Unique Equivalent Martingale Measure Research Articles

Equilibrium in Complete Market with Jump-Diffusion Processes

This paper establishes the existence of a solution to the optimization problem. Supposing that risk assets pay continuous dividend regarded as the function of time. It is established that the behaviour model of the stock pricing process is jump-diffusion driven by a count process. We give a characterization of the optimal portfolio by means of the value function and the equivalent martingale measure defined by the utility function. The unique equivalent martingale measure, the unique optimal consumption and portfolio pair and the corresponding wealth process are deduced. We provide a simple characterization of an equilibrium market and discuss existence and uniqueness of equilibrium in the economy.

Consumption Optimization and Equilibrium

This paper studies the problem of consumption optimization and equilibrium in discontinuous time financial markets. It is established that the behavior model of the stock pricing process is jump-diffusion driven by a count process. It is proved that the existence of unique optimal consumption and portfolio pair and unique equivalent martingale measure by stochastic analysis methods. The unique equivalent martingale measure, the unique optimal consumption and portfolio pair and the corresponding wealth process are deduced. Finally we provide a simple characterization of an equilibrium market.

Wealth Optimization Models with Stochastic Volatility and Continuous Dividends

This paper study the problem of wealth optimization.It is established that the behavior model of the stock pricing process is jump-diffusion driven by a count process and stochastic volatility. Supposing that risk assets pay continuous dividend regarded as the function of time. It is proved that the existence of an optimal portfolio and unique equivalent martingale measure by stochastic analysis methods. The unique equivalent martingale measure ,the optimal wealth process, the value function and the optimal portfolio are deduced. Key words: Jump-Diffusion process; Stochastic volatility; Dividends; Incomplete financial market; Wealth optimization

Maximization of Wealth in a Jump-Diffusion Model

This paper study the problem of wealth optimization when jump-diffusion asset price model being driven by a count process that more general than Poisson process. It is found unique equivalent martingale measure, we employ the conventional stochastic analysis methods. It is proved that the existence of an optimal portfolio and consumption process. The optimal wealth process, the value function, the optimal portfolio and consumption process are given.

A law of large numbers approach to valuation in life insurance

The classical Principle of Equivalence ensures that a life insurance company can accomplish that the mean balance per policy converges to zero almost surely for an increasing number of independent policyholders. By certain assumptions, this idea is adapted to the general case with stochastic financial markets. The implied minimum fair price of general life insurance policies is then uniquely determined by the product of the assumed unique equivalent martingale measure of the financial market with the physical measure for the biometric risks. The approach is compared with existing related results. Numeric examples are given.

Wealth optimization in an incomplete market driven by a jump-diffusion process

We consider a small investor who operates within an incomplete market driven by a diffusion with jumps. His behavior facing the market risks is defined by a utility function. We prove the existence of an optimal portfolio and we characterize this portfolio. The optimal strategy determines a unique equivalent martingale measure which is identified with the Davis’ probability. We find explicit formulae for the optimal portfolio, the wealth and the value function in some particular examples of utility functions. Moreover, we prove that the range of the viable prices is reduced by the use of a utility function. At last, we compare the value function with the value function evalued in a market driven by a continuous asset.

A simple regime switching term structure model

Abstract. We extend the short rate model of Vasicek (1977) to include jumps in the local mean. Conditions ensuring existence of a unique equivalent martingale measure are given, implying that the model is arbitrage-free and complete. We develop efficient numerical methods for computation of zero coupon bond prices, illustrate how the model is easily calibrated to market data and show how other interest rate derivatives can be priced.

The Minimal Entropy Martingale Measure and the Valuation Problem in Incomplete Markets

Let χ be a family of stochastic processes on a given filtered probability space (Ω, F, (Ft)t∈T, P) with T⊆R+. Under the assumption that the set Me of equivalent martingale measures for χ is not empty, we give sufficient conditions for the existence of a unique equivalent martingale measure that minimizes the relative entropy, with respect to P, in the class of martingale measures. We then provide the characterization of the density of the minimal entropy martingale measure, which suggests the equivalence between the maximization of expected exponential utility and the minimization of the relative entropy.

On hedging in finite security markets

A market is considered where trading can take place only at discrete time points, the trading frequency cannot grow without bound, and the number of states of nature is finite. The main objectives of the paper are to show that the market can be completed also with highly correlated risky assets, and to describe an efficient algorithm to compute a self-financing hedging strategy. The algorithm consists off-line of a backwards recursion and on-line of the solution, in each period, of a system of linear equations; it is a consequence of a proof where, using a well-known mathematical property, it is shown that uniqueness of the martingale measure implies completeness also in our setting. The significance of ‘multistate’ models versus the familiar binomial model is discussed and it is shown how the evolution of prices of the (correlated) risky assets can be chosen so that a given probability measure is already the unique equivalent martingale measure.

Pricing contingent claims on stocks driven by Lévy processes

We consider the problem of pricing contingent claims on a stock whose price process is modelled by a geometric Levy process, in exact analogy with the ubiquitous geometric Brownian motion model. Because the noise process has jumps of random sizes, such a market is incomplete and there is not a unique equivalent martingale measure. We study several approaches to pricing options which all make use of an equivalent martingale measure that is in different respects closest to the underlying canonical measure, the main ones being the Follmer-Schweizer minimal measure and the martingale measure which has minimum relative entropy with respect to the canonical measure. It is shown that the minimum relative entropy measure is that constructed via the Esscher transform, while the Follmer-Schweizer measure corresponds to another natural analogue of the classical Black-Scholes measure.

A note on the existence of unique equivalent martingale measures in a Markovian setting

Simple sufficient conditions for the existence of a unique equivalent martingale measure are provided. Furthermore, these conditions give us a handle on situations where an equivalent martingale measure cannot exist. The existence of a unique equivalent martingale measure is of relevance to problems in mathematical finance. Two examples of models for which the question of existence was unresolved are studied. By means of our results existence of a unique equivalent measure up to an explosion time is proved.