Fast Fourier Transform for Discrete Asian Options European Finance Management Association Lugano June 2001 Eric Ben-Hamou

June 2001Lugano 2001 ConferenceSlide N°2 Outline Motivations Description of the methods Log-Normal density Non-Log-Normal density Conclusion

June 2001Lugano 2001 ConferenceSlide N°3 Motivations When pricing a derivative, one should keep in mind: Within a model, the quality of the approximation. model risk, reflected by the uncertainty on the model parameters objectivity of the model So need to examine the risk of a certain derivative, here Asian option.

June 2001Lugano 2001 ConferenceSlide N°4 Asian option’s characteristics academic case is geometric Brownian motion continuous time average: no easy-closed forms (Geman Yor 93 Madan Yor 96) discrete averaging (see Vorst 92 Turnbull and Wakeman 91 Levy 92 Jacques 96 Bouazi et al. 98 Milevsky Posner 97 Zhang 98 Andreasen 98) empirical literature suggests fat-tailed distribution (Mandelbrot 63 Fama 65) and smile literature (Kon 84 Jorion 88 Bates 96 Dupire 94 Derman et al 94 see Dumas et al. 95) stochastic volatility (Hull and White 87 Heston 93)

June 2001Lugano 2001 ConferenceSlide N°5 Asian option risk Main risks: jump of delta risk, or reset risk at each fixing (Vorst 92 Turnbull and Wakeman 91 Levy 92…) distribution risk underlying the jump of delta at each fixing dates… other issues like modeling of discrete dividends (Benhamou Duguet 2000) Good method: tackles discret averaging non lognormal densities

June 2001Lugano 2001 ConferenceSlide N°6 Aim: Assume that returns are iid with a well known density (known numerically) Target of the method: get the density of any type of sum of underlying at different dates.. with

June 2001Lugano 2001 ConferenceSlide N°7 Standard hypotheses:… complete markets and no arbitrage. If the density of is known then simple numerical integration. Density of a sum of independent variables is the convolution of the individual densities FFT O(Nlog 2 (N) ) typically N=2 p so that p2 p (like binomial tree)

June 2001Lugano 2001 ConferenceSlide N°8 New insights Old method first offered by Caverhill Clewlow 92 but inefficient and only for lognormal densities One way of improving it is to systematically re-center the convolution at each outcome Second, we examine the impact of non- lognormal law and see important changes in the delta..

June 2001Lugano 2001 ConferenceSlide N°9 Factorization Hodges (90) Recursive scheme

June 2001Lugano 2001 ConferenceSlide N°10 Algorithm 1 Result Algorithm 2 Result

June 2001Lugano 2001 ConferenceSlide N°11 leads to This imposes at each step to interpolate the densities obtained at the previous step. interpolation

June 2001Lugano 2001 ConferenceSlide N°12 Efficiency of the algorithm

June 2001Lugano 2001 ConferenceSlide N°13 Grid specification (lognormal case) Centered grid with 2 12 =4096 points per grid Simpson numerical integration for the final procedure interpolation done by standard cubic spline Standard FFT algorithm as described in Press et al. Proxy for the mean efficient for volatility lower than 30%

June 2001Lugano 2001 ConferenceSlide N°14 Non log-normal case Smile can be seen as a proof of non log- normal densities with fatter tails. (excess kurtosis and skewness) Lognormal case already requires numerical methods So instead use of Student distributions, Pareto, generalized Pareto, power-laws distributions.. Here took Student density

June 2001Lugano 2001 ConferenceSlide N°15 Student additional advantage to converge to normal distribution… which gives the geometric Brownian motion Assumptions: follows a Student density of df normal case Example of Student law

June 2001Lugano 2001 ConferenceSlide N°16 Student law Variance is exactly density is given by the Gamma function

June 2001Lugano 2001 ConferenceSlide N°17 Non log-normal densities

June 2001Lugano 2001 ConferenceSlide N°18 Delta hedging Strong impact on the delta whereas small impact on the price. Price effect offsets by the overpricing of the lognormal approximation This justifies the use of lognormal approximation as a way of incorporating fat tails… but wrong for the delta

June 2001Lugano 2001 ConferenceSlide N°19 Results Long maturity before expiry short maturity before expiry

June 2001Lugano 2001 ConferenceSlide N°20 Conclusion Offered an efficient method for the pricing of discrete Asian options with non lognormal laws. Shows that fat tails impacts much more for the Greeks than the price future work: adaptation to floating strike option use of other fat tailed distribution raises the issue of deriving an efficient methodology for deriving densities from Market prices