In this paper we investigate the asymptotics of forward-start options and the forward implied volatility smile in the Heston model as the maturity approaches zero. We prove that the forward smile for out-ofthe-money options explodes and compute a closed-form high-order expansion detailing the rate of the explosion.Furthermore the result shows that the square-root behaviour of the variance process induces a singularity such that for certain parameter configurations one cannot obtain high-order out-of-the-money forward smile asymptotics. In the at-the-money case a separate model-independent analysis shows that the small-maturity limit is well defined for any Itô diffusion. The proofs rely on the theory of sharp large deviations (and refinements) and incidentally we provide an example of degenerate large deviations behaviour.smile σ t,τ (k) is then defined (see also [9,30]) as the unique solution to C obs (t, τ, k) = C BS (τ, k, σ t,τ (k)). The forward smile is a generalisation of the spot implied volatility smile, and the two are equal when t = 0.Asymptotics of the spot implied volatility surface have received a large amount of attention over the past decade. These results have helped shape calibration methodologies based on arbitrage-free approximation of the spot smile in a large variety of models. Small-maturity asymptotics have been studied by Berestycki-Busca-Florent [8] using PDE methods for continuous time diffusions and by Henry-Labordère [28] using heat kernel expansions. Forde et al. [18] and Jacquier et al. [32] derived small and large-maturity asymptotics in Date: November 8, 2018. 2010 Mathematics Subject Classification. 60F10, 91G99, 91G60.
Abstract. 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 Lévy models. This expansion applies to both small and large maturities and is based solely on the knowledge 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 show (i) that the small-maturity exploding behaviour of forward smiles depends on whether the quadratic variation of the underlying is bounded or not, and (ii) that the forward-start date also has to be rescaled in order to obtain non-trivial small-maturity asymptotics.
We consider the fractional Heston model originally proposed by Comte, Coutin and Renault [12]. Inspired by recent groundbreaking work on rough volatility [2, 6, 24, 26] which showed that models with volatility driven by fractional Brownian motion with short memory allows for better calibration of the volatility surface and more robust estimation of time series of historical 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 smile.
We consider the fractional Heston model originally proposed by Comte, Coutin and Renault [12]. Inspired by recent ground-breaking work on rough volatility [2,6,24,26] which showed that models with volatility driven by fractional Brownian motion with short memory allows for better calibration of the volatility surface and more robust estimation of time series of historical 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 smile.Date: August 10, 2017.2010 Mathematics Subject Classification. 60F10, 91G99, 91B25.
Abstract. 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 Lévy models. This expansion applies to both small and large maturities and is based solely on the knowledge 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 show (i) that the small-maturity exploding behaviour of forward smiles depends on whether the quadratic variation of the underlying is bounded or not, and (ii) that the forward-start date also has to be rescaled in order to obtain non-trivial small-maturity asymptotics.
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