Motivated by a common mathematical finance topic, we discuss the reciprocal of the exit time from a cone of planar Brownian motion which also corresponds to the exponential functional of Brownian motion in the framework of planar Brownian motion. We prove a conjecture of Vakeroudis and Yor (2012) concerning infinite divisibility properties of this random variable and present a novel simple proof of the result of DeBlassie (1987), (1988) concerning the asymptotic behavior of the distribution of the Bessel clock appearing in the skew-product representation of planar Brownian motion, as t→∞. We use the results of the windings approach in order to obtain results for quantities associated to the pricing of Asian options.

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.

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.

Spectral expansion techniques have been extensively exploited for the pricing of exotic options. In this paper, we present novel applications of spectral methods for the quantitative risk management of variable annuity guaranteed benefits such as guaranteed minimum maturity benefits and guaranteed minimum death benefits. The objective is to find efficient and accurate solution methods for the computation of risk measures, which is the key to determining risk-based capital according to regulatory requirements. Our example calculations show that two spectral methods used in this paper are highly efficient and numerically more stable than conventional known methods. Hence these approaches are more suitable for intensive calculations involving death benefits.

We generalize, in terms of power series, the celebrated Geman-Yor formula for the pricing of Asian options in the framework of spectrally negative Lévy-driven assets. We illustrate our result by providing some new examples.

Barrieu, Rouault and Yor (2004) determined asymptotics for the logarithm of the distribution function of the Hartman-Watson distribution. We determine the asymptotics of the density. This refinement can be applied to the pricing of Asian options in the Black-Scholes model.

One approach to the computation of the price of an Asian option involves the Hartman–Watson distribution. However, numerical problems for its density occur for small values. This motivates the asymptotic study of its distribution function.

Sums of log-normals frequently appear in a variety of situations, including engineering and financial mathematics. In particular, the pricing of Asian or basket options is directly related to finding the distributions of such sums. There is no general explicit formula for the distribution of sums of log-normal random variables. This paper looks at the limit distributions of sums of log-normal variables when the second parameter of the log-normals tends to zero or to infinity; in financial terms, this is equivalent to letting the volatility, or maturity, tend either to zero or to infinity. The limits obtained are either normal or log-normal, depending on the normalization chosen; the same applies to the reciprocal of the sums of log-normals. This justifies the log-normal approximation, much used in practice, and also gives an asymptotically exact distribution for averages of log-normals with a relatively small volatility; it has been noted that all the analytical pricing formulae for Asian options perform poorly for small volatilities. Asymptotic formulae are also found for the moments of the sums of log-normals. Results are given for both discrete and continuous averages. More explicit results are obtained in the case of the integral of geometric Brownian motion.

The aim of this paper is to investigate discrete approximations of the exponential functional of Brownian motion (which plays an important role in Asian options of financial mathematics) with the help of simple, symmetric random walks. In some applications the discrete model could be even more natural than the continuous one. The properties of the discrete exponential functional are rather different from the continuous one: typically its distribution is singular with respect to Lebesgue measure, all of its positive integer moments are finite and they characterize the distribution. On the other hand, using suitable random walk approximations to Brownian motion, the resulting discrete exponential functionals converge almost surely to the exponential functional of Brownian motion; hence their limit distribution is the same as in the continuous case, namely that of the reciprocal of a gamma random variable, and so is absolutely continuous with respect to Lebesgue measure. In this way, we also give a new and elementary proof of an earlier result by Dufresne and Yor.

This paper studies the law of any real powers of the integral of geometric Brownian motion over finite time intervals. As its main results, an apparently new integral representation is derived and its interrelations with the integral representations for these laws originating by Yor and by Dufresne are established. In fact, our representation is found to furnish what seems to be a natural bridge between these other two representations. Our results are obtained by enhancing the Hartman-Watson Ansatz of Yor, based on Bessel processes and the Laplace transform, by complex analytic techniques. Systematizing this idea in order to overcome the limits of Yor's theory seems to be the main methodological contribution of the paper.

There are two types of Asian options in the financial markets which differ according to the role of the average price. We give a symmetry result between the floating- and fixed-strike Asian options. The proof involves a change of numéraire and time reversal of Brownian motion. Symmetries are very useful in option valuation, and in this case the result allows the use of more established fixed-strike pricing methods to price floating-strike Asian options.

We give 2 explicit formulae for the hedging portfolio of Asian options. One is based on the usual Lognormal approximation, and the other on an Inverse Gaussian approximation. Both give excellent results as replicating strategies when the parameters of the model are in a reasonable range.