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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.

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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.

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    $\begingroup$ The statement "and plug it into the COS-formula you cited" is not entirely correct. The characteristic function given in the answer is that of the log return process $X_t$ or the log asset price $\ln S_t$. The original formulation of payoff coefficients in the Fang & Oosterlee paper is based on the characteristic function and cumulants of $\ln \left( S_t / K \right)$ however; see Section 3.1. This is a simple transformation, everything stays normal but its also easy to miss; see also arxiv.org/abs/2005.13248. $\endgroup$ – LocalVolatility Jun 17 at 10:03
  • $\begingroup$ @LocalVolatility You are, of course, absolutely right! Thanks for pointing that out. I never saw the reason why Fang and Oosterlee did that because you can do the calculations with the characteristic function of $\ln(S_T)$, which seems the natural choice to me. But one only needs the adjustment $\phi_{c+X}(u)=e^{iuc}\phi_X(u)$ to switch between the two characteristic functions. Thanks for the paper, I haven’t seen that one yet! $\endgroup$ – Kevin Jun 17 at 10:41
  • $\begingroup$ @KeSchn Am I right to say that to apply COS method, one needs to know the distribution of underlying? For example, in Black-Scholes, we assume that underlying follows a lognormal distribution. So, we can find its characteristic function. $\endgroup$ – Idonknow Jun 17 at 12:58
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    $\begingroup$ @KeSchn Thanks for your resourceful and insighful answers. I am a Pure Maths PhD trying to break into quantitative finance after my graduation. Recently I started reading papers on numerical option pricing methods and try to implement them, just to ensure I understand them as applications and theory are equally important in job, I think. $\endgroup$ – Idonknow Jun 17 at 13:15
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    $\begingroup$ @Idonknow You're welcome! :) Duffie et al. (2000) is from Econometrica, Heston (1993) from the RFS. The Heston case is a special case of the AJD model from Duffie et al. (2000). Cont and Tankov (2004, ``Financial Modelling with Jump Processes'') is a book on jump processes which illustrates how to find the characteristic function for Lévy processes, again without knowing the density function, see also Lewis (2001, SSRN). $\endgroup$ – Kevin Jun 17 at 14:24

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