Minimal Entropy Martingale Measure Research Articles

PRICING FOR GEOMETRIC MARKED POINT PROCESSES UNDER PARTIAL INFORMATION: ENTROPY APPROACH

The problem of the arbitrage-free pricing of a European contingent claim B is considered in a general model for intraday stock price movements in the case of partial information. The dynamics of the risky asset price is described through a marked point process Y, whose local characteristics depend on some unobservable jump diffusion process X. The processes Y and X may have common jump times, which means that the trading activity may affect the law of X and could be also related to the presence of catastrophic events. Risk-neutral measures are characterized and in particular, the minimal entropy martingale measure is studied. The problem of pricing under restricted information is discussed, and the arbitrage-free price of the claim B w.r.t. the minimal entropy martingale measure is computed by using filtering techniques.

The Minimal Entropy Martingale Measures for Exponential Additive Processes

In this paper, we will consider exponential additive processes as a financial market model. Under a mild condition, we will determine the minimal entropy martingale measures (MEMMs) for the exponential additive processes. To this end, we will prepare several results on the exponential moment of additive processes and integrals based on them. As an application of our result, we will deduce optimal strategy for exponential utility maximization problem. We will also investigate our result through several examples, such as time-dependent versions of double Poisson model, Merton model and Kou model.

The Minimal Entropy and the Convergence of the p-Optimal Martingale Measures in a General Jump Model

We first consider the minimal entropy martingale measure in a general jump model introduced in Kohlmann and Xiong (International Journal of Pure and Applied Mathematics, 37(3):321–348) and give a description of this measure (MEMM) as the solution of a backward martingale equation (BME). To relate the (MEMM) to the p-optimal martingale measure (p-OMM) we consider the convergence of the solution of the BME associated with the (p-OMM) to the solution of the BME associated with the MEMM. Under some assumptions, we prove the convergence of the p-OMM to the MEMM both in entropy and strongly in L 1(P). As an application, we consider the exp-optimal utility of an investor with utility function U exp(x) = −exp(− k 0 x), and as q↑∞, we show that the q-optimal terminal wealth of an investor with utility converges to the exp-optimal terminal wealth of an investor with utility function U exp(x) strongly in L r (P) for a large enough r.

On q-optimal martingale measures in exponential Lévy models

We give a sufficient condition to identify the q-optimal signed and the q-optimal absolutely continuous martingale measures in exponential Levy models. As a consequence, we find that in the one-dimensional case, the q-optimal equivalent martingale measures may exist only if the tails for upward jumps are extraordinarily light. Moreover, we derive the convergence of q-optimal signed, resp. absolutely continuous, martingale measures to the minimal entropy martingale measure as q approaches one. Finally, some implications for portfolio optimization are discussed.

On Convergence to the Exponential Utility Problem with Jumps

We derive an explicit portfolio for the exponential utility maximization problem via an approximation approach for exponential Lévy processes (mainly discussing − e −x (exponential problem) and (2m-th problem)). A result by Jeanblanc et al. (Annals of Applied Probability, 2007) is applied: The convergence of q-optimal martingale measures to the minimal entropy martingale measure. Except for conditions on the existence of the q-optimal measures, we replace technical assumptions by minor integrability conditions. We obtain convergence of the portfolios of the 2m-th to the exponential problem. The influence of jump intensity and jump size distribution upon the portfolio, in comparison to the continuous case, is discussed.

A nonparametric approach for European option valuation

A nonparametric approach for European option valuation is proposed in this paper, which adopts a purely jump model to describe the price dynamics of the underlying asset, and the minimal entropy martingale measure for those jumps is used as the pricing measure of this market. A simple Monte Carlo simulation method is proposed to calculate the price of derivatives under this risk neural measure. And the volatility of the spot market can be renewed automatically without particular specification in the proposed method. The performances of the proposed method are compared to that of the Black–Scholes formula in an artificial world and the real world. The results of our investigations suggest that the proposed method is a valuable method.

Option Pricing for Time-Change Exponential Lévy Model Under Memm

The purpose of this article is to study the rational evaluation of European options price when the underlying price process is described by a time-change Levy process. European option pricing formula is obtained under the minimal entropy martingale measure (MEMM) and applied to several examples of particular time-change Levy processes. It can be seen that the framework in this paper encompasses the Black-Scholes model and almost all of the models proposed in the subordinated market.

Minimal fq-martingale measures for exponential Lévy processes

Let L be a multidimensional Lévy process under P in its own filtration. The fq-minimal martingale measure Qq is defined as that equivalent local martingale measure for $\mathcal {E}(L)$ which minimizes the fq-divergence E[(dQ/dP)q] for fixed q∈(−∞, 0)∪(1, ∞). We give necessary and sufficient conditions for the existence of Qq and an explicit formula for its density. For q=2, we relate the sufficient conditions to the structure condition and discuss when the former are also necessary. Moreover, we show that Qq converges for q↘1 in entropy to the minimal entropy martingale measure.

The relative entropy in CGMY processes and its applications to finance

The CGMY market model generates infinite equivalent martingale measures (EMM). In order to price options, we need an adequate method to choose one EMM. This paper presents the relative entropy for CGMY processes, and apply it to choosing an EMM called the model preserving minimal entropy martingale measure.

The minimal entropy martingale measure for general Barndorff-Nielsen/Shephard models

We determine the minimal entropy martingale measure for a general class of stochastic volatility models where both price process and volatility process contain jump terms which are correlated. This generalizes previous studies which have treated either the geometric L\'{e}vy case or continuous price processes with an orthogonal volatility process. We proceed by linking the entropy measure to a certain semi-linear integro-PDE for which we prove the existence of a classical solution.

Esscher transforms and the minimal entropy martingale measure for exponential Lévy models

In this paper we offer a systematic survey and comparison of the Esscher martingale transform for linear processes, the Esscher martingale transform for exponential processes, and the minimal entropy martingale measure for exponential Lévy models, and present some new results in order to give a complete characterization of those classes of measures. We illustrate the results with several concrete examples in detail.

Option pricing and Esscher transform under regime switching

We consider the option pricing problem when the risky underlying assets are driven by Markov-modulated Geometric Brownian Motion (GBM). That is, the market parameters, for instance, the market interest rate, the appreciation rate and the volatility of the underlying risky asset, depend on unobservable states of the economy which are modelled by a continuous-time Hidden Markov process. The market described by the Markov-modulated GBM model is incomplete in general and, hence, the martingale measure is not unique. We adopt a regime switching random Esscher transform to determine an equivalent martingale pricing measure. As in Miyahara [33], we can justify our pricing result by the minimal entropy martingale measure (MEMM).

The density process of the minimal entropy martingale measure in a stochastic volatility model with jumps

The density process of the minimal entropy martingale measure in a stochastic volatility model with jumps

An entropy approach to the Stein and Stein model with correlation

We outline a martingale duality method for determining the minimal entropy martingale measure in a general continuous semimartingale model, and provide the relevant verification results. This method is illustrated by a detailed case study of the Stein and Stein stochastic volatility model driven by two correlated Brownian motions. It turns out that in case the mean reversion level and the correlation coefficient are nonzero, an investor who can use trading strategies adapted to the Brownian filtration may achieve a higher expected exponential utility from terminal wealth than an investor who can only observe the price process.

MINIMAL ENTROPY–HELLINGER MARTINGALE MEASURE IN INCOMPLETE MARKETS

This paper defines an optimization criterion for the set of all martingale measures for an incomplete market model when the discounted price process is bounded and quasi‐left continuous. This criterion is based on the entropy–Hellinger process for a nonnegative Doléans–Dade exponential local martingale. We develop properties of this process and establish its relationship to the relative entropy “distance.” We prove that the martingale measure, minimizing this entropy–Hellinger process, is unique. Furthermore, it exists and is explicitly determined under some mild conditions of integrability and no arbitrage. Different characterizations for this extremal risk‐neutral measure as well as immediate application to the exponential hedging are given. If the discounted price process is continuous, the minimal entropy–Hellinger martingale measure simply is the minimal martingale measure of Föllmer and Schweizer. Finally, the relationship between the minimal entropy–Hellinger martingale measure (MHM) and the minimal entropy martingale measure (MEM) is provided. We also give an example showing that in contrast to the MHM measure, the MEM measure is not robust with respect to stopping.

A PDE representation of the density of the minimal entropy martingale measure in stochastic volatility markets

Under general conditions stated in Rheinländer [An entropy approach to the stein/stein model with correlation. Preprint, 2003, ETH Zürich.], we prove that in a stochastic volatility market the Radon–Nikodym density of the minimal entropy martingale measure (MEMM) can be expressed in terms of the solution of a semilinear PDE. The semilinear PDE is suggested by the dynamic programming approach to the utility indifference pricing problem of contingent claims. One of our main results is the existence and uniqueness of a classical solution of the semilinear PDE in the case of a general stochastic volatility model with additive noise correlated with the asset price. Our results are applied to the Stein–Stein and Heston stochastic volatility models.

Pricing Jump Risk with Utility Indifference

In an incomplete market, option prices depend on investors' utility functions. In this paper, we establish the connection between risk preference and optimal hedging strategy, and price options according to the principle of utility indifference. Taking the exponential utility function, we completely characterize the risk-neutral valuation for jump-diffusion processes. By using a recent result of duality by Delbaen (2000) we prove that pricing measure for the risk neutral valuation is just the equivalent minimal entropy martingale measure. We show that risk aversion contributes a price spread from the risk neutral price. We also show that, however, risk-neutral valuation does not correspond to any practical hedging strategy. Minimal variance hedging strategy is discussed. Parallel analysis is carried over to discrete setting with multi-nomial random walks, and efficient numerical methods are developed. Numerical examples show that our model reproduces crash-o-phobia and other features of market prices of options.

On the Convergence of the p-Optimal Martingale Measures to the Minimal Entropy Martingale Measure

In an incomplete financial market where asset prices are continuous semimartingales, we establish the convergence of the p-optimal martingale measures to the minimal entropy martingale measure as p tends to 1. The result is achieved exploiting the theory of BMO-martingales and semimartingale backward equations.

Absolutely continuous optimal martingale measures

In this paper, we consider the problem of maximizing the expected utility of terminal wealth in the framework of incomplete financial markets. In particular, we analyze the case where an economic agent, who aims at such an optimization, achieves infinite wealth with strictly positive probability. By convex duality theory, this is shown to be equivalent to having the minimal-entropy martingale measure Q^ non-equivalent to the historical probability P (what we call the absolutely-continuous case). In this anomalous case, we no longer have the representation of the optimal wealth as the terminal value of a stochastic integral, stated in Schachermayer [9] for the case of Q^ ∼ P (i.e. the equivalent case). Nevertheless, we give an approximation of this terminal wealth through solutions to suitably-stopped problems, solutions which still admit the integral representation introduced in [9]. We also provide a class of examples fitting to the absolutely-continuous case.

From the Minimal Entropy Martingale Measures to the Optimal Strategies for the Exponential Utility Maximization: the Case of Geometric Lévy Processes

In this article, we will consider a multi-dimensional geometric L'evy process as a financial market model. We will first determine the minimal entropy martingale measure (MEMM); we will next derive the optimal strategy for the exponential utility maximization of terminal wealth concretely from the representation of the MEMM.