The minimal entropy and minimal martingale measures are shown to be related by an Esscher transform, involving the mean-variance trade-off, in an incomplete diffusion model containing a traded stock and a correlated non-traded stochastic factor. The coefficients of the diffusions are measurable with respect to the Brownian motion driving the non-traded factor, as is typical in stochastic volati...

Let (Ω,A, μ) be a Lebesgue space and T : Ω → Ω an ergodic measure preserving automorphism with positive entropy. We show that there is a bounded and strictly stationary martingale difference sequence defined on Ω with a common non-degenerate lattice distribution satisfying the central limit theorem with an arbitrarily slow rate of convergence and not satisfying the local limit theorem. A simila...

We consider the problem of valuation of certain Asian options in the geometric jump-diffusion models with continuously dividend-paying assets. With the sources of diffusion risks and two primitive tradeable assets, the market in this model is, in general, incomplete, and so, there are more than one equivalent martingale measures and no-arbitrage prices. For this jump-diffusion model, we adopt t...

This paper considers the American option pricing problem under the stochastic volatility models. In particular, we introduce the GARCHmodel with two heavy-tailed distributions: classical tempered stable (CTS) and normal tempered stable (NTS) distribution. Then we apply the Markov chain approach to compute the prices of American style options under these two models. Minimal entropy provides a co...

In this paper, a modified formula for European options and a set of complete convergence proofs for the solution that cover the entire time horizon of a European option contact are presented under the Heston model with minimal entropy martingale measure. Although He & Zhu [5] worked on this model, they only provided a converged solution with a condition imposed on the time to expiry. The new so...

We construct a continuous bounded stochastic process (St)t2[0;1] which admits an equivalent martingale measure but such that the minimal martingale measure in the sense of Follmer and Schweizer does not exist. This example also answers (negatively) a problem posed by I. Karatzas, J. P. Lehozcky and S. E. Shreve as well as a problem posed C. Stricker.

In a discrete-time financial market setting, the paper relates various concepts introduced for dynamic portfolios (both in discrete and in continuous time). These concepts are: value preserving portfolios, numeraire portfolios, interest oriented portfolios, and growth optimal portfolios. It will turn out that these concepts are all associated with a unique martingale measure which agrees with t...

In an incomplete financial market in which the dynamics of the asset prices is driven by a d-dimensional continuous semimartingale X, we consider the problem of pricing European contingent claims embedded in a power utility framework. This problem reduces to identifying the p-optimal martingale measure, which can be given in terms of the solution to a semimartingale backward equation. We use th...