Minimal Martingale Measure Research Articles

On the Structure of General Mean-Variance Hedging Strategies

We provide a new characterisation of mean-variance hedging strategies in a general semimartingale market. The key point is the introduction of a new probability measure P* which turns the dynamic asset allocation problem into a myopic one. The minimal martingale measure relative to P* coincides with the variance-optimal martingale measure relative to the original probability measure P.

Stochastic orders in dynamic reinsurance markets

We consider a dynamic reinsurance market, where the traded risk process is driven by a compound Poisson process and where claim amounts are unbounded. These markets are known to be incomplete, and there are typically infinitely many martingale measures. In this case, no-arbitrage pricing theory can typically only provide wide bounds on prices of reinsurance claims. Optimal martingale measures such as the minimal martingale measure and the minimal entropy martingale measure are determined, and some comparison results for prices under different martingale measures are provided. This leads to a simple stochastic ordering result for the optimal martingale measures. Moreover, these optimal martingale measures are compared with other martingale measures that have been suggested in the literature on dynamic reinsurance markets.

An approximation pricing algorithm in an incomplete market: A differential geometric approach

The minimal distance equivalent martingale measure (EMM) defined in Goll and Ruschendorf (2001) is the arbitrage-free equilibrium pricing measure. This paper provides an algorithm to approximate its density and the fair price of any contingent claim in an incomplete market. We first approximate the infinite dimensional space of all EMMs by a finite dimensional manifold of EMMs. A Riemannian geometric structure is shown on the manifold. An optimization algorithm on the Riemannian manifold becomes the approximation pricing algorithm. The financial interpretation of the geometry is also given in terms of pricing model risk.

Performance of utility-based strategies for hedging basis risk

The performance of optimal strategies for hedging a claim on a non-traded asset is analysed. The claim is valued and hedged in a utility maximization framework, using exponential utility. A traded asset, correlated with that underlying the claim, is used for hedging, with the correlation ρ typically close to 1. Using a distortion method (Zariphopoulou 2001 Finance Stochastics 5 61–82) we derive a nonlinear expectation representation for the claim’s ask price and a formula for the optimal hedging strategy. We generate a perturbation expansion for the price and hedging strategy in powers of ε2 =1−ρ2. The terms in the price expansion are proportional to the central moments of the claim payoff under the minimal martingale measure. The resulting fast computation capability is used to carry out a simulation-based test of the optimal hedging program, computing the terminal hedging error over many asset price paths. These errors are compared with those from a naive strategy which uses the traded asset as a proxy for the non-traded one. The distribution of the hedging error acts as a suitable metric to analyse hedging performance. We find that the optimal policy improves hedging performance, in that the hedging error distribution is more sharply peaked around a non-negative profit. The frequency of profits over losses is increased, and this is measured by the median of the distribution, which is always increased by the optimal strategies. An empirical example illustrates the application of the method to the hedging of a stock basket using index futures.

Minimal martingale measures for jump diffusion processes

We consider an incomplete market model whose stock price fluctuation is given by a jump diffusion process. For this model, we calculate the density process of the minimal martingale measure. Also, we state the relation to a locally risk-minimizing strategy.

Minimal martingale measures for jump diffusion processes

We consider an incomplete market model whose stock price fluctuation is given by a jump diffusion process. For this model, we calculate the density process of the minimal martingale measure. Also, we state the relation to a locally risk-minimizing strategy.

A reduced form model for ESO valuation

In this paper we extend a reduced form model for the valuation of employee share options (ESOs) to incorporate employee departure, and company takeover. We also allow for performance linked vesting and other exotic features specific to ESOs. We clarify the assumptions underlying the reduced form model, and discuss their implications. We analyze the probabilistic structure of the model which includes an explicit characterization of the set of equivalent martingale measures, as well as the computation of the variance optimal martingale measure and the minimal martingale measure. Moreover, we deduce an additive decomposition of the relative entropy. Particular ESO specifications are studied emphasizing different aspects of the proposed framework. In this context, we also provide strict no-arbitrage bounds for ESO prices by applying optimal stopping. Furthermore, possible limitations of the proposed model are explored by examining departures from the crucial assumptions of no-arbitrage, i.e. by considering the effects of the employee having inside information.

A Comparison of Option Prices Under Different Pricing Measures in a Stochastic Volatility Model with Correlation

This paper investigates option prices in an incomplete stochastic volatility model with correlation. In a general setting, we prove an ordering result which says that prices for European options with convex payoffs are decreasing in the market price of volatility risk. As an example, and as our main motivation, we investigate option pricing under the class of q-optimal pricing measures. Using the ordering result, we prove comparison theorems between option prices under the minimal martingale, minimal entropy and variance-optimal pricing measures. If the Sharpe ratio is deterministic, the comparison collapses to the well known result that option prices computed under these three pricing measures are the same. As a concrete example, we specialise to a variant of the Heston model for which the Sharpe ratio is increasing in volatility. For this example we are able to deduce option prices are decreasing in the parameter q. Numerical solution of the pricing pde corroborates the theory and shows the magnitude of the differences in option price due to varying q. Choice of q is shown to influence the level of the implied volatility smile for options of varying maturity.

Option pricing with discrete rebalancing

We consider option pricing when dynamic portfolios are discretely rebalanced. The portfolio adjustments only occur after fixed relative changes in the stock price. The stock price follows a marked point process (MPP) and the market is incomplete. We first characterise the equivalent martingale measures. An explicit pricing formula based on the minimal martingale measure (MMM) is then provided together with the hedging strategy underlying portfolio adjustments. Two examples illustrate our pricing framework: a jump process driven by a latent geometric Brownian motion and a marked Poisson process. We establish the convergence to the Black–Scholes model when the triggering price increment shrinks to zero. For the empirical application, we use IBM, France Telecom (FT) and CAC 40 intraday transaction data, and compare option prices given by the marked Poisson model, the Black–Scholes model and observed option prices.

Option Pricing with Discrete Rebalancing

We consider option pricing when dynamic portfolios are discretely rebalanced. The portfolio adjustments only occur after fixed relative changes in the stock price. The stock price follows a marked point process and the market is incomplete. We first characterize the equivalent martingale measures. An explicit pricing formula based on the minimal martingale measure is then provided together with the hedging strategy underlying portfolio adjustments. Two examples illustrate our pricing framework: a jump process driven by a latent geometric Brownian motion and a marked Poisson process. We establish the convergence to the Black-Scholes model when the triggering price increment shrinks to zero. For the empirical application we use IBM, France Telecom and CAC 40 intraday transaction data, and compare option prices given by the marked Poisson model, the Black-Scholes model and observed option prices.

Generalised Sharpe Ratios and Asset Pricing in Incomplete Markets *

The paper presents an incomplete market pricing methodology generating asset price bounds conditional on the absence of attractive investment opportunities in equilibrium. The paper extends and generalises the seminal article of Cochrane and Saa-Requejo who pioneered option pricing based on the absence of arbitrage and high Sharpe Ratios. Our contribution is threefold: We base the equilibrium restrictions on an arbitrary utility function, obtaining the Cochrane and Saa-Requejo analysis as a special case with truncated quadratic utility. We extend the definition of Sharpe Ratio from quadratic utility to the entire family of CRRA utility functions and restate the equilibrium restrictions in terms of Generalised Sharpe Ratios which, unlike the standard Sharpe Ratio, provide a consistent ranking of investment opportunities even when asset returns are highly non-normal. Last but not least, we demonstrate that for Ito processes the Cochrane and Saa-Requejo price bounds are invariant to the choice of the utility function, and that in the limit they tend to a unique price determined by the minimal martingale measure.

A Unified Characterization of 𝑞-Optimal and Minimal Entropy Martingale Measures by Semimartingale Backward Equations

Abstract We give a unified characterization of 𝑞-optimal martingale measures for 𝑞 ∈ [0, ∞) in an incomplete market model, where the dynamics of asset prices are described by a continuous semimartingale. According to this characterization the variance-optimal, the minimal entropy and the minimal martingale measures appear as the special cases 𝑞 = 2, 𝑞 = 1 and 𝑞 = 0 respectively. Under assumption that the Reverse Hölder condition is satisfied, the continuity (in 𝐿1 and in entropy) of densities of 𝑞-optimal martingale measures with respect to 𝑞 is proved.

Convergence of Jump-Diffusion Modelsto the Black–Scholes Model

We consider a jump-diffusion model for asset price which is described as a solution of a linear stochastic differential equation driven by a Lévy process. Such a market is incomplete and there are many equivalent martingale measures. We price a contingent claim with respect to the minimal martingale measure and construct a hedging strategy for the contingent claim in the locally risk-minimizing sense. We study the problem of convergence of option prices jointly with the costs from the locally risk-minimizing strategies when the jump-diffusion models converge to the Black–Scholes model.

A Semimartingale Backward Equation and the Variance-Optimal Martingale Measure under General Information Flow

We consider a financial market model, where the dynamics of asset prices are given by an Rd-valued continuous semimartingale and the information flow is right-continuous. Using the dynamic programming approach we express the variance-optimal martingale measure in terms of the value process of a suitable optimization problem and show that this value process uniquely solves the corresponding semimartingale backward equation. We consider two extreme cases when this equation admits an explicit solution. In particular, we give necessary and sufficient conditions in order that the variance-optimal martingale measure coincides with the minimal martingale measure as well as with the martingale measure appearing in the second extreme case.

Mean-Variance Hedging for Discontinuous Semimartingales

Mean-variance hedging is well-known as one of hedging methods for incomplete markets. Our end is leading to mean-variance hedging strategy for incomplete market models whose asset price process is given by a discontinuous semimartingale and whose mean-variance trade-off process is not deterministic. In this paper, on account, we focus on this problem under the following assumptions: (1) the local martingale part of the stock price process is a process with independent increments; (2) a certain condition restricting the number and the size of jumps of the asset price process is satisfied; (3) the mean-variance trade-off process is uniformly bounded; (4) the minimal martingale measure coincides with the variance-optimal martingale measure.

Minimal Martingale Measures for Discrete-time Incomplete Financial Markets

In this note, we give a characterization of the minimal martingale measure for a general discrete-time incomplete financial market. Then we concretely work out the minimal martingale measure for a specific discrete-time market model in which the assets' returns in different times are independent.

Exponential Hedging and Entropic Penalties

We solve the problem of hedging a contingent claim B by maximizing the expected exponential utility of terminal net wealth for a locally bounded semimartingale X. We prove a duality relation between this problem and a dual problem for local martingale measures Q for X where we either minimize relative entropy minus a correction term involving B or maximize the Q‐price of B subject to an entropic penalty term. Our result is robust in the sense that it holds for several choices of the space of hedging strategies. Applications include a new characterization of the minimal martingale measure and risk‐averse asymptotics.

Claim pricing and hedging under market incompleteness and “mean–variance” preferences

This paper derives bid and ask prices of a general contingent claim in an incomplete market for an investor who can also trade in a given asset and in bonds. The investor's preferences are modelled by a quadratic derived utility function. The ask (resp. bid) price is determined in such a way that the investor is indifferent between the two following policies: (i) selling (resp. buying) the claim at this price and investing optimally in the asset and in bonds; (ii) investing optimally in the asset and in bonds, without trading the contingent claim. The ask (resp. bid) price thus derived is the expectation of the final payoff of the claim under the signed martingale measure that M. Schweizer [Mathematics of Operation Research 20 (1995) 1–32] called “minimal martingale measure”, plus (resp. minus) a quantity that is directly proportional to: (i) the number of claims being traded; (ii) a coefficient of risk-aversion; (iii) the smallest variance of the error of replication due to hedging one single claim. The results we derive apply to very general claims, hence to portfolios of claims.

The p-optimal martingale measure in continuous trading models

We discuss the p-optimal martingale measure for p∈(1,∞) in continuous incomplete markets whose stock price is fluctuated by a d-dimensional continuous semimartingale. In this paper, we treat two simple models. One is a model where the mean-variance trade-off process is deterministic. Another is a model where the minimal martingale measure coincides with the minimal entropy martingale measure. In these models, we prove that the p-optimal martingale measure coincides with the minimal martingale measure under some conditions.

The Relations between Minimal Martingale Measure and Minimal Entropy Martingale Measure

We consider incomplete markets, where each risky asset fluctuation is a continuous semimartingale, and study a subset of Equivalent Local Martingale Measures in which Minimal Martingale Measure minimizes relative entropy.We also discuss, as special cases, some models with the risky assetfluctuation represented as a solution of some stochastic differential equations.Finally, we mention that the predictable representation property is essentialin order that Minimal Martingale Measure coincides with Minimal Entropy Martingale Measure.