Lévy Processes, Probability Densities, and Fourier Option Pricing
Summary
Lévy processes do not necessarily have probability densities, and even when a density exists it may not have a closed-form expression. The document contrasts this with familiar exponential Lévy examples, including geometric Brownian motion and a jump-diffusion, which have closed-form densities and option prices. The variance gamma process is also cited as having closed-form expressions involving special functions.
For many other models, pricing methods use the characteristic function and evaluate a Fourier transform numerically. The discussion points to Fourier-based approaches for exponential Lévy models and notes that related techniques also apply to many stochastic volatility models, such as Heston. Applicability depends on the method: some extensions require independent increments and therefore do not cover stochastic volatility. The account is conceptual and does not give implementation details or compare numerical accuracy across methods.
Key ideas
- Lévy processes do not necessarily have a probability density.
- A density may exist without a convenient closed-form expression.
- Fourier pricing can use a model’s characteristic function to evaluate option prices numerically.
- Some exponential Lévy models have closed-form densities and option prices.
- Fourier methods also apply to many stochastic volatility models, though some extensions require independent increments.
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Full text
# Are Lévy processes absolutely continuous? # Are Lévy processes absolutely continuous? If $X_t$ is a Lévy process, is it absolutely continuous? Meaning, does it have a density? ## Answer by Kevin (score 3) https://quant.stackexchange.com/a/51177 The simple answer is ''no''. Lévy processes do not necessarily have a density. An accessible (and excellent) book on the topic was written by Cont and Tankov (2004). The introduction to chapter 11.1.3 reads as > Contrary to the classical Black-Scholes case, in exponential Lévy models there are no explicit formulae for call option prices, because the probability density of a Lévy process is typically not known in closed form. However, the characteristic function of this density can be expressed in terms of elementary functions for the majority of Lévy processes discussed in the literature. This has led to the development of Fourier-based option pricing methods for exponential Lévy models. In these methods, one needs to evaluate one Fourier transform numerically. These Fourier methods were developed by Carr and Madan (1999) who discuss the fast Fourier transform, Lewis (2001) who consideres generalised Fourier transforms of option payoffs and Bakshi and Madan (2000) who provide deep economical intuition and come up with a Black-Scholes like formula. In some cases, you may find closed-form option prices. Obviously, the geometric Brownian motion from Black and Scholes (1973) is an exponential Lévy process. So is the jump-diffusion from Kou (2002) and even for the variance gamma process, you have ''closed form'' solutions (using modified Bessel functions and other special functions). Note that the above models also have a closed form density function. But in general, you have to use Fourier methods. Look at the CGMY model from Carr et al. (2002) which generalises the variance gamma process and cannot be described by a single SDE. Such Lévy processes are typically characterised by a Lévy triplet (deterministic mean, variance (diffusion part) and jump component). Note that Fourier methods are not restricted to exponential Lévy models. They also apply to most stochastic volatility models, most notably the Heston (1993) model. Some extensions of the basic Fourier methods however do require independent increments and exclude stochastic volatility models.
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