Minimal Hellinger martingale measures of order <Emphasis Type="Italic">q</Emphasis>

This paper proposes an extension of the minimal Hellinger martingale measure (MHM hereafter) concept to any order q≠1 and to the general semimartingale framework. This extension allows us to provide a unified formulation for many optimal martingale measures, including the minimal martingale measure of Föllmer and Schweizer (here q=2). Under some mild conditions of integrability and the absence of arbitrage, we show the existence of the MHM measure of order q and describe it explicitly in terms of pointwise equations in ? ^d . Applications to the maximization of expected power utility at stopping times are given. We prove that, for an agent to be indifferent with respect to the liquidation time of her assets (which is the market’s exit time, supposed to be a stopping time, not any general random time), she is forced to consider a habit formation utility function instead of the original utility, or equivalently she is forced to consider a time-separable preference with a stochastic discount factor.     

Mean-variance hedging for continuous processes: New proofs and examples

Let be a special semimartingale of the form and denote by the mean-variance tradeoff process of . Let be the space of predictable processes for which the stochastic integral is a square-integrable semimartingale. For a given constant and a given square-integrable random variable , the mean-variance optimal hedging strategy by definition minimizes the distance in between and the space . In financial terms, provides an approximation of the contingent claim by means of a self-financing trading strategy with minimal global risk. Assuming that is bounded and continuous, we first give a simple new proof of the closedness of in and of the existence of the F?llmer-Schweizer decomposition. If moreover is continuous and satisfies an additional condition, we can describe the mean-variance optimal strategy in feedback form, and we provide several examples where it can be computed explicitly. The additional condition states that the minimal and the variance-optimal martingale measures for should coincide. We provide examples where this assumption is satisfied, but we also show that it will typically fail if is not deterministic and includes exogenous randomness which is not induced by .     

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.Received: March 2004, Mathematics Subject Classification (2000): 62P05, 60J75, 60G44JEL Classification: G10     

GARCH options via local risk minimization

We apply a quadratic hedging scheme developed by Föllmer, Schweizer, and Sondermann to European contingent products whose underlying asset is modeled using a GARCH process and show that local risk-minimizing strategies with respect to the physical measure do exist, even though an associated minimal martingale measure is only available in the presence of bounded innovations. More importantly, since those local risk-minimizing strategies are in general convoluted and difficult to evaluate, we introduce Girsanov-like risk-neutral measures for the log-prices that yield more tractable and useful results. Regarding this subject, we focus on GARCH time series models with Gaussian innovations and we provide specific sufficient conditions concerning the finiteness of the kurtosis, under which those martingale measures are appropriate in the context of quadratic hedging. When this equivalent martingale measure is adapted to the price representation we are able to recover the classical pricing formulas of Duan and Heston and Nandi, as well as hedging schemes that improve the performance of those proposed in the literature.     

Minimal Entropy Martingale Measures of Jump Type Price Processes in Incomplete Assets Markets

We consider the incomplete assets market and assume that the market has no-arbitrage. Then there are many equivalent martingale measures associated with the market. Among them, a probability measure which minimizes the relative entropy with respect to the original probability measure P, has a special importance. Such a measure is called the minimal entropy martingale measure (MEMM). In a previous paper, we have proved the existence theorem of the MEMM for the price processes given in the form of the diffusion type stochastic differential equation. In this article we discuss the MEMM of the jump type price processes, or especially of the log Lévy processes, and we give the explicit form of MEMM.     

On <Emphasis Type="Italic">q</Emphasis>-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 Lévy 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. C.N. gratefully acknowledges financial support by UniCredit, Markets and Investment Banking. However, this paper does not reflect the opinion of UniCredit, Markets and Investment Banking, it is the personal view of the authors.     

Option pricing and Esscher transform under regime switching

Summary 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).We would like to thank the referees for many helpful and insightful comments and suggestions.Correspondence to: R. J. Elliott     

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. The q-optimal pricing measure is related to the marginal utility indifference price of an agent with constant relative risk aversion. 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 specialize to a variant of the Hull-White or 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.JEL Classification: D52, G13     

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.     

Minimal <Emphasis Type="Italic">q</Emphasis>-entropy martingale measures for exponential time-changed Lévy processes

We consider structure preserving measure transforms for time-changed Lévy processes. Within this class of transforms preserving the time-changed Lévy structure, we derive equivalent martingale measures minimizing relative q-entropy. They combine the corresponding transform for the Lévy process with an Esscher transform on the time change. Structure preservation is found to be an inherent property of minimal q-entropy martingale measures under continuous time changes, whereas it imposes an additional restriction for discontinuous time changes.     

Efficient discretization of stochastic integrals

Sharp asymptotic lower bounds on the expected quadratic variation of the discretization error in stochastic integration are given when the integrator admits a predictable quadratic variation and the integrand is a continuous semimartingale with nondegenerate local martingale part. The theory relies on inequalities for the kurtosis and skewness of a general random variable which are themselves seemingly new. Asymptotically efficient schemes which attain the lower bounds are constructed explicitly. The result is directly applicable to a practical hedging problem in mathematical finance; for hedging a payoff which is replicated by a continuous-time trading strategy, it gives an asymptotically optimal way to choose discrete rebalancing dates and portfolios with respect to transaction costs. The asymptotically efficient strategies in fact reflect the structure of the transaction costs. In particular, a specific biased rebalancing scheme is shown to be superior to unbiased schemes if the transaction costs follow a convex model. The problem is discussed also in terms of exponential utility maximization.