From Characteristic Functions and Fourier Transforms to PDFs/CDFs and Option Prices

From Characteristic Functions and Fourier Transforms to PDFs/CDFs and Option Prices Liuren Wu Zicklin School of Business, Baruch College Fall, 2007 Liuren Wu Fourier Transforms Option Pricing, Fall, 2007 1 / 21

From a dynamic process to a terminal distribution If we specify a dynamic process for X and its initial conditions, we can compute the distribution of X over a certain time horizon, [t, T ]. The reverse is not true. Given a conditional distribution for X over [t, T ], there can be many processes that can generate this distribution There are many routes to Rome. Knowing the dynamic process is important for hedging practices Dynamic hedging is more likely to work if the underlying process is continuous instead of discontinuous. The objective of this note: Given a dynamic process for X, derive the probability density (PDF) of X and price options based on integration of terminal payoffs over the probability densities (instead of based on dynamic hedging arguments and PDEs). Liuren Wu Fourier Transforms Option Pricing, Fall, 2007 2 / 21

Formalizing the idea Assume X is a Markov process: Any information about X up to time t is summarized by X t. Let f (X T X t ) denote the conditional distribution of X T conditional on time-t information (filtration F t ) under the risk-neutral measure Q. Let Π(X T ) denote the payoff of a contingent claim (derivative), which we assume is a function of X T. Then, its time-t value is p t = E Q t [e R ] T r t s ds Π(X T ) = e R T r t s ds Π(X T )f (X T X t )dx T Instead of solving PDEs, binomial trees, or simulating the process, we focus on doing the integration based on the risk-neutral densities. In most of the examples, I assume deterministic interest rates: p t = e rτ Π(X T )f (X T X t )dx T X where r now is the time-t continuously compounded spot rate of maturity T t. X Liuren Wu Fourier Transforms Option Pricing, Fall, 2007 3 / 21

The road map For most models (processes) discussed in this class on security prices S t, we can derive the characteristic function or Fourier transform of the log security return (ln S T /S t ) (semi-)analytically. Given the characteristic function (CF), we just need one numerical integration to obtain the probability density function (PDF) or cumulative density function (CDF). Given the Fourier transforms (FT), we just need one numerical integration to obtain the value of vanilla options. The integration is one-dimensional in both cases no matter how many dimensions/factors the security price S t is composed of. We can apply fast Fourier inversion (FFT) to make the numerical integration (for both PDF and option prices) very fast. Solving PDEs becomes difficult as the dimension increases. Simulation works (slowly) for high-dimensional cases and is reserved for pricing exotics. Liuren Wu Fourier Transforms Option Pricing, Fall, 2007 4 / 21

Characteristic function: definition In probability theory, the characteristic function (CF) of any random variable X completely defines its probability distribution. On the real line it is given by the following formula: φ X (u) E [ e iux ] = e iux f X (x)dx = e iux df X (x), u R where u is a real number, i is the imaginary unit, and E denotes the expected value, f X (x) denotes the probability density function (PDF), and F X (x) denotes the cumulative density function (CDF). CF is well defined on the whole real line (u). For option pricing, we extend the definition to the complex plane, u D C, where D denotes the subset of the complex plane on which the expectation is well defined. φ X (u) under this extended definition is called the generalized Fourier transform. Ω Liuren Wu Fourier Transforms Option Pricing, Fall, 2007 5 / 21

Example: The Black-Scholes model Under BSM, the log security return follows, s t ln S t /S 0 = (µ 12 ) σ2 t + σw t The return is normally distributed with mean ( µ 1 2 σ2) t and variance σ 2 t. ( ) The PDF is f s (x) = 1 2πσ 2 t exp (x (µ 1 2 σ2 )t) 2 2σ 2 t. The CF is: φ s (u) = E [ ] e iust = e iumean 1 2 u2variance = e (iuµt iu 1 2 σ2 t 1 2 u2 σ 2 t) = e (iuµt 1 2 σ2 (iu+u 2 )t) Under Q, µ = r q. Liuren Wu Fourier Transforms Option Pricing, Fall, 2007 6 / 21

The inversion: From CF to PDF and CDF There is a bijection between CDF and CFs: Two distinct probability distributions never share the same CF. Given a CF φ, it is possible to reconstruct the corresponding CDF: 1 F X (y) F X (x) = lim τ 2π +τ In general this is an improper integral... Another form of the inversion F X (x) = 1 2 + 1 2π The inversion formula for PDF: f X (x) = 1 2 + 0 τ e iux e iuy φ X (u)du iu e iux φ X ( u) e iux φ X (u) du. iu e iux φ X (u)du = 1 π 0 e iux φ X (u)du. All the integrals here should be understood as a principal value if there is no separate convergence at the limits. Exercise: Based on the BSM model, try a simple discretization of the integral and see how close you can get to the analytical soln. Can you think of some better ways to improve the accuracy and speed of the numerical integration? Liuren Wu Fourier Transforms Option Pricing, Fall, 2007 7 / 21

Proofs Preliminary results: For fixed x, e iux = cos ux + i sin ux, 1 R e iuζ π iu du = π 1 R sin uζ du = sgn(ζ), u 2 R 0 sin uζ R du = sgn(ζ), π u sgn(y x)df (y) = R x df (y) + R x df (y) = 1 2F (x), φ(u)&φ( u) are complex conjugates. CDF inversion: I = Hence, F (x) = 1 2 + 1 2π I. PDF inversion: = = = e iux φ X ( u) e iux φ X (u) du 0 iu e iux e iuz e iux e iuz df (z)du 0 iu 2 sin u(x z) df (z)du 0 u Z 2 sin u(x z) dudf (z) = πsgn(x z)df (z) 0 u = π(2f (x) 1) f (x) = F (x) = 1 2π 0 e iux φ ( u) + e iux φ (u) du = 1 π e iux φ (u) du 0 Liuren Wu Fourier Transforms Option Pricing, Fall, 2007 8 / 21

Fourier transforms and inversions of European options Take a European call option as an example. We perform the following rescaling and change of variables: c(k) = c(k, t)/s 0 = e rt E Q [ 0 (e s t e k ] )1 st k. We represent the option price as percentage of the spot, and we quote the price in terms of log strike instead of strike. We can derive the Fourier transform of the call option in terms of the Fourier transform (CF) of the log return ln S t /S 0. Hence, if we know the CF of the return, we would know the transform of the option. Then, we can use numerical inversion to obtain option prices directly. There are two ways of doing this. Liuren Wu Fourier Transforms Option Pricing, Fall, 2007 9 / 21

I. The CDF analog Treat c (k) = e rt E Q 0 as a CDF. The option transform: χ I c(u) [( e s t e k) 1 st k] = e rt ( e s t e k) 1 st xdf (s) e iuk dc(k) = e rt φ s (u i) iu + 1, u R. Thus, if we know the CF of the return, φ s (u), we know the transform of the option, χ I c(u). The inversion formula is analogous to the inversion of a CDF: c (x) = 1 2 e qt + 1 2π 0 e iux χ I c ( u) e iux χ I c (u) du. iu Note the slight difference from the CDF inversion. Use quadrature methods for the numerical integration. The literature often writes: c (x) = e qt Q1 (x) e rt e x Q 2 (x). Then, we must invert twice. References: Duffie, Pan, Singleton, 2000, Transform Analysis and Asset Pricing for Affine Jump Diffusions, Econometrica, 68(6), 1343 1376. Singleton, 2001, Estimation of Affine Asset Pricing Models Using the Empirical Characteristic Function, Journal of Econometrics, 102, 111-141. Liuren Wu Fourier Transforms Option Pricing, Fall, 2007 10 / 21

Proofs: The option transform χ I c (u) e iuk dc (k) = e iuk Z c (k) c (k) iue iuk dk Checking the boundary conditions, we have c ( ) = 0 (when strike is infinity) and c () = e rt F t /S t = e qt when the strike is zero. Hence, e iu c ( ) = 0 and we will carry the other non-convergent limit e iu e qt. χ I c (u) = e iu e qt c (k) iue iuk dk = e iu e qt iue rt " e st e k # 1 s t k df (s) e iuk dk Z " = e iu e qt iue rt e st e k # 1 s t k eiuk dk df (s) Z " Z s = e iu e qt iue rt t e iuk+st e (iu+1)k # dk df (s) = 2 e iu e qt iue rt 4 e s t eiuk iu e 3 (iu+1)k st 5 df (s). iu + 1 We need to check the boundary again. lim k e (iu+1)k = 0 given the real component e. The other boundary is non-convergent e s t e iu, which we pull out and take the expectation to have iue rt e s t e iu df (s) = e iu e rt F t /S t = e iu e qt, iu which cancels out the other nonconvergent term. Z " # χ I e (iu+1)st c (u) = iue rt e(iu+1)s t df (s) = e rt e (iu+1)s t rt e φ (u i) df (s) =. iu iu + 1 iu + 1 iu + 1 Liuren Wu Fourier Transforms Option Pricing, Fall, 2007 11 / 21

Proofs: The option transform inversion The proof for the option transform inversion is similar to that for the CDF, except for the boundary conditions: Z x sgn (y x) dc (y) = dc (y) + dc (y) = c (x) + c () + c ( ) c (x) = e qt 2c (x). x In the case of CDF, F ( ) = 1, and F () = 0. In the case of call option prices, c ( ) = 0 (when strike is infinity), and c () = e rt F t /S t = e qt (when strike is zero). Hence, the inversion formula is I e iux χ (u) e iux χ ( u) du 0 iu = (as before) = πsgn (x z) df (z) = π e qt 2c (x). Thus, c (x) = 1 2 e qt + 1 2π I. Liuren Wu Fourier Transforms Option Pricing, Fall, 2007 12 / 21

II. The PDF analog Treat c(k) analogous to a PDF. The option transform: χ II c (z) e izk c(k)dk = e rt φ s (z i) (iz) (iz + 1) with z = u iα, α D R + for the option transform to be well defined. The range of α depends on payoff structure and model. The exact value choice of α is a numerical issue. Carr and Madan (1999, Journal of Computational Finance) refer to α as the dampening coefficient. Given the transform on return φ s(u), we know the transform on call. The inversion is analogous to that for a PDF: c(k) = 1 2 iα+ iα e izk χ II c (z)dz = e αk π 0 e iuk χ II c (u iα)du. References: Carr&Wu, Time-Changed Levy Processes and Option Pricing, JFE, 2004, 17(1), 113 141. Liuren Wu Fourier Transforms Option Pricing, Fall, 2007 13 / 21

Proofs χ II (z) e izk c (k) dk = e rt " = e rt " = e rt " Z s t 2 = e rt 4 e s t eizk iz e st e k 1 s t k df (s) # e izk dk e st e k # 1 s t k eizk dk df (s) e izk+st e (iz+1)k # dk df (s) e(iz+1)k iz + 1 3 st 5 df (s) We need to consider the boundary conditions at k =. lim k e (iz+1)k = 0 as long as the real component of izu is greater than 1. lim k e izk = 0 as long as the real component of iz is greater than 0. Hence, taken together, we need the real component of iz to be greater than zero. If we write z = u iα, with both u and α real, we have iz = iu + α. Hence, we need α > 0 for the above boundary condition to converge. Given that u i > 0, we have Z " χ II (z) = e rt e (iz+1)st iz e(iz+1)s t iz + 1 # df (s) = e rt e (iz+1)s t rt e φ (z i) df (s) =. iu (iz + 1) iz (iz + 1) Liuren Wu Fourier Transforms Option Pricing, Fall, 2007 14 / 21

Fast Fourier Transform (FFT) FFT is an efficient algorithm for computing discrete Fourier coefficients. The discrete Fourier transform is a mapping of f = (f 0,, f N 1 ) on the vector of Fourier coefficients d = (d 0,, d N 1 ), such that d j = 1 N 1 f k e jk 2π N i, j = 0, 1,, N 1. N k=0 FFT allows the efficient calculation of d if N is an even number, say N = 2 n, n N. The algorithm reduces the number of multiplcations in the required N summations from an order of 2 2n to that of n2 n 1, a very considerable reduction. By a suitable discretization, we can approximate the inversion of a PDF (also option price) in the above form to take advantage of the computational efficiency of FFT. Liuren Wu Fourier Transforms Option Pricing, Fall, 2007 15 / 21

Return PDF inversion Compare the PDF inversion with the FFT form: f X (x) = 1 π 0 e iux φ X (u)du. d j = 1 N 1 f k e jk 2π N i N k=0 Discretize the integral: f X (x) 1 π N k=0 e iu k x φ X (u k ) u Set η = u, u k = ηk. Set x j = b + λj with λ = 2π/(ηN) being the return grid and b being a parameter that controls the return range. To center return around zero, set b = λn/2. The PDF becomes f X (x j ) 1 N 1 f k e jk 2π N i, f m = N N π eiu k b φ X (u k )η. (1) k=0 with j = 0, 1,, N 1. The summation has the FFT form and can hence be computed efficiently. Liuren Wu Fourier Transforms Option Pricing, Fall, 2007 16 / 21

Call value inversion Compare the call inversion (method II) with the FFT form: c(k) = e νk π 0 e iuk χ II c (u iν)du. Discretize the integral: c(k) e νk π Set η = u, u m = ηm. d j = 1 N 1 f m e jm 2π N i N m=0 N m=0 e iumk χ II c (u m iν) u Set k j = b + λj with λ = 2π/(ηN) being the return grid and b being a parameter that controls the return range. To center return around zero, set b = λn/2. The call value becomes c(k j ) 1 N 1 f m e jm 2π N i, N m=0 f m = N π e νk j +iu mb χ II c (u m )η. with j = 0, 1,, N 1. The summation has the FFT form and can hence be computed efficiently. Liuren Wu Fourier Transforms Option Pricing, Fall, 2007 17 / 21

FFT implementation To implement the FFT, we need to fix the following parameters N = 2 n : The number of summation grids. Setting it to be the power of 2 can speed up the FFT calculation. η = u: The discrete summation grid width. The smaller the grid, the better the approximation. However, given N, η also determines the strike grid λ = 2π/(ηN). The finer the summation grid η, the coarser the strike spacing returned from the FFT calculation. There is a trade off: If we want to have more option value calculated at a finer grid of strikes, we would need to use a coarser summation grid and hence less accuracy. The lower and upper bound truncation b = λn/2 is also determined by the summation grid choice. FFT generates option values at N strikes simultaneously. However, if the strike grid is larger, many of the returned strikes are out of the interesting region. Liuren Wu Fourier Transforms Option Pricing, Fall, 2007 18 / 21

Fractional FFT Fractional FFT (FRFT) separates the integration grid choice from the strike grids. With appropriate control, it can generate more accurate option values given the same amount of calculation. The method can efficiently compute, N 1 d j = f m e jmζi, j = 0, 1,..., N 1, m=0 for any value of the parameter ζ. The standard FFT can be seen as a special case for ζ = 2π/N. Therefore, we can use the FRFT method to compute, c(k j ) 1 N 1 f m e jmηλi, N m=0 f m = N π e νk j +iu mb χ II c (u m )η. without the trade-off between the summation grid η and the strike spacing λ. We require ηλ = 2π/N under standard FFT. Liuren Wu Fourier Transforms Option Pricing, Fall, 2007 19 / 21

Fractional FFT implementation Let d = D(f, ζ) denote the FRFT operation, with D(f) = D(f, 2π/N) being the standard FFT as a special case. An N-point FRFT can be implemented by invoking three 2N-point FFT procedures. Define the following 2N-point vectors: ( ( ) ) N 1 y = f n e iπn2 ζ, n=0 (0)N 1 n=0, (2) ( ( ) N 1 z = e iπn2 ζ, (e ) ) N 1 iπ(n n)2 α. (3) n=0 The FRFT is given by, D k (h, ζ) = (e ) N 1 iπk2 ζ k=0 n=0 D 1 k (D j (y) D j (z)), (4) where D 1 k ( ) denotes the inverse FFT operation and denotes element-by-element vector multiplication. Liuren Wu Fourier Transforms Option Pricing, Fall, 2007 20 / 21

Fractional FFT implementation Due to the multiple application of the FFT operations, an N-point FRFT procedure demands a similar number of elementary operations as a 4N-point FFT procedure. Given the free choices on λ and η, FRFT can be applied more efficiently. Using a smaller N with FRFT can achieve the same option pricing accuracy as using a much larger N with FFT. The accuracy improvement is larger when we have a better understanding of the model and model parameters so that we can set the boundaries more tightly. Caveat: The more freedom also asks for more discretion and caution in applying this method to generate robust results in all situations. This concern becomes especially important for model estimation, during which the trial model parameters can vary greatly. Reference: Chourdakis, 2005, Option pricing using fractional FFT, JCF, 8(2). Liuren Wu Fourier Transforms Option Pricing, Fall, 2007 21 / 21