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

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...

Mimimal Relative Entropy Martingale Measure of Birth and Death Process

In this article, we investigate the MEMM (Minimal relative Entropy Martingale Measure) of Birth and Death processes and the MEMM of generalized Birth and Death processes. We see that the existence problem of the MEMM is reduced to the problem of solving the corresponding Hamilton-Jacobi-Bellman equation.

Implied and Local Volatilities under Stochastic Volatility

For asset prices that follow stochastic-volatility diffusions, we use asymptotic methods to investigate the behavior of the local volatilities and Black–Scholes volatilities implied by option prices, and to relate this behavior to the parameters of the stochastic volatility process. We also give applications, including risk-premium-based explanations of the biases in some näıve pricing and hedg...

The minimal entropy martingale measure in a market of traded financial and actuarial risks

In arbitrage-free but incomplete markets, the equivalent martingale measure Q for pricing traded assets is not uniquely determined. A possible approach when it comes to choosing a particular pricing measure is to consider the one that is ‘closest’to the physical probability measure P, where closeness is measured in terms of relative entropy. In this paper, we determine the minimal entropy marti...

Normalization for Implied Volatility

We study specific nonlinear transformations of the Black-Scholes implied volatility to show remarkable properties of the volatility surface. Model-free bounds on the implied volatility skew are given. Pricing formulas for the European options which are written in terms of the implied volatility are given. In particular, we prove elegant formulas for the fair strikes of the variance swap and the...

G–Brownian Motion and Dynamic Risk Measure under Volatility Uncertainty