On the Short-Time Behavior of the Implied Volatility for Jump-Diffusion Models With Stochastic Volatility

Abstract

In this paper we use the Malliavin calculus techniques to obtain an expression for the short-time behavior of the at-the-money implied volatility skew for a generalization of the Bates model, where the volatility does not need to be neither a difussion, nor a Markov process, as the examples in section 7 show. This expression depends on the derivative of the volatility in the sense of Malliavin calculus.

A regularity structure for rough volatility

“…Note the stark contrast to the idea of “trending” fractional volatility, which amounts to taking . The evidence for the rough regime (recent calibration suggests H as low as 0.05) is now overwhelming—both under the physical and the pricing measure (see Alòs, León, & Vives, ; Bayer, Friz, & Gatheral, ; Forde & Zhang, ; Fukasawa, , ; Gatheral, Jaisson, & Rosenbaum, ; Mijatović & Tankov, ). It should be noted, however, that these different regimes can be easily mixed, so that rough volatility governs the short time behavior, while trending volatility affects the long time behavior; we refer to Comte and Renault (), Comte, Coutin, and Renault (), Alòs and Yang (), and Bennedsen, Lunde, and Pakkanen () for more information on this.…”

“…Alòs et al. () and Fukasawa (Fukasawa, , )—that in the previously considered simple rough volatility models, now writing σ(.) instead of f (.…”

A Hull and White formula for a stochastic volatility Lévy model with infinite activity

“…If the volatility process is independent from price jumps, we have D N,− s,y u(s−, y) = 0 and we obtain V t = E t (BS(t, X t , v t )) that generalizes the formula in [3]. As in the previous remark, only in the finite variation case we can recuperate exactly the formula in [3]. This formula covers Bates model and any correlated model with any type of Lévy jumps in the price process.…”

“…On the stochastic interval [T j , T j+1 [ we can apply the anticipative Itô formula for continuous process presented in [5] and proceed as in [3]. Then we have that…”

“…On other hand, as it was shown in [8], any generalization of Black-Scholes model, from the case of deterministic volatility to different cases of stochastic volatility, give a pricing formula, also called Hull and White formula, that depends not on the current volatility but on the future average volatility. The fact that the future average volatility is a non adapted process suggests the use of anticipative calculus techniques as the natural tool to deal with anticipative processes as done in [3] and [2]. In these papers, a general jump diffusion model with no precise assumption on the dynamics of the volatility process is analyzed.…”