Estimating the Hurst parameter from short term volatility swaps: a Malliavin calculus approach

Abstract

This paper is devoted to studying the difference between the fair strike of a volatility swap and the at-the-money implied volatility (ATMI) of a European call option. It is well-known that the difference between these two quantities converges to zero as the time to maturity decreases. In this paper, we make use of a Malliavin calculus approach to derive an exact expression for this difference. This representation allows us to establish that the order of the convergence is different in the correlated and in the uncorrelated case, and that it depends on the behavior of the Malliavin derivative of the volatility process. In particular, we will see that for volatilities driven by a fractional Brownian motion, this order depends on the corresponding Hurst parameter H. Moreover, in the case H ≥ 1/2, we develop a model-free approximation formula for the volatility swap, in terms of the ATMI and its skew.

Rough multifactor volatility for SPX and VIX options

“…The first theorem states that the small-time limit of the implied volatility is equal to the limit of the forward volatility. This is well known for Markovian stochastic volatility models in [5,10] and in a one-factor setting [2]. To streamline the call to the assumptions, we shall group them using mixed subscript notations, for example (H 123 ) corresponds to (H…”

“…Note that we did not assume the limit of E[u 0 ] to be finite. The proof, in Appendix 6.2.1, builds on arguments from [5,Proposition 3.1]. We then turn our attention to the ATM skew, defined in 2.1.…”

“…This is required to prove the curvature formula using Clark-Ocone formula (2) and three times the anticipative Itô formula. • These assumptions can be replaced by direct conditions on φ when only considering the stock price as underlying martingale price process, as in [3,4,5].…”

“…Proofs of the main results.6.2.1. Proof of Theorem 3.1: Level.To prove this result, we draw insights from the proofs of [2, Theorem 8] and[5, Proposition 3.1]. By definition…”

Continuation value computation using Malliavin calculus under general volatility stochastic process for American option pricing

“…Yamada [28] provided an approximation scheme for multidimensional Stratonovich stochastic differential equations using Malliavin calculus, and applied it to the SABR model. Yamada and Yamamoto [29] then constructed a second-order discretization scheme that they applied to price European option under the SABR model, while Alòs and Shiraya [3] studied volatility swaps and European options. Alòs and Lorite [2] have recently reviewed the application of Malliavin calculus in finance, but ignored American option pricing.…”