A fractional Black-Scholes model with stochastic volatility and European option pricing

Highlights

• European option pricing is studied under a FMLS model with stochastic volatility.

• We have successfully solved a fractional partial differential equation system.

• The derived pricing formula is truly explicit, involving no Fourier inversion.

Abstract

In this paper, we introduce the stochastic volatility into the FMLS (finite moment log-stable) model to capture the effect of both jumps and stochastic volatility. However, this additional stochastic source adds another degree of complexity in seeking for analytical formula when pricing European options, as the involved FPDE (fractional partial differential equation) system governing option prices is now of three dimensions. Albeit difficult, we have still managed to present an analytical solution expressed in terms of Fourier cosine series, after a two-step solution procedure is developed for the target FPDE system. This solution is different from the most literature as it is truly explicit, involving no Fourier inversion. It is also shown through the numerical experiments that it converges very rapidly and has potential to be applied in practice.

Introduction

The Black-Scholes (B-S) model enjoys the great popularity ever since it has been proposed in 1973 (Black & Scholes, 1973), as a result of the model simplicity and tractability. However, the simplified assumption that the underlying price follows the log-normal distribution under this particular model is not consistent with real market observations, making it unable to capture the main features, such as skewness (Peiro, 1999) and fat tails (Rachev, Menn, & Fabozzi, 2005), exhibited by asset returns. Such inconsistency between the model and reality has certainly caused pricing biases, a typical example of which is the so-called “volatility smile” shown by the implied volatility (Dupire et al., 1994). Therefore, this has attracted a lot of research interest into developing various modifications to the B-S model.

One of the most popular approaches is to introduce an additional stochastic source in modeling the underlying price, making the volatility another random variable. Such extensions, albeit appealing, break down the analytical tractability of the B-S model in most cases, and numerical methods had to be resorted to at the very early stage (Scott, 1987, Wiggins, 1987), which hinders the potential practical applications of these models due to the problem of the inefficiency. Later on, a few pricing models that admit analytical pricing formula for European options were proposed (Hull and White, 1987, Stein and Stein, 1991), but these are still not satisfactory due to unrealistic assumptions associated with the volatility. Fortunately, a model that is equipped with a closed-form pricing formula for European options and at the same time satisfies a wide range of basic properties possessed by the volatility was proposed by Heston (1993). This model has gained a large amount of attention from both academic researchers as well as market practitioners, and He and Zhu (2016) went even further to present another formula under this model with a different equivalent martingale measure.

Another popular trend is to incorporate jumps in the underlying dynamics. A natural choice is to directly add a jump component to the B-S model, formulating the jump diffusion models (Merton, 1976, Kou, 2002). A more general jump structure for the underlying price is built by pure-jump Lévy processes with infinite activity, and in these models, it is not necessary to introduce a Brownian motion, as their infinitely small jumps near the origin at any time interval can act as a counterpart to the continuous part of jump-diffusion models. Some well-known examples belonging to this category include the Variance-Gamma model (Madan, Carr, & Chang, 1998), Normal Inverse Gaussian process (Rydberg, 1997) and Carr-Geman-Madan-Yor (CGMY) model (Carr, Geman, Madan, & Yor, 2002). In particular, the FMLS (finite moment log-stable) model proposed by Carr and Wu (2003) has received a lot of attention because it is not only able to capture the high-frequency empirical probability distribution of the S&P data as well as volatility smirks at different maturities, it, unlike many other Lévy processes, also ensures all moments of the underlying are finite, and thus guarantees the existence of an equivalent martingale measure and fitness of option prices at all maturities.

Unfortunately, it has been pointed out by a number of authors that purely Lévy processes or traditional stochastic volatility models are not sufficient to reflect real market observations (Yamazaki, 2014, Carr et al., 2003). In order to capture time series of underlying prices and reproduce implied volatility in plain-vanilla option market, Carr et al. (2003) established a general framework combining stochastic volatility and Lévy models together, leading to the time changed Lévy processes. In addition to the flexibility in reflecting market reality, the framework is also tractable in analytically computing European option prices, and thus many studies have been carried out under this framework (Carr et al., 2012, Tour et al., 2018, Umezawa and Yamazaki, 2015, Zeng and Kwok, 2016, Gong and Zhuang, 2017, Sabino, 2020, Cui et al., 2019). Despite its popularity, the price of European options under Lévy models is usually written in the form of inverse Fourier transform, using the characteristic function, whereas numerical inversion of Fourier transform is not a trivial task that needs to be dealt with very carefully and often requires the use of some special techniques (Carr & Madan, 1999). In this sense, such kind of solutions is not truly explicit.

In this paper, we proceed in a different way; instead of following the general framework of time changed Lévy processes, we directly make the volatility term of the FMLS model another random variable, formulating the stochastic volatility FMLS model. By making use of the fractional partial differential equation (FPDE), an explicit and analytical pricing formula for European options is successfully derived, which does not require numerically inverting the Fourier transform. This particular formula is written in a Fourier cosine series form, and its various properties are shown through numerical experiments.

The rest of the paper is organized as follows. In Section 2, we briefly introduce the stochastic volatility FMLS model and present the FPDE system governing option prices. In Section 3, the derivation details are discussed. In Section 4, numerical examples and useful discussions are shown, followed by some concluding remarks in the last section.