General Optimized Lower and Upper Bounds for Discrete and Continuous Arithmetic Asian Options

We develop accurate analytical pricing formulae for discretely and continuously monitored arithmetic Asian options under general stochastic asset models, including exponential Lévy models, stochastic volatility models, and the constant elasticity of variance diffusion. The payoff of the arithmetic Asian option depends on the arithmetic average price of the underlying asset monitored over a pre-specified period. Unluckily, the pricing of arithmetic Asian options does not admit true analytical solutions, even under the lognormal model, as the distribution law of the arithmetic average is not known analytically. For more than two decades, much effort has been put into the research on efficient methodologies for computing the price of this option or, in general, expected values of functionals of the average value, under different model assumptions for the underlying. Developing such methods is of considerable practical importance as arithmetic averages see wide application in many fields of finance. The main objective of this research is to present a simple, accurate and fast pricing formula in the form of a lower bound for arithmetic Asian options allowing flexible modelling of the underlying asset price dynamics, filling this way an important long-standing gap in the literature. Our proposed method is distinguished from other pricing methodologies for Asian options due to a number of appealing features. First, it can be applied flexibly to a wide range of non-Gaussian models, such as pure jump Lévy models, Merton’s normal and Cai and Kou’s generalized hyperexponential jump diffusions, models with/out jumps in the asset price/volatility dynamics, and the CEV diffusion, without restricting to models admitting time changed Brownian (Lévy) representations which may not be always common or straightforward to use. Second, we provide interesting theoretical findings related to the pricing of Asian options in the CEV