The problem of pricing arithmetic Asian options is nontrivial, and has attracted much interest over the last two decades. This paper provides a method for calculating bounds on option prices and approximations to option deltas in a market where the underlying asset follows a geometric Lévy process. The core idea is to find a highly correlated, yet more tractable proxy to the event that the option finishes in-the-money. The paper provides a means for calculating the joint characteristic function of the underlying asset and proxy processes, and relies on Fourier methods to compute prices and deltas. Numerical studies show that the lower bound provides accurate approximations to prices and deltas, while the upper bound provides good though less accurate results.

The local volatility model is a well-known extension of the Black–Scholes constant volatility model, whereby the volatility is dependent on both time and the underlying asset. This model can be calibrated to provide a perfect fit to a wide range of implied volatility surfaces. The model is easy to calibrate and still very popular in foreign exchange option trading. In this paper, we address a question of validation of the local volatility model. Different stochastic models for the underlying asset can be calibrated to provide a good fit to the current market data, which should be recalibrated every trading date. A good fit to the current market data does not imply that the model is appropriate, and historical backtesting should be performed for validation purposes. We study delta hedging errors under the local volatility model using historical data from 2005 to 2011 for the AUD/USDimplied volatility. We performed backtests for a range of option maturities and strikes using sticky delta and theoretically correct delta hedging. The results show that delta hedging errors under the standard Black–Scholes model are no worse than those of the local volatility model. Moreover, for the case of in- and at-the-money options, the hedging error for the Black–Scholes model is significantly better.

We discuss simulation of sensitivities or Greeks of multi-asset European style options under a special Lévy process model: that is, the subordinated Brownian motion model. The Malliavin calculus method combined with Monte Carlo and quasi-Monte Carlo methods is used in the simulations. Greeks are expressed in terms of the expectations of the option payoff functions multiplied by the weights involving Malliavin derivatives for multi-asset options. Numerical results show that the Malliavin calculus method is usually more efficient than the finite difference method for options with nonsmooth payoffs. The superiority of the former method over the latter is even more significant when both are combined with quasi-Monte Carlo methods.

We study the optimal proportional reinsurance and investment problem in a general jump-diffusion financial market. Assuming that the insurer’s surplus process follows a jump-diffusion process, the insurer can purchase proportional reinsurance from the reinsurer and invest in a risk-free asset and a risky asset, whose price is modelled by a general jump-diffusion process. The insurance company wishes to maximize the expected exponential utility of the terminal wealth. By using techniques of stochastic control theory, closed-form expressions for the value function and optimal strategy are obtained. A Monte Carlo simulation is conducted to illustrate that the closed-form expressions we derived are indeed the optimal strategies, and some numerical examples are presented to analyse the impact of model parameters on the optimal strategies.