The Small-Maturity Smile for Exponential Lévy Models

Abstract

We derive a small-time expansion for out-of-the-money call options under an exponential Lévy model, using the small-time expansion for the distribution function given in , combined with a change of numéraire via the Esscher transform. In particular, we find that the effect of a non-zero volatility σ of the Gaussian component of the driving Lévy process is to increase the call price by (1)) as t → 0, where ν is the Lévy density. Using the small-time expansion for call options, we then derive a small-time expansion for the implied volatilityσ 2 t (k) at log-moneyness k, which sharpens the first order estimateσ. Our numerical results show that the second order approximation can significantly outperform the first order approximation. Our results are also extended to a class of time-changed Lévy models. We also consider a small-time, small log-moneyness regime for the CGMY model, and apply this approach to the small-time pricing of at-the-money call options; we show that for Y ∈ (1, 2), limt→0 t −1/Y E(St − S0)+ = S0E * (Z+) and the corresponding at-the-money implied volatilityσt(0) satisfies limt→0σt (0), where Z is a symmetric Ystable random variable under P * and Y is the usual parameter for the CGMY model appearing in the Lévy density ν(x) = Cx −1−Y e −M x 1 {x>0} + C|x| −1−Y e −G|x| 1 {x<0} of the process.

Asymptotics of Forward Implied Volatility

“…The large-maturity forward smile asymptotic is given in the following proposition, proved in Section 5.1. When t = 0 in (2.11) and (2.13) below, we recover-and improve-the asymptotics in [16], [18], [19], [20], [21]. It is interesting to note that the (strict) martingale property (Λ 0 (1) = 0) is only required in Proposition 2.12 below and not in Proposition 2.10 and Theorem 2.4.…”

The Large-Maturity Smile for the Sabr and Cev-Heston Models

“…Kt γ → ∞ as t → ∞, so this is a large-time, large-strike regime for the CEV model, similar to the large-time, large-strike regime for the Heston and exponential Lévy models discussed in [9,12].…”

Pathwise large deviations for the rough Bergomi model

“…The main shortcoming of such stochastic volatility models, however, is that they are unable to capture the true steepness of the implied volatility smile close to maturity. While choosing to add jumps to stock price models, for example modelling the stock price process as an exponential Lévy process, does indeed produce steeper implied volatility smiles [17], the question of the presence of jumps in stock price processes remains controversial [6,12].…”