Chapter 16: Fourier-Based Option Analysis
A modern field of derivative valuation is based on Fourier inversion, with several techniques now available to value derivative contracts with asset dynamics that are more complex than the lognormal distribution. The Fourier inversion of the characteristic function approach developed by Heston (1993) and Bakshi and Madan (2000) generates a result with a form similar to the classic Black–Scholes analytical equation (Schmelzle, 2010). Regrettably, the Bakshi and Madan method as well as related techniques developed later are not suitable to efficient and accurate numerical calculation (Chourdakis, 2008). An alternate approach was developed by Carr and Madan (1999) that allows the use of the highly efficient fast Fourier transform (FFT) algorithms to transform the damped European option price.
16.2 Risk-Neutral Valuation
The stochastic price process of an asset is described by the probability measure. The simplest approach to value derivatives is based on assets that are free of arbitrage opportunities within the risk-neutral probability measure, . Assuming that an equivalent probability measure exists, the discounted process will form a martingale
where the time-dependent risk-free rate, , creates a time-dependent bond price ...