Minimal martingale measure on a finite probability space

Risk measurement and Implied volatility under Minimal Entropy Martingale Measure for Levy process

This paper focuses on two main issues that are based on two important concepts: exponential Levy process and minimal entropy martingale measure. First, we intend to obtain   risk measurement such as value-at-risk (VaR) and conditional value-at-risk (CvaR) using Monte-Carlo methodunder minimal entropy martingale measure (MEMM) for exponential Levy process. This Martingale measure is used for the...

A Minimality Property of the Minimal Martingale Measure

Let X be a continuous adapted process for which there exists an equivalent local martingale measure (ELMM). The minimal martingale measure P̂ is the unique ELMM for X with the property that local P -martingales strongly orthogonal to the P -martingale part of X are also local P̂ -martingales. We prove that if P̂ exists, it minimizes the reverse relative entropy H(P |Q) over all ELMMs Q for X. A co...

In discrete time a local martingale is a martingale under an equivalent probability measure

1 Result and Discussion We consider a discrete-time infinite horizon model with an adapted d-dimensional process S = (S t) given on a stochastic basis (Ω, F, F = (F t) t=0,1,... , P). The notations used: M(P), M loc (P) and P are the sets of d-dimensional martingales, local martingales and predictable (i.e. (F t−1)-adapted) processes; H · S t = j≤t H j ∆S j. To our knowledge, this result was ne...

On the Minimal Martingale Measureand

The Minimal Entropy Martingale Measure and Hedging in Incomplete Markets

The intent of these essays is to study the minimal entropy martingale measure, to examine some new martingale representation theorems and to discuss its related Kunita-Watanabe decompositions. Such problems arise in mathematical finance for an investor who is confronted with the issues of pricing and hedging in incomplete markets. We adopt the standpoint of a ra­ tional investor who principally...