Minimal Martingale Measure Research Articles

Pricing Asian Options on Assets Driven by a Combined Geometric Brownian Motion and a Geometric Compound Poisson Process

In the paper we study the pricing of Asian options when the price dynamics of the underlying asset are driven by a combined geometric Brownian motion and a geometric compound Poisson process. With the presence of the jump effect, the market in this model is (in general) incomplete, and that therefore there are no unique hedging prices. For this model, we adopt the minimal martingale measure introduced by Follmer and Schweizer [7] as the risk-neutral pricing measure. We then present a partial integro-differential equation (PIDE) whose solution leads to Asian option prices.

The Minimal Entropy Martingale Measure (MEMM) for a Markov-Modulated Lévy Model

This paper deals with the characterization problem of the minimal entropy martingale measure (MEMM) for a Markov-modulated exponential Levy model. This model is characterized by the presence of a factor modulating the risky asset price movements between different regimes. We circumvent the difficulty caused by the modulator factor by using the special properties of Markov additive process. Our work generalizes some previous works in the literature which have treated either the exponential Levy case or the exponential-additive case.

Risk Minimization for a Filtering Micromovement Model of Asset Price

The classical option hedging problems have mostly been studied under continuous-time or equally spaced discrete-time models, which ignore two important components in the actual price: random trading times and market microstructure noise. In this paper, we study optimal hedging strategies for European derivatives based on a filtering micromovement model of asset prices with the two commonly ignored characteristics. We employ the local risk-minimization criterion to develop optimal hedging strategies under full information. Then, we project the hedging strategies on the observed information to obtain hedging strategies under partial information. Furthermore, we develop a related nonlinear filtering technique under the minimal martingale measure for the computation of such hedging strategies.

An Equilibrium Approach to Indifference Pricing

The utility indifference framework has received a lot of attention, because it is based on a utility maximization principle, which is one of the most fundamental principles of economics, for pricing a contingent claim. The price based on utility indifference framework is the maximum or minimum (in some cases, threshold) price for each investor. Therefore, the price is the indicator for the investor to join the market of the contingent claim. Our purpose is to expand the view of utility indifference framework, that is, to deduce the equilibrium price in the utility indifference framework. We attain the result that, under the setting of exponential utility, the equilibrium price will be uniquely evaluated by minimal entropy martingale measure.

The Föllmer–Schweizer decomposition: Comparison and description

This paper proposes two main contributions concerning the Föllmer–Schweizer decomposition (called hereafter the FS-decomposition). First we completely elaborate the relationship between this decomposition and the Galtchouk–Kunita–Watanabe decomposition under the minimal martingale measure. The difference between these two decompositions is highlighted in a very practical example, and the martingale tools that enhance this difference are illustrated in the semimartingale framework as well. The second main contribution focuses on the description of the FS-decomposition using the predictable characteristics.

Modeling and Pricing of Variance and Volatility Swaps for Local Semi-Markov Volatilities in Financial Engineering

We consider a semi-Markov modulated security market consisting of a riskless asset or bond with constant interest rate and risky asset or stock, whose dynamics follow gemoetric Brownian motion with volatility that depends on semi-Markov process. Two cases for semi-Markov volatilities are studied: local current and local semi-Markov volatilities. Using the martingale characterization of semi-Markov processes, we find the minimal martingale measure for this incomplete market. Then we model and price variance and volatility swaps for local semi-Markov stochastic volatilities.

RISK-NEUTRAL MEASURES AND PRICING FOR A PURE JUMP PRICE PROCESS

This article considers the asset price movements in a financial market when risky asset prices are modeled by marked point processes. Their dynamics depend on an underlying event arrivals process, modeled again by a marked point process. Taking into account the presence of catastrophic events, the possibility of common jump times between the risky asset price process and the arrivals process is allowed. By setting and solving a suitable control problem, the characterization of the minimal entropy martingale measure is obtained. In a particular case, a pricing problem is also discussed.

Option pricing in bilateral Gamma stock models

Abstract In the framework of bilateral Gamma stock models we seek for adequate option pricing measures, which have an economic interpretation and allow numerical calculations of option prices. Our investigations encompass Esscher transforms, minimal entropy martingale measures, p-optimal martingale measures, bilateral Esscher transforms and the minimal martingale measure. We illustrate our theory by a numerical example.

Pricing of Variance and Volatility Swaps with Semi-Markov Volatilities

We consider a semi-Markov modulated market consisting of a riskless asset or bond, B, and a risky asset or stock, S, whose dynamics depend on a semi-Markov process x. Using the martingale characterization of semi-Markov processes, we note the incompleteness of semi-Markov modulated markets and find the minimal martingale measure. We price variance (Theorem 1) and volatility (Theorem 2) swaps for stochastic volatilities driven by the semi-Markov processes. We also discuss some extensions of the obtained results such as local semi-Markov volatility, Dupire formula for the local semi-Markov volatility and residual risk associated with the swap pricing.

Evaluation of the MEMM, Parameter Estimation and Option Pricing for Geometric Lévy Processes

In this paper, we shall propose a useful approach to evaluate concretely the MEMM (minimal entropy martingale measure) for the typical geometric Levy processes such as compound Poisson, stable, VG (Variance Gamma), CGMY (Carr-Geman-Madan-Yor), NIG (Normal Inverse Gaussian), etc. In addition, we shall estimate the parameters of geometric Levy processes and value the European call option and Asian call option using the Nikkei financial data.

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.

Minimal martingale measure: Pricing and hedging in a pure jump model under restricted information

Financial asset price movements are considered in a model with partial information. The risky assets’ price is modeled as a marked point process. The underlying event arrival process, which is again a marked point process, is assumed to be unobserved by the market agents. Taking into account the presence of catastrophic events, the possibility of common jump times between the risky assets’ price process and the arrival process is allowed. The problem of characterizing the equivalent martingale measures is investigated and in particular the minimal martingale measure is studied. The hedging and the pricing problem of a European contingent claim are discussed. The arbitrage-free price is identified with the conditional expectation with respect to the observations under the minimal martingale measure.

Local Risk-Minimization for Payment Processes: Application to Insurance Contracts with Surrender Option or to Default Sensitive Contingent Claims

In this paper, we extend the local risk-minimization theory, initially developed by Schweizer for a single payoff, to a payment stream. We basically follow Moller's approach in when he extended the risk-minimization theory to payment processes.In a second step, we apply this extended theory to insurance contracts with surrender option or equivalently, to default sensitive contingent claims. This part can be seen as an extension of Biagini and Cretarola on the local risk-minimization of defaultable claims. The main differences with the works of these authors are, firstly, that the discounted financial assets prices follow a general continuous s-dimensional semimartingale and, secondly, that we do not assume that the so-called (H)-hypothesis necessarily holds.As a by-product, we also study the impact of an enlargement of filtration on the minimal martingale measure.

On the Esscher transforms and other equivalent martingale measures for Barndorff-Nielsen and Shephard stochastic volatility models with jumps

We compute and then discuss the Esscher martingale transform for exponential processes, the Esscher martingale transform for linear processes, the minimal martingale measure, the class of structure preserving martingale measures, and the minimum entropy martingale measure for stochastic volatility models of the Ornstein–Uhlenbeck type as introduced by Barndorff-Nielsen and Shephard. We show that in the model with leverage, with jumps both in the volatility and in the returns, all those measures are different, whereas in the model without leverage, with jumps in the volatility only and a continuous return process, several measures coincide, some simplifications can be made and the results are more explicit. We illustrate our results with parametric examples used in the literature.

Exponential utility indifference valuation in two Brownian settings with stochastic correlation

We study the exponential utility indifference valuation of a contingent claimBin an incomplete market driven by two Brownian motions. The claim depends on a nontradable asset stochastically correlated with the traded asset available for hedging. We use martingale arguments to provide upper and lower bounds, in terms of bounds on the correlation, for the valueVBof the exponential utility maximization problem with the claimBas random endowment. This yields an explicit formula for the indifference valuebofBat any time, even with a fairly general stochastic correlation. Earlier results with constant correlation are recovered and extended. The reason why all this works is that, after a transformation to the minimal martingale measure, the valueVBenjoys a monotonicity property in the correlation between tradable and nontradable assets.

Exponential utility indifference valuation in two Brownian settings with stochastic correlation

We study the exponential utility indifference valuation of a contingent claim B in an incomplete market driven by two Brownian motions. The claim depends on a nontradable asset stochastically correlated with the traded asset available for hedging. We use martingale arguments to provide upper and lower bounds, in terms of bounds on the correlation, for the value VB of the exponential utility maximization problem with the claim B as random endowment. This yields an explicit formula for the indifference value b of B at any time, even with a fairly general stochastic correlation. Earlier results with constant correlation are recovered and extended. The reason why all this works is that, after a transformation to the minimal martingale measure, the value VB enjoys a monotonicity property in the correlation between tradable and nontradable assets.

A locally risk-minimizing hedging strategy for unit-linked life insurance contracts in a Lévy process financial market

In [Riesner, M., 2006. Hedging life insurance contracts in a Lévy process financial market. Insurance Math. Econom. 38, 599–608] the (locally) risk-minimizing hedging strategy for unit-linked life insurance contracts is determined in an incomplete financial market driven by a Lévy process. The considered risky asset is not a martingale under the original measure and therefore, a change of measure to the minimal martingale measure is performed. The goal of this paper is to show that the risk-minimizing hedging strategy under the new martingale measure which is found in the paper cited above is not the locally risk-minimizing strategy under the original measure. Finally, the real locally risk-minimizing strategy is derived and a relationship between the number of risky assets held in the proposed portfolio cited in the above-mentioned paper and the one proposed here is given.

Risk Minimizing Option Pricing in a Regime Switching Market

We study option pricing in a regime switching market where the risk free interest rate, growth rate and the volatility of a stock depends on a finite state Markov chain. Using a minimal martingale measure we show that the risk minimizing option price satisfies a system of Black–Scholes partial differential equations with weak coupling.

Nonlinear stochastic integrals for hyperfinite Lévy processes

I develop a notion of nonlinear stochastic integrals for hyperfinite Levy processes and use it to find exact formulas for expressions which are intuitively of the form $$\sum_{s=0}^t\phi(\omega,dl_{s},s)$$ and $$\prod_{s=0}^t\psi(\omega,dl_{s},s)$$ , where l is a Levy process. These formulas are then applied to geometric Levy processes, infinitesimal transformations of hyperfinite Levy processes, and to minimal martingale measures. Some of the central concepts and results are closely related to those found in S. Cohen’s work on stochastic calculus for processes with jumps on manifolds, and the paper may be regarded as a reworking of his ideas in a different setting and with totally different techniques.