Characteristic Functions for the COS Option Pricing Method

# How to find characteristic function in Fourier Cosine method (COS method) by Fang and Oosterlee


# How to find characteristic function in Fourier Cosine method (COS method) by Fang and Oosterlee












Fang and Oosterlee (2009) introduced Fourier-Cosine method (COS method) in their paper. The formula to price an option is approximately $$e^{-r\Delta t} \sum_{k=0}^{N-1}' Re\left\{ \phi\left( \frac{k\pi}{b-a}; x \right) e^{-ik\pi \frac{a}{b-a}} \right\} V_k$$ where $\phi$ is the characteristic function of the probability density function of the underlying and $V_k$'s are cosine series coefficients of payoff at maturity.

The authors proposed that to apply formula above to price option, one just need to find $V_k.$

However, I have difficulty finding characteristic function instead.

It can be shown easily that characteristic function always exists. But I do not know how to calculate it, say, European call option under Black-Scholes assumption.

## Answer by Kevin (score 3, accepted)

https://quant.stackexchange.com/a/54983

For Fourier methods, you always need the characteristic function of the log-asset price $\ln(S_t)$. In the Black-Scholes model, $\ln(S_t)\sim N\left(\ln(S_0)+\left(r-\delta-\frac{1}{2}\sigma^2\right)t,\sigma^2t\right)$. It is well-known that the characteristic function of $X\sim N(m,s^2)$ is given by $$\phi_X(u)=\exp\left(imu-\frac{1}{2}s^2u^2\right).$$ You can derive this by a simple integration exercise. As you said, it's the Fourier transform of the Gaussian bell curve. This function is, of course, complex valued.

As @LocalVolatility pointed out, you may need the characteristic function of $\ln\left(\frac{S_T}{K}\right)=\ln(S_T)-\ln(K)$. In general, for any constant $c$ and integrable random variable $X$, we have $$\phi_{X+c}(u)=e^{iuc}\phi_X(u).$$

Fang and Oosterlee derive $V_k$ for some European-options and demonstrate a way of estimating $a,b$ based on the cumulants of the distribution. Having found all of this, the implementation is very easy. According to Hirsa (2013), the COS method is the ``fastest known Fourier-based method''!

Carr and Wu (2004) and Lewis (2001) list characteristic functions for many different exponential Lévy processes (e.g. Merton, Kou, NIG, VG, CGMY, ...). Stochastic volatility models such as Heston (recall ``little Heston trap''!), double Heston, 4/2 have closed-form characteristic functions as well. Even characteristic functions of rough volatility models can be approximated. Some models however do not have a known characteristic function (e.g. CEV, local volatility). So, you cannot use the COS method for these models.