and Heston models, the density function f satisfies, f (y) = 1 2π R e iyw ˆf (w) dw = 1 2π R e iyw ˆfLevy (w)e iwx dw = f Levy (y x). Our pricing formula becomes: v(x, t) e r(t t) k 2 k=k 1 c m,k V α m,k (x). For n K strikes: Compute the density coefficients only once (c m,k does not depend on x). Compute the pay-off coefficients n K times ( V m,k α depends on x). The pay-off coefficients are computed in less CPU time than the density coefficients. We can use different wavelets (Haar, B-splines) and then V α m,k are computed analytically, which is even faster. (20) Luis Ortiz-Gracia (CRM) SWIFT 21 / 33

22 Outline 1 Option pricing and wavelets Luis Ortiz-Gracia (CRM) SWIFT 22 / 33

23 We aim at showing the high efficiency and robustness of SWIFT: Fast convergence. Multiple strikes valuation. Long maturity options Fat-tailed densities. Size of the integration interval. Scale of approximation: how do we fix m? Luis Ortiz-Gracia (CRM) SWIFT 23 / 33

24 Example 1: fast convergence: cash-or-nothing under GBM with parameters, S 0 = 100, r = 0.1, T = 0.1, and volatility σ = (21) 0.01 K=80 K=100 K= e-06 1e-08 absolute error 1e-10 1e-12 1e-14 1e-16 1e-18 1e scale (m) Luis Ortiz-Gracia (CRM) SWIFT 24 / 33

25 Example 1: fast convergence: cash-or-nothing under GBM with parameters, S 0 = 100, r = 0.1, T = 0.1, and volatility σ = (22) m K k 1 k 2 j j Error CPU time (milli-seconds) e e e e e Luis Ortiz-Gracia (CRM) SWIFT 25 / 33

26 Example 2: multiple strikes valuation: 21 strikes 1 strike (K = 100) scale m CPU time (milli-seconds) Max. absolute error 2.04e e e 06 scale m CPU time (milli-seconds) Absolute error 4.78e e e 07 Table : Simultaneous valuation of 21 European call options (top) with strikes ranging from 50 to 150, and valuation of only one European call option (bottom) under the Heston dynamics with parameters S 0 = 100, µ = 0, λ = , η = , ū = , u 0 = , ρ = The reference values have been computed using the COS method with terms and L = 10. Luis Ortiz-Gracia (CRM) SWIFT 26 / 33

27 Example 3: long maturity options: Method Error (T = 50) Error (T = 100) SWIFT (m = 0) 1.91e e-05 COS (N = 35) 4.98e e+02 SWIFT (m = 1) 7.78e e-06 COS (N = 70) 2.79e e-05 Table : Absolute errors corresponding to the valuation of a call option under the GBM dynamics by means of the SWIFT and COS methods with parameters S 0 = 100, K = 120, r = 0.1, σ = The reference values have been computed using the Black-Scholes formulae: (T = 50) and (T = 100). Remarks: COS: how to increase N? SWIFT: is not sensible w.r.t. longer maturities. By removing part of the sum, SWIFT reaches better results (local wavelet behaviour). Luis Ortiz-Gracia (CRM) SWIFT 27 / 33

28 Example 4: fat-tailed densities: [a, b] f ( ) k1 2 m f ( ) k2 2 m Error (area) [ 1, 1] 5.97e e e-01 [ 2, 2] 1.15e e e-01 [ 5, 5] 1.30e e e-01 [ 10, 10] 1.27e e e-02 [ 20, 20] 1.10e e e-09 [ 32.83, 25.19] 1.02e e e-15 Table : Absolute error of the computed area under the recovered density by the SWIFT method at scale m = 0 corresponding to the CGMY dynamics with parameters S 0 = 100, K = 110, r = 0.1, q = 0.05, T = 5, C = 1, G = 5, M = 5, Y = 1.5. The interval in the last row of the table has been computed with the help of cumulants: [ 32.83, 25.19]. It might be difficult to accurately determine [a, b] (COS is sensitive). SWIFT allows us to control the density mass. Luis Ortiz-Gracia (CRM) SWIFT 28 / 33

29 Example 5: size of the integration interval: COS SWIFT 1 COS SWIFT e absolute error 1e-06 1e-07 1e-08 absolute error e-06 1e-09 1e-10 1e-08 1e-11 1e e L L Figure : Absolute errors corresponding to the valuation of a cash-or-nothing call (left) and a European call (right) under the GBM dynamics by means of the SWIFT (m = 3) and COS (N = 40) methods with parameters S 0 = K = 100, r = 0.1, T = 1, σ = The reference values have been computed using the Black-Scholes formulae: (cash-or-nothing) and (call). Luis Ortiz-Gracia (CRM) SWIFT 29 / 33

30 Example 6: scale of approximation: how do we fix m? 2 CGMY Y=1.5 m=0 GBM m=2 CGMY Y=0.1 m=4 1 CGMY Y=1.5 m=0 GBM m=2 CGMY Y=0.1 m= density 1 Fourier transform x w Figure : Density function (left) and characteristic function (right). CGMY Y=1.5: engineering accuracy (m = 0) and machine accuracy (m = 1). GBM: engineering accuracy (m = 2) and machine accuracy (m = 4). CGMY Y=0.1: engineering accuracy (m = 4). Rule-of-thumb: m = 0, 1 (light-tails in Fourier domain) and m = 2, 3, 4 (fat-tails in Fourier domain). Obviously: convergence with m + 1 m. COS: wide range of values for N. Luis Ortiz-Gracia (CRM) SWIFT 30 / 33

31 Outline 1 Option pricing and wavelets Luis Ortiz-Gracia (CRM) SWIFT 31 / 33

32 In this work we advocated the use of Shannon wavelets within the European option pricing framework. Specifically, we have focused on the discounted expected pay-off pricing formula, where the price of the option is computed by integrating the pay-off function multiplied by the density function. We have presented a novel method based on Shannon wavelets to recover the density function from its characteristic function and we have called it SWIFT. The density coefficients as well as the pay-off coefficients are computed using Vieta s formula. The presented numerical method shows a high accuracy and robustness, as well as a fast convergence to the solution at low scales with only a few terms in the expansion. SWIFT is efficient for the valuation of long- and short-maturity contracts and it does not need to rely on an a-priori truncation of the entire real line. In case that we perform an a-priori truncation (to facilitate the use of an FFT) the method is not affected by the width of the interval for the approximation and it automatically computes the number of coefficients needed for the new interval. Efficient computation of risk measures at portfolio level and pricing of exotic derivatives are object of future research. Luis Ortiz-Gracia (CRM) SWIFT 32 / 33

33 Thank you for your attention Luis Ortiz-Gracia (CRM) SWIFT 33 / 33