Using the large deviation principle (LDP) for a re-scaled fractional Brownian motion B H t where the rate function is defined via the reproducing kernel Hilbert space, we compute small-time asymptotics for a correlated fractional stochastic volatility model of the form dSt = Stσ(Yt)(ρdWt + ρdBt), dYt = dB H t where σ is α-Hölder continuous for some α ∈ (0, 1]; in particular, we show that t H− 1 2 log St satisfies the LDP as t → 0 and the model has a well-defined implied volatility smile as t → 0, when the log-moneyness k(t) = xt 1 2 −H . Thus the smile steepens to infinity or flattens to zero depending on whether H ∈ (0,, 1). We also compute large-time asymptotics for a fractional local-stochastic volatility model of the form: ) and µ > 0 as t → ∞.
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 Gärtner-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 smallmaturity 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).
Abstract. Using the Gärtner-Ellis theorem from large deviations theory, we characterize the leadingorder 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 [40] 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.
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