Asymptotics of forward implied volatility

We prove here a general closed-form expansion formula for forward-start options and the forward implied volatility smile in a large class of models, including the Heston stochastic volatility and time-changed exponential L\'evy models. This expansion applies to both small and large maturities and is based solely on the properties of the forward characteristic function of the underlying process. The method is based on sharp large deviations techniques, and allows us to recover (in particular) many results for the spot implied volatility smile. In passing we (i) show that the forward-start date has to be rescaled in order to obtain non-trivial small-maturity asymptotics, (ii) prove that the forward-start date may influence the large-maturity behaviour of the forward smile, and (iii) provide some examples of models with finite quadratic variation where the small-maturity forward smile does not explode.Comment: 37 pages, 13 figure

Asymptotics of forward implied volatility

We study asymptotics of forward-start option prices and the forward implied volatility smile using the theory of sharp large deviations (and refinements). In Chapter 1 we give some intuition and insight into forward volatility and provide motivation for the study of forward smile asymptotics. We numerically analyse no-arbitrage bounds for the forward smile given calibration to the marginal distributions using (martingale) optimal transport theory. Furthermore, we derive several representations of forward-start option prices, analyse various measure-change symmetries and explore asymptotics of the forward smile for small and large forward-start dates. In Chapter 2 we derive a general closed-form expansion formula (including large-maturity and `diagonal' small-maturity asymptotics) for the forward smile in a large class of models including the Heston and Schobel-Zhu stochastic volatility models and time-changed exponential Levy models. In Chapter 3 we prove that the out-of-the-money small-maturity forward smile explodes in the Heston model and a separate model-independent analysis shows that the at-the-money small-maturity limit is well defined for any Ito diffusion.Chapter 4 provides a full characterisation of the large-maturity forward smile in the Heston model. Although the leading-order decay is provided by a fairly classical large deviations behaviour, the algebraic expansion providing the higher-order terms depends highly on the parameters, and different powers of the maturity come into play. Classical (Ito diffusions) stochastic volatility models are not able to capture the steepness of small-maturity (spot) implied volatility smiles. Models with jumps, exhibiting small-maturity exploding smiles, have historically been proposed as an alternative. A recent breakthrough was made by Gatheral, Jaisson and Rosenbaum, who proposed to replace the Brownian driver of the instantaneous volatility by a short-memory fractional Brownian motion, which is able to capture the short-maturity steepness while preserving path continuity. In Chapter 5 we suggest a different route, randomising the Black-Scholes variance by a CEV-generated distribution, which allows us to modulate the rate of explosion (through the CEV exponent) of the implied volatility for small maturities. The range of rates includes behaviours similar to exponential Levy models and fractional stochastic volatility models. As a by-product, we make a conjecture on the small-maturity forward smile asymptotics of stochastic volatility models, in exact agreement with the results in Chapter 3 for Heston.Open Acces

Large-Maturity Regimes of the Heston Forward Smile

We provide a full characterisation of the large-maturity forward implied volatility smile in the Heston model. Although the leading decay is provided by a fairly classical large deviations behaviour, the algebraic expansion providing the higher-order terms highly depends on the parameters, and different powers of the maturity come into play. As a by-product of the analysis we provide new implied volatility asymptotics, both in the forward case and in the spot case, as well as extended SVI-type formulae. The proofs are based on extensions and refinements of sharp large deviations theory, in particular in cases where standard convexity arguments fail.Comment: 32 pages, 16 figures New Section 5 providing more (financial) intuition

Asymptotics for Exponential Levy Processes and their Volatility Smile: Survey and New Results

Exponential L\'evy processes can be used to model the evolution of various financial variables such as FX rates, stock prices, etc. Considerable efforts have been devoted to pricing derivatives written on underliers governed by such processes, and the corresponding implied volatility surfaces have been analyzed in some detail. In the non-asymptotic regimes, option prices are described by the Lewis-Lipton formula which allows one to represent them as Fourier integrals; the prices can be trivially expressed in terms of their implied volatility. Recently, attempts at calculating the asymptotic limits of the implied volatility have yielded several expressions for the short-time, long-time, and wing asymptotics. In order to study the volatility surface in required detail, in this paper we use the FX conventions and describe the implied volatility as a function of the Black-Scholes delta. Surprisingly, this convention is closely related to the resolution of singularities frequently used in algebraic geometry. In this framework, we survey the literature, reformulate some known facts regarding the asymptotic behavior of the implied volatility, and present several new results. We emphasize the role of fractional differentiation in studying the tempered stable exponential Levy processes and derive novel numerical methods based on judicial finite-difference approximations for fractional derivatives. We also briefly demonstrate how to extend our results in order to study important cases of local and stochastic volatility models, whose close relation to the L\'evy process based models is particularly clear when the Lewis-Lipton formula is used. Our main conclusion is that studying asymptotic properties of the implied volatility, while theoretically exciting, is not always practically useful because the domain of validity of many asymptotic expressions is small.Comment: 92 pages, 15 figure

The Small and Large Time Implied Volatilities in the Minimal Market Model

This paper derives explicit formulas for both the small and large time limits of the implied volatility in the minimal market model. It is shown that interest rates do impact on the implied volatility in the long run even though they are negligible in the short time limit.Comment: 50 pages, 4 figures, typo on page 18 correcte