The Esscher Transform and the Minimal Martingale Measure

In this chapter we present two ways to choose a unique martingale measure in an incomplete market. The first way is to use an extended version of the Esscher transform, which implies that we restrict the class of martingale measures. The second way is to use the minimal martingale measure, that is, the measure which minimizes the norm of the associated Girsanov kernel. We exemplify the two methods and discuss the economic significance.

Moment-Based Density Approximation Techniques as Applied to Heavy-tailed Distributions

Several advances are made in connection with the approximation and estimation of heavy-tailed distributions. It is first explained that on initially applying the Esscher transform to heavy-tailed density functions such as the Pareto, Studentt and Cauchy, said densities can be approximated by employing a certain moment-based methodology. Alternatively, density approximants can be obtained by appropriately truncating such distributions or mapping them onto finite supports. These techniques are then extended to the context of density estimation, their validity being demonstrated by means of simulation studies. As well, illustrative actuarial examples are presented.

Series representation of the pricing formula for the European option driven by space-time fractional diffusion

In this paper, we show that the price of an European call option, whose underlying asset price is driven by the space-time fractional diffusion, can be expressed in terms of rapidly convergent double-series. This series formula is obtained from the Mellin-Barnes representation of the option price with help of residue summation in ℂ2. We also derive the series representation for the associated risk-neutral factors, obtained by Esscher transform of the space-time fractional Green functions.

Nonparametric Estimation of Risk-Neutral Distribution via the Empirical Esscher Transform

This paper introduces an empirical version of the Esscher transform for nonparametric option pricing. Traditional parametric methods require the formulation of an explicit risk-neutral model and are operational only for a few probability distributions for the returns of the underlying asset. In our proposal, we make only mild assumptions on the price kernel and there is no need for the formulation of the risk-neutral model. First, we simulate sample paths for the returns under the physical measure P. Then, based on the empirical Esscher transform, the sample is reweighted, giving rise to a risk-neutralized sample from which derivative prices can be obtained by a weighted sum of the options’ payoffs in each path. We analyze our proposal in experiments with artificial and real data.

A New Class of Distributions Based on Hurwitz Zeta Function with Applications for Risk Management

This paper constructs a new family of distributions, which is based on the Hurwitz zeta function, which includes novel distributions as well important known distributions such as the normal, gamma, Weibull, Maxwell-Boltzmann and the exponential power distributions. We provide the n-th moment, the Esscher transform and premium and the tail conditional moments for this family.