Multiscale Stochastic Volatility for Equity, Interest Rate, and Credit Derivatives

Abstract

Building upon the ideas introduced in their previous book, Derivatives in Financial Markets with Stochastic Volatility, the authors study the pricing and hedging of financial derivatives under stochastic volatility in equity, interest-rate, and credit markets. They present and analyze multiscale stochastic volatility models and asymptotic approximations. These can be used in equity markets, for instance, to link the prices of path-dependent exotic instruments to market implied volatilities. The methods are also used for interest rate and credit derivatives. Other applications considered include variance-reduction techniques, portfolio optimization, forward-looking estimation of CAPM 'beta', and the Heston model and generalizations of it. 'Off-the-shelf' formulas and calibration tools are provided to ease the transition for practitioners who adopt this new method. The attention to detail and explicit presentation make this also an excellent text for a graduate course in financial and applied mathematics.

Option pricing with Lévy-Stable processes generated by Lévy-Stable integrated variance

“…Thus dW t dZ t = ρdt and we can write Z t = ρW t + 1 − ρ 2Z t , whereZ t is independent of W t . The correlation parameter ρ is also known in the literature as the leverage effect and empirical studies suggest that ρ < 0 [FPS00]. In our case the notion of 'correlation' does not apply because for Lévy-Stable random variables, as given that moments of second and higher order do not exist, nor do correlations.…”

American options under stochastic volatility: control variates, maturity randomization & multiscale asymptotics

“…A common theme in all the proposed stochastic volatility models is mean-reversion of a stochastic factor driving the volatility of the underlying process. A detailed discussion of the mean-reverting nature of these models is provided by Fouque et al [13] where empirical references are given for fast and slow factors in volatility fluctuations. Hence, in [13], the authors proposed multiscale stochastic volatility models which capture both the separation of scales and mean-reverting characteristics of the volatility process.…”

“…In the Heston model, the European option price appears as a Fourier inversion integral which can be efficiently calculated using numerical methods. Fouque et al [13] provide an approximation for European option prices for calibration in the multiscale model which is reasonably accurate. Relatively, little has been done to address the problem of American option pricing in stochastic volatility setting.…”

Option pricing under a stressed-beta model

“…Stochastic volatility models form a rich class of models which, in particular, generate the observed skews of implied volatilities [see for instance Fouque et al (2000)]. In general, unlike in the Black-Scholes model, there is no explicit option pricing formula, one exception being the Heston model (Heston 1993) for which European options are given semi-explicitly, up to an inverse Fourier transform.…”

“…5, we show how to incorporate in the model a fast mean-reverting stochastic volatility component in the dynamics of the market price, in order to account for the skew of implied volatility in the options on the market as well. We use a singular perturbation method introduced in Fouque et al (2000) and also employed in Fouque and Kollman (2009).…”