Large Deviation Principle for Volterra Type Fractional Stochastic Volatility Models

Abstract

We study fractional stochastic volatility models in which the volatility process is a positive continuous function σ of a continuous Gaussian process B. Forde and Zhang established a large deviation principle for the log-price process in such a model under the assumptions that the function σ is globally Hölder-continuous and the process B is fractional Brownian motion. In the present paper, we prove a similar small-noise large deviation principle under weaker restrictions on σ and B. We assume that σ satisfies a mild local regularity condition, while the process B is a Volterra type Gaussian process. Under an additional assumption of the self-similarity of the process B, we derive a large deviation principle in the small-time regime. As an application, we obtain asymptotic formulas for binary options, call and put pricing functions, and the implied volatility in certain mixed regimes.AMS 2010 Classification: 60F10, 60G15, 60G18, 60G22, 41A60, 91G20.

Pathwise Asymptotics for Volterra Type Stochastic Volatility Models

“…The principal result we use is Chaganty Theorem (see Theorem 2.3 in [6]), where a large deviation principle for joint and marginal distributions is stated. In this way the same results contained in [13], [17] and [18] can be obtained in a more general context, see Section 7. In the stochastic volatility models of interest, the dynamic of the asset price process (S t ) t∈[0,T ] is modeled by the following equation…”

“…However, the increments of the Riemann-Liouville fractional Brownian motion lack the stationarity property. Condition (b) for this process, with α = 2H, was established in Lemma 7 in [17]. a-th fold integrated Brownian motion.…”

“…Notice that when H = 1/2 the fractional Brownian motion reduces to the Wiener process. Condition (b) for this process, with α = min{2H, 1}, was established in [28] and Lemma 8 in [17]. Fractional Ornstein-Uhlenbeck process.…”

“…(see, e.g., Proposition A.1 in [8]). Condition (b) for this process, with α = min{2H, 1}, was established in Lemma 10 in [17]. Note that this is not a self similar process, therefore large deviations for small-time cannot be deduced from large deviations for small-noise.…”

“…The function Ψ is finite on C 0 [0, T ] 2 and, for f ∈ H Remark 6.20. Let D L be defined as in (17). Then thanks to Remarks 5.1 and 5.3, for f ∈ D L there exist constants µ L , σ L and σ L (depending on L) such that, for f ∈ D L and t ∈ [0, T ], we have…”