We extend previous large deviations results for the randomised Heston model to the case of moderate deviations. The proofs involve the Gärtner–Ellis theorem and sharp large deviations tools.

Options with extendable features have many applications in finance and these provide the motivation for this study. The pricing of extendable options when the underlying asset follows a geometric Brownian motion with constant volatility has appeared in the literature. In this paper, we consider holder-extendable call options when the underlying asset follows a mean-reverting stochastic volatility. The option price is expressed in integral forms which have known closed-form characteristic functions. We price these options using a fast Fourier transform, a finite difference method and Monte Carlo simulation, and we determine the efficiency and accuracy of the Fourier method in pricing holder-extendable call options for Heston parameters calibrated from the subprime crisis. We show that the fast Fourier transform reduces the computational time required to produce a range of holder-extendable call option prices by at least an order of magnitude. Numerical results also demonstrate that when the Heston correlation is negative, the Black–Scholes model under-prices in-the-money and over-prices out-of-the-money holder-extendable call options compared with the Heston model, which is analogous to the behaviour for vanilla calls.

The main purpose of this paper is to present a novel analytical approach for pricing discretely sampled gamma swaps, defined in terms of weighted variance swaps of the underlying asset, based on Heston’s two-factor stochastic volatility model. The closed-form formula obtained in this paper is in a much simpler form than those proposed in the literature, which substantially reduces the computational burden and can be implemented efficiently. The solution procedure presented in this paper can be adopted to derive closed-form solutions for pricing various types of weighted variance swaps, such as self-quantoed variance and entropy swaps. Most interestingly, we discuss the validity of the current solutions in the parameter space, and provide market practitioners with some remarks for trading these types of weighted variance swaps.

We develop a simplified analytical approach for pricing discretely-sampled variance swaps with the realized variance, defined in terms of the squared log return of the underlying price. The closed-form formula obtained for Heston’s two-factor stochastic volatility model is in a much simpler form than those proposed in literature. Most interestingly, we discuss the validity of our solution as well as some other previous solutions in different forms in the parameter space. We demonstrate that market practitioners need to be cautious, making sure that their model parameters extracted from market data are in the right parameter subspace, when any of these analytical pricing formulae is adopted to calculate the fair delivery price of a discretely-sampled variance swap.

In this paper we investigate the asymptotic behaviors of the finite- and infinite-time ruin probabilities for a Poisson risk model with stochastic investment returns which constitute a general adapted càdlàg process and heavy-tailed claim sizes which are bivariate upper tail independent. The results of this paper show that the asymptotic ruin probabilities are dominated by the extreme of insurance risk but not by that of investment risk. As applications of the results, we discuss four special cases when the investment returns are determined by a fractional Brownian motion, an integrated Vasicek model, an integrated Cox–Ingersoll–Ross model, and the Heston model.

We prove that the Heston volatility is Malliavin differentiable under the classical Novikov condition and give an explicit expression for the derivative. This result guarantees the applicability of Malliavin calculus in the framework of the Heston stochastic volatility model. Furthermore, we derive conditions on the parameters which assure the existence of the second Malliavin derivative of the Heston volatility. This allows us to apply recent results of Alòs (2006) in order to derive approximate option pricing formulae in the context of the Heston model. Numerical results are given.