Asymptotic formulae for implied volatility in the Heston model

TL;DR: In this article, a tail expansion for the Heston density was derived and a new parameter called critical slope was defined in a model free manner, which drives the second and higher order terms in tail-and implied volatility expansions.

TL;DR: In this paper, it was shown that the implied volatility has a uniform (in log moneyness x) limit as the maturity tends to infinity, given by an explicit closed-form formula, for x in some compact neighborhood of zero in the class of affine stochastic volatility models.

TL;DR: In this paper, a general closed-form expansion formula for forward-start options and the forward implied volatility smile was proposed, which applies to both small and large maturities, based solely on the properties of the forward characteristic function of the underlying process.

TL;DR: 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 is proved.

TL;DR: In this paper, the authors present a characterization of continuous local martingales with respect to Brownian motion in terms of Markov properties, including the strong Markov property, and a generalized version of the Ito rule.

TL;DR: A classic reference, intended for graduate students mathematicians, physicists, and engineers, this book can be used both as the basis for instructional courses and as a reference tool as discussed by the authors, and it can be found in many libraries.

TL;DR: A First Course Algebraic methods in Markov Chains Ratio Theorems of Transition Probabilities and Applications Sums of Independent Random Variables as a Markov Chain Order Statistics, Poisson Processes, and Applications Continuous Time Markov chains Diffusion Processes Compounding Stochastic Processes Fluctuation Theory of Partial Sum of Independent Identically Distributed Random Variable Queueing Processes Miscellaneous Problems Index as discussed by the authors.

TL;DR: The Black-Scholes theory of derivative pricing has been applied to derivatives as discussed by the authors, where the rate of mean-reverting stochastic volatility has been estimated for European derivatives.