# In what cases characteristic function of (log-)price process is known?

Hey I know that we can use characteristic function of log-price process to price different options. But when we know the characteristic function? I know that we can take Levy processes and constant interest rate, but what if I want to add stochastic volatility and stochastic interest rate to my model? And can we in some way get the CF from SDEs (for example Heston model + stochastic interest rate)? I would like to know the pros and cons of using Fourier methods in option pricing instead of MC simulation and in what cases it's not possible to use Fourier methods (or when MC simualations are better choice).

Duffie et al. (2000) show how to obtain the characteristic function of the log asset price in a fairly general affine jump diffusion model. Among others this includes the Black-Scholes (1973) model, the Heston (1993) model, the Bates (1996) model, the Merton (1976) model and the Kou (2002) model. This case also allows you to add stochastic interest rates. Bakshi, Cao and Chen (1997) use characteristic functions to price options in SVJ-I models (and find stochastic interest rates to be a minor feature of equity option pricing models).

The Lévy–Khintchine theorem tells us the characteristic function of the log asset price in exponential Lévy models. In addition to the previous jump diffusions, further examples include the variance gamma model, the CGMY model, and normal inverse Gaussian model and the Meixner model.

El Euch and Rosenbaum (2019) illustrate how to even approximate the characteristic function of the log asset price in rough volatility models.

The better question is for what models we do not know the characteristic function. Examples include the CEV model, the SABR model and local volatility models. In these cases, simulations are the way to go.

As a final note, characteristic functions capture the information about the probability distribution of the asset price at a particular point in time. They are therefore extremely popular to price path-independent options. Whilst there exist generalisations to handle path-dependency and early-exercise, Monte Carlo simulations are perhaps more popular in these cases.

• Great answer! Maybe one little addition about SABR and CEV. It is quite common for these two models to express vol in terms of the Black model and to use a Black pricer for European options. So simulation is not necessary for these options. And could you point us to a source about generalisations to handle path-dependency, cause I wasn't aware of it and would like to learn more about it. Jul 14, 2021 at 5:22
• @JohnDoe Lord et al. (2008) propose the CONV method to price Bermudan-style options, Fang and Oosterlee (2009) apply the COS method to discrete barrier options and Bermudan options (American options can be interpolated from them). Zhang and Oosterlee (2013) and Zhang and Oosterlee (2014) study Asian options (with early exercise features). Jul 14, 2021 at 9:07
• @JohnDoe As you see, ‪Cornelis Oosterlee has done some great work in this area. The idea with Fourier methods is to study the discrete problem via backward induction (particularly for Lévy processes with independent increments) as this only requires information about the probability distribution at a specific moment in time. The continuous analogues (like American options and continuously observed barriers or averages) are obtained by (Richardson) extrapolation. Jul 14, 2021 at 9:10