Antoine Jacquier

SMALL-TIME ASYMPTOTICS FOR IMPLIED VOLATILITY UNDER THE HESTON MODEL

We rigorize the work of Lewis (2007) and Durrleman (2005) on the small-time asymptotic behavior of the implied volatility under the Heston stochastic volatility model (Theorem 2.1). We apply the Gartner-Ellis theorem from large deviations theory to the exponential affine closed-form expression for the moment generating function of the log forward price, to show that it satisfies a small-time large deviation principle. The rate function is computed as Fenchel-Legendre transform, and we show that this can actually be computed as a standard Legendre transform, which is a simple numerical root-finding exercise. We establish the corresponding result for implied volatility in Theorem 3.1, using well known bounds on the standard Normal distribution function. In Theorem 3.2 we compute the level, the slope and the curvature of the implied volatility in the small-maturity limit At-the-money, and the answer is consistent with that obtained by formal PDE methods by Lewis (2000) and probabilistic methods by Durrleman (2004).

The small-time smile and term structure of implied volatility under the Heston model

We characterize the asymptotic smile and term structure of implied volatility in the Heston model at small maturities. Using saddlepoint methods we derive a small-maturity expansion formula for call option prices, which we then transform into a closed-form expansion (including the leading-order and correction terms) for implied volatility. This refined expansion reveals the relationship between the small-expiry smile and all Heston parameters (including the pair in the volatility drift coefficient), sharpening the leading-order result of Forde and Jacquier [Int. J. Theor. Appl. Finance, 12 (2009), pp. 861--876], which found the relationship between the zero-expiry smile and the diffusion coefficients.

Arbitrage-Free SVI Volatility Surfaces

In this article, we show how to calibrate the widely used SVI parameterization of the implied volatility smile in such a way as to guarantee the absence of static arbitrage. In particular, we exhibit a large class of arbitrage-free SVI volatility surfaces with a simple closed-form representation. We demonstrate the high quality of typical SVI fits with a numerical example using recent SPX options data.

The Large Maturity Smile for the Heston Model

Using the Gärtner–Ellis theorem from large deviations theory, we characterise the leading-order behaviour of call option prices under the Heston model, in a new regime where the maturity is large and the log-moneyness is also proportional to the maturity. Using this result, we then derive the implied volatility in the large-time limit in the new regime, and we find that the large-time smile mimics the large-time smile for the Barndorff–Nielsen normal inverse Gaussian model. This makes precise the sense in which the Heston model tends to an exponential Lévy process for large times. We find that the implied volatility smile does not flatten out as the maturity increases, but rather it spreads out, and the large-time, large-moneyness regime is needed to capture this effect. As a special case, we provide a rigorous proof of the well-known result by Lewis (Option Valuation Under Stochastic Volatility, Finance Press, Newport Beach, 2000) for the implied volatility in the usual large-time, fixed-strike regime, at leading order. We find that there are two critical strike values where there is a qualitative change of behaviour for the call option price, and we use a limiting argument to compute the asymptotic implied volatility in these two cases.

Convergence of Heston to SVI

In this short note, we prove by an appropriate change of variables that the SVI implied volatility parameterization presented in Gatherals book and the large-time asymptotic of the Heston implied volatility agree algebraically, thus confirming a conjecture from Gatheral as well as providing a simpler expression for the asymptotic implied volatility in the Heston model. We show how this result can help in interpreting SVI parameters.

Asymptotic formulae for implied volatility in the Heston model

In this paper, we prove an approximate formula expressed in terms of elementary functions for the implied volatility in the Heston model. The formula consists of the constant and first-order terms in the large maturity expansion of the implied volatility function. The proof is based on saddlepoint methods and classical properties of holomorphic functions.

Small-Time Asymptotics for an Uncorrelated Local-Stochastic Volatility Model

Abstract We add some rigour to the work of Henry-Labordère (2009; Analysis, Geometry, and Modeling in Finance: Advanced Methods in Option Pricing (London and New York: Chapman & Hall)), Lewis (2007; Geometries and Smile Asymptotics for a Class of Stochastic Volatility Models. Available at http://www.optioncity.net (accessed 28 May 2011)) and Paulot (2009; Asymptotic implied volatility at the second order with application to the SABR model, Working Paper, Available at papers.ssrn.com/sol3/papers.cfm?abstract_id=1413649 (accessed 11 June 2011)) on the small-time behaviour of a local-stochastic volatility model with zero correlation at leading order. We do this using the Freidlin—Wentzell (FW) theory of large deviations for stochastic differential equations (SDEs), and then converting to a differential geometry problem of computing the shortest geodesic from a point to a vertical line on a Riemmanian manifold, whose metric is induced by the inverse of the diffusion coefficient. The solution to this variable endpoint problem is obtained using a transversality condition, where the geodesic is perpendicular to the vertical line under the aforementioned metric. We then establish the corresponding small-time asymptotic behaviour for call options using Hölders inequality, and the implied volatility (using a general result in Roper and Rutkowski (forthcoming, A note on the behavior of the Black–Scholes implied volatility close to expiry, International Journal of Thoretical and Applied Finance). We also derive a series expansion for the implied volatility in the small-maturity limit, in powers of the log-moneyness, and we show how to calibrate such a model to the observed implied volatility smile in the small-maturity limit.

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 Levy 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 nontrivial small-maturity asymptotics, (ii) prove that the forward-start date may influence the large-maturity behavior of the forward smile, and (iii) provide some examples of models with finite quadratic variation where the small-maturity forward smile does not explode.

Asymptotic Behaviour of the Fractional Heston Model

We consider here the fractional version of the Heston model originally proposed by Comte, Coutin and Renault. Inspired by some recent ground-breaking work by Gatheral, Jaisson and Rosenbaum, who showed that fractional Brownian motion with short memory allows for a better calibration of the volatility surface (as opposed to the classical econometric approach of long memory of volatility), we provide a characterisation of the short- and long-maturity asymptotics of the implied volatility smile. Our analysis reveals that the short-memory property precisely provides a jump-type behaviour of the smile for short maturities, thereby fixing the well-known standard inability of classical stochastic volatility models to fit the short-end of the volatility skew.

Large Deviations and Stochastic Volatility with Jumps: Asymptotic Implied Volatility for Affine Models

Let