A Novel Fourier Transform B-spline Method for Option Pricing*

Transcription

1 A Novel Fourier Transform B-spline Method for Option Pricing* Paper available from SSRN: Gareth G. Haslip, FIA PhD Cass Business School, City University London October 23 rd 2013 Financial Engineering Workshop Series, Cass Business School * Research conducted jointly with Vladimir K. Kaishev, Professor, Cass Business School, City University London

2 Executive Summary Efficient methods for pricing European options are an important requirement for the risk-neutral calibration of asset models to the observed implied volatility surface ( inverse problem ) An effective option pricing method for calibration purposes should exhibit two key features: 1. Fast calculation time across many strike prices and maturities 2. Robustness and accuracy across the entire parameter space, moneyness, and maturities Our option pricing framework is called the Fourier Transform B-spline Method (FTBS) First application of B-spline interpolation theory in the field of derivative pricing Provides accurate and extremely efficient closed-form representation of the European option price under an inverse Fourier transform Ideally suited to the calibration of asset price models as it provides fast and accurate pricing of options across multiple strike prices and the parameter space We compare the FTBS method with five other state-of-the-art option pricing methods (FFT, FRFT, IAC, COS, and CONV), which we have diligently implemented to the best of our ability in C++ using identical hardware FTBS method is the preferable approximation method, relative to the other methods considered, for computing option prices across multiple strikes prices for the VG process, and for the Heston model and KoBoL (CGMY) model, for precision up to the level of 1E-6 2

3 Assumptions and Asset Models Applicable to the FTBS Method Notation and Model Assumptions Let with be defined on a continuous-time probability space Asset price evolves according to an exponential stochastic process: where is the risk-free interest rate and is the continuous income yield provided by the asset Assume is a semimartingale process with respect to the filtration and is the riskneutral measure under the assumption of no arbitrage so that is a martingale Require closed-form expression for characteristic function Wide range of asset models considered: Asset Models (Jump) Diffusion Models Double exponential jump diffusion of Kou (2002) Mixed exponential jump diffusion of Cai and Kou (2011) Pure Jump Lévy Processes Variance gamma of Madan and Seneta (1990) KoBoL (CGMY) model of Boyarchenko and Levendorskiĭ (2000) 3 Affine Processes Stochastic volatility model of Heston (1993) Stochastic volatility jump diffusion of Bates (1996)

4 The FTBS Pricing Method for European-Style Options Starting point is the Fourier transform based pricing integral in the complex domain We utilise the Lewis-Lipton formula provided by Lewis (2001) and Lipton (2002) This provides a semi-closed form pricing formula for any European-style option (1) where is the strike price, is the expiry time and This holds provided the MGF of exists for all with and Using (1) the price of a European call option can be evaluated using numerical integration However, this is relatively slow and is not suitable for real-time option pricing applications Also integral must be truncated at suitably large number which introduces truncation error Note that the method is also applicable for alternatives for the contour of integration parameter For simplicity of the presentation, and ease of use of the method, we have chosen to use the above form with = 0.5 This works reasonably well for a wide choice of models and parameters Note that we do not consider pricing deep out-of-the-money options in this paper 4

5 Strike-Separable Pricing Formula The following simplified version of (1) provides the foundations for the FTBS method: (2) where and Dependent on parameter k Recall Trigonometric Function Therefore dependent on strike price and time to maturity 5 Characteristic Function and are completely independent of option contract features contained in k Only dependent on choice of log-return process through the characteristic function Integration is now carried out over the unit interval which avoids truncation error in (1) Separation of the integral trigonometric component dependent on strike price and components which are independent of the strike price is fundamental to the FTBS method We now use optimal B-spline interpolation to approximate and

6 Splines, B-splines and Divided Differences The function is sampled at interpolation sites with and Spline function approximation: Let and partition the interval by the points called knots Define spline function of order, degree as a piece-wise polynomial function that coincides with a polynomial of degree between the knots The polynomial pieces are smoothly joined at the knots so that the spline is n-2 times continuously differentiable on the interval 6

7 Splines, B-splines and Divided Differences To introduce the set of B-spline basis functions, we add the interval : additional knots at each end of Interpolation Sites Knots Given the interpolation sites, the optimal choice of internal knots is provided by De Boor (2001) as being approximately the averages of the interpolation sites:, ( ) We have tested this choice numerically and it performs as well as the theoretical optimum 7

8 Splines, B-splines and Divided Differences Polynomial splines form on represented as: form a linear space of functions, an element of which is (3) where are constant coefficients,, and are B-spline basis functions The B-spline basis functions are defined on as an nth order divided difference where Notation denotes the nth order divided difference calculated recursively as: where and are pairwise distinct (alternative formula applies otherwise) 8

9 Splines, B-splines and Divided Differences B-spline basis functions have a very useful properties, in particular the Peano representation of a divided difference, which is an essential tool in our methodology for computing integrals: (4) Using the Peano representation we will be able to compute numerically in closed-form by replacing and by their B-spline approximants and We will see that the Peano representation enables us to obtain efficient and accurate approximations to option prices Use quadratic splines ( ) which are known to often produce better fits than cubic splines 9

10 Derivation of the FTBS Pricing Formula for European Options Overview 1. Approximate: and 2. Substitute: (5) 3. Apply Peano s representation (4) to compute the integrals in closed-form: Peano s formula: Integrate three times 10

11 FTBS Pricing Formula for European Options (Main Result) Let, and, be the sets of knots and linear coefficients of quadratic spline interpolants, and where and then (6) where for, and if this simplifies to In practice, we can use the same interpolation sites and knots for both integrals, since functions and are based on the real and imaginary parts of the same characteristic function A similar result is provided in the paper for alternative choices of the contour parameter v 11

12 FTBS Error Bound for European Options (Main Result) The absolute error of the FTBS European option price is bounded by (7) where: are the constants obtained from the optimal spline interpolation ( ) following Gaffney and Powell (1976) of and The bound in (7) converges to zero as the mesh sizes rate, where, i.e. go to zero at a Therefore, by increasing the number of data sites used in the interpolation, any level of accuracy can be obtained for the option price 12

13 Computational Aspects of FTBS Method Extremely efficient pricing method across the quantum of strike prices: Coefficients, depend only on the choice of stochastic model and time to maturity (they are only computed once at the first strike price in the list of strike prices considered) Calculation of the divided differences of, is independent of model choice and they are pre-computed and stored in a data file at a fine resolution over all possible values of Computing simply requires evaluating two sum products over (say) 50 values Important advantage of the FTBS method is the elimination of truncation error which comes from the necessity to truncate the limit at infinity in the original expression (1) This means that it is no longer necessary to carefully identify a truncation point for each asset price model by considering how quickly the characteristic function decays to zero Location of interpolation sites is important to extract information from functions and as efficiently as possible Uniform interpolation sites ensures full model and parameter independence However, greater efficiency is achieved by targeting regions where functions have greater curvature In practice the allocation 60% to [0, 0.2], 20% to [0.2, 0.6] and 20% to [0.6, 1] worked effectively for a large range of models 13

14 Computational Aspects of FTBS Method Pre-computation of the divided differences for the chosen interpolation sites is recommend to maximise efficiency. Recall: Consider the parameter ranges:,, and This implies where Divide uniformly with points with width Pre-compute the divided differences : 10% for,, and where 10 E.g. for = 1E-5 and 100 interpolation sites, less than 1GB storage is required to store the divided differences for all possible combinations of contract parameters 14 0% Alternatively, for calibration applications, the required divided differences can be evaluated at the beginning of the calibration process which takes just one to two milliseconds

15 Numerical Results of FTBS Method Performance and numerical accuracy assessed relative to state-of-the-art existing methods Our main numerical comparison considers the computation time, for a specified level of precision, of pricing European options across multiple strikes prices (31 options) As a second comparison, we consider the efficiency of the FTBS method for pricing an European option at a single strike price A like-for-like comparison is provided for all comparison methods we have diligently implemented each pricing method to the best of our ability in C++, using identical hardware This enables an completely objective comparison of the FTBS method s performance Comparison Pricing Methods Asset Price Model Multiple Strikes Single Strike Fast Fourier Transform of Carr and Madan (1999) - VG, Heston, Fractional Fast Fourier Transform (FRFT) of Chourdakis (2005) - KoBoL (CGMY) Cosine (COS) method of Fang and Oosterlee (2008) Convolution (CONV) method of Lord et al. (2009) - Integration-along-cut and Inverse FFT of Levendorskiĭ and Xie (2012) - Finally, to validate the robustness of the FTBS method across options of all tenors and moneyness and across the entire parameter space, an inverse calibration problem is tested VG 15

16 Like-for-like Comparisons (Multiple Strike Prices, Precision 1E-5): Parameter Sets Low, Bench, and High. Calculation time reported for pricing 31 options. FTBS method dominates all comparison pricing methods across range of asset price models and parameter sets at precision level 1E-5 (maximum absolute error across all option prices) Calculation times for FTBS method very competitive: < 1 microsecond per option for VG process < 2 microsecond per option for Heston model < 3 microsecond per option for KoBoL (CGMY) model 16

17 KoBoL (CGMY) CPU time (log) Heston CPU time (log) VG CPU time (log) Like-for-like Comparisons (Multiple Strike Prices): Parameter Sets Low, Bench, and High. Calculation time reported for pricing 31 options. 100 Low Bench High E-04 1.E-05 1.E-06 1.E E-08 1.E-09 1.E-04 1.E-05 1.E-06 1.E E-08 1.E-09 1.E-04 1.E-05 1.E-06 1.E-07 1.E-08 1.E E-04 1.E-05 1.E-06 1.E E-08 1.E-09 1.E-04 1.E-05 1.E-06 1.E E-08 1.E-09 1.E-04 1.E-05 1.E-06 1.E-07 1.E-08 1.E E-04 1E-05 1E-06 1E-07 Precision (log) E-08 1E E-04 1E-05 1E-06 1E-07 1E-08 1E-09 Precision (log) FTBS 17 COS FFT FRFT 1E-04 1E-05 1E-06 1E-07 Precision (log) 1E-08 1E-09

18 Like-for-like Comparisons (Single Strike Price): Comparison of precision and computation times for pricing a single European option under the VG process Outperforms all comparator methods except IAC at all levels of precision considered 18

19 Risk-Neutral Calibration Inverse problem to identify risk-neutral model parameters from cross-sectional option price data Provides an effective test of robustness across options of all tenors and moneyness, and across the entire parameter space of the asset model Comparison to reference calibrations of Schoutens et al. (2005) using 144 European call option prices with maturities ranging from less than one month to 5.16 years Calibrations using FFT and FRFT configurations of Chourdakis (2005) FFT and FRFT methods demonstrated instability for Bates model over full parameter space Interesting to note Bates model offers two sets of parameters with very different characteristics 19

20 Conclusion Fourier Transform B-spline pricing method for European options has been demonstrated to provide superior precision / computation times to state-of-the art comparison methods Efficiency of FTBS method is explained by a combination of factors: Optimal spline interpolation is able to reproduce arbitrary functions with a high degree of precision using only a modest number of interpolation sites (e.g. 50) The Peano representation of a divided difference enables the pricing integral to be evaluated in closed-form as a linear combination of spline coefficients and divided differences The spline interpolation is independent of strike price and hence only is carried out a single time for the first strike price considered (when pricing over a quantum of strike prices) The divided differences are model independent and can be pre-computed These factors combined mean that computing European option prices across the quantum of strike prices becomes a three stage process: 1. Load pre-computed divided differences into memory (e.g. at beginning of the day) 2. Perform spline interpolation to functions and to obtain spline coefficients 3. Compute each option price as a sum product of spline coefficients and divided differences requires just 4p addition and multiplication operations to compute 20

21 References Bates, D.S Jumps and stochastic volatility: Exchange rate processes implicit in Deutsche mark options. Review of Financial Studies 9(1) Black, F., M. Scholes The pricing of options and corporate liabilities. Journal of Political Economy Boyarchenko, S. I., and Levendorskiĭ, S. On rational pricing of derivative securities for a family of nongaussian processes. Preprint 98/7, Institut für Mathematik, Universität Potsdam, (1998). Boyarchenko, S. I., and Levendorskiĭ, S. Option pricing for truncated Lévy processes. International Journal of Theoretical and Applied Finance, 3 (2000), Boyarchenko, S.I., S. Levendorskiĭ Non-Gaussian Merton-Black-Scholes Theory. World Scientific Singapore. Cai, N., SG Kou Option pricing under a mixed-exponential jump diffusion model. Management Science 57(11) Carr, P., and Madan, D. Option valuation using the fast Fourier transform. Journal of Computational Finance, 2(1999), Carr, P., H. Geman, D.B. Madan, M. Yor The fine structure of asset returns: An empirical investigation. The Journal of Business 75(2) Chourdakis, K. Option pricing using the fractional FFT. Journal of Computational Finance, 8 (2005), De Boor, C A practical guide to splines, vol. 27. Springer Verlag. Dufresne, D., J. Garrido, M. Morales Fourier inversion formulas in option pricing and insurance. Methodology and Computing in Applied Probability 11(3) Eberlein, E., Glau, K., and Papapantoleon, A. Analysis of Fourier transform valuation formulas and applications. Applied Mathematical Finance, 17 (2010),

22 References Fang, F., CW Oosterlee A novel pricing method for European options based on Fourier cosine series expansions. SIAM Journal on Scientific Computing Gaffney, P., M. Powell Optimal interpolation. Numerical Analysis Heston, S.L A closed-form solution for options with stochastic volatility with applications to bond and currency options. Review of Financial Studies 6(2) Kou, SG A jump-diffusion model for option pricing. Management Science Levendorskiĭ, S., J. Xie Fast pricing and calculation of sensitivities of out-of-the-money European options under Lévy processes. Journal of Computational Finance 15(3) Lewis, A. Option valuation under stochastic volatility. Option Valuation under Stochastic Volatility, (2000). Lewis, A. A simple option formula for general jump-diffusion and other exponential Lévy processes. Unpublished working paper, (2001). Lipton, A The vol smile problem. Risk magazine 15(2) Lord, R., F. Fang, F. Bervoets, CW Oosterlee A fast and accurate FFT-based method for pricing earlyexercise options under Lévy processes. SIAM Journal on Scientific Computing 30(4) Madan, D. B., and Seneta, E. The variance gamma (VG) model for share market returns. Journal of business, (1990), Schoutens, W., E. Simons, J. Tistaert Model risk for exotic and moment derivatives. Exotic Option Pricing and Advanced Lévy Models