The time average of geometric Brownian motion plays a crucial role in the pricing of Asian options in mathematical finance. In this paper we consider the asymptotics of the discrete-time average of a geometric Brownian motion sampled on uniformly spaced times in the limit of a very large number of averaging time steps. We derive almost sure limit, fluctuations, large deviations, and also the asymptotics of the moment generating function of the average. Based on these results, we derive the asymptotics for the price of Asian options with discrete-time averaging in the Black–Scholes model, with both fixed and floating strike.

We present a rigorous study of the short maturity asymptotics for Asian options with continuous-time averaging, under the assumption that the underlying asset follows the constant elasticity of variance (CEV) model. The leading order short maturity limit of the Asian option prices under the CEV model is obtained in closed form. We propose an analytical approximation for the Asian options prices which reproduces the exact short maturity asymptotics, and demonstrate good numerical agreement of the asymptotic results with Monte Carlo simulations and benchmark test cases for option parameters relevant for practical applications.

The number of crossings of the implied volatility function with a fixed level is bounded above by the number of crossings of the risk-neutral density with the density of a log-normal distribution with the same mean as the forward price. It is bounded below by the number of convex payoffs priced equally by the two densities. We discuss the implications of these bounds for the implied volatility in the N-component Gaussian mixture model, with particular attention to the possibility of W-shaped smiles. We show that the implied volatility in this model crosses any level at most $2(N-1)$ times. We show that a bimodal density need not produce a W-shaped smile, and a unimodal density can produce an oscillatory smile. We give monotonicity properties of the implied volatility in Gaussian mixtures under stochastic orderings of the location parameters and volatilities of the mixture components. For some of these results we make use of a novel convexity property of the Black-Scholes price at one strike with respect to the price at another strike. The combined constraints from density crossings and extreme strike asymptotics restrict the allowed shapes of the implied volatility. As an application we discuss a symmetric N = 3 Gaussian mixture model which generates three possible smile shapes: U-shaped, W-shaped and an oscillatory shape with two minima and two maxima.