Asymptotics of Implied Volatility in Local Volatility Models

Abstract

Using an expansion of the transition density function of a one-dimensional time inhomogeneous diffusion, we obtain the first-and second-order terms in the short time asymptotics of European call option prices. The method described can be generalized to any order. We then use these option prices approximations to calculate the firstand second-order deviation of the implied volatility from its leading value and obtain approximations which we numerically demonstrate to be highly accurate.

Small-Time Asymptotics of Option Prices and First Absolute Moments

“…Due to their importance for model calibration and testing, small-time asymptotics of option prices have received considerable attention in recent years; see [2,5,6,7,10,11,12,13,14,15,16,17,22,24]. A survey of recent literature is given in the introduction of Forde et al [16].…”

Options with Extreme Strikes

“…However, the majority of current research in this area has focused on short maturity asymptotics and implied volatility for European options; see, e.g., [1][2][3][4][5] for the local volatility models, [6][7][8][9][10] for the exponential Lévy models, [11][12][13][14][15][16][17] for the stochastic volatility models and [18,19] for model-free frameworks.…”

Option Pricing in the Moderate Deviations Regime

“…An expansion of the pdf q(·, t) of S t has been worked out in Gatheral et al [22]. They assume growth conditions on σ and its derivatives, which can be alleviated by the principle of not feeling the boundary (Appendix A of [22]).…”

“…Generic time-inhomogeneous local volatility models dS t = σ(S t , t)S t dW t could be treated very similarly, using the heat kernel expansion in Section 3 of [22], itself taken from Yosida [38].…”

“…They assume growth conditions on σ and its derivatives, which can be alleviated by the principle of not feeling the boundary (Appendix A of [22]). Proposition 2.1 of [22] says that…”