Marek Musiela

An example of indifference prices under exponential preferences

Abstract.The aim herein is to analyze utility-based prices and hedging strategies. The analysis is based on an explicitly solved example of a European claim written on a nontraded asset, in a model where risk preferences are exponential, and the traded and nontraded asset are diffusion processes with, respectively, lognormal and arbitrary dynamics. Our results show that a nonlinear pricing rule emerges with certainty equivalent characteristics, yielding the price as a nonlinear expectation of the derivative’s payoff under the appropriate pricing measure. The latter is a martingale measure that minimizes its relative to the historical measure entropy.

A valuation algorithm for indifference prices in incomplete markets

Abstract.A probabilistic iterative algorithm is constructed for indifference prices of claims in a multiperiod incomplete model. At each time step, a nonlinear pricing functional is applied that isolates and prices separately the two types of risk. It is represented solely in terms of risk aversion and the pricing measure, a martingale measure that preserves the conditional distribution of unhedged risks, given the hedgeable ones, from their historical counterparts.

Portfolio choice under dynamic investment performance criteria

A new dynamic criterion for measuring the performance of self-financing investment strategies is introduced. To this aim, a family of stochastic processes defined on [0, ∞) and indexed by a wealth argument is used. Optimality is associated with their martingale property along the optimal wealth trajectory. The optimal portfolios are constructed via stochastic feedback controls that are functionally related to differential constraints of fast diffusion type. A multi-asset Ito-type incomplete market model is used.

Portfolio Choice under Space-Time Monotone Performance Criteria

The class of time-decreasing forward performance processes is analyzed in a portfolio choice model of Ito-type asset dynamics. The associated optimal wealth and portfolio processes are explicitly constructed and their probabilistic properties are discussed. These formulae are, in turn, used in analyzing how the investors preferences can be calibrated to the market, given his desired investment targets.

Investment and Valuation Under Backward and Forward Dynamic Exponential Utilities in a Stochastic Factor Model

We introduce a new class of dynamic utilities that are generated forward in time. We discuss the associated value functions, optimal investments, and indifference prices and we compare them with their traditional counterparts, implied by backward dynamic utilities.

Stochastic Partial Differential Equations and Portfolio Choice

We introduce a stochastic partial differential equation which describes the evolution of the investment performance process in portfolio choice models. The equation is derived for two formulations of the investment problem, namely, the traditional one (based on maximal expected utility of terminal wealth) and the recently developed forward formulation. The novel element in the forward case is the volatility process which is up to the investor to choose. We provide various examples for both cases and discuss the differences and similarities between the different forms of the equation as well as the associated solutions and optimal processes.

Optimal Asset Allocation under Forward Exponential Performance Criteria

This work presents a novel concept in stochastic optimization, namely, the notion of forward performance. As an application, we analyze a portfolio management problem with exponential criteria. Under minimal model assumptions we explicitly construct the forward performance process and the associated optimal wealth and asset allocations. For various model parameters, we recover a range of investment policies that correspond to distinct financial applications.

INITIAL INVESTMENT CHOICE AND OPTIMAL FUTURE ALLOCATIONS UNDER TIME-MONOTONE PERFORMANCE CRITERIA

The paper offers a new perspective on optimal portfolio choice by investigating how and to what extent knowledge of an investors desirable initial investment choice can be used to determine his future optimal portfolio allocations. Optimality of investment decisions is built on the so-called forward investment performance criteria and, in particular, on the time-monotone ones. It is shown that for this class of forward criteria the desired initial allocations completely characterize the future optimal investment strategies. The analysis uses the connection between a nonlinear equation, satisfied by the local risk tolerance, and the backward heat equation. Complete solutions are provided as well as various examples.

Chapter One. The Single Period Binomial Model

René Carmona: René Carmona is published by Princeton University Press and copyrighted,

An Introduction to Financial Derivatives

We shall first review briefly the most important kinds of financial contracts, traded either on exchanges or over-the-counter (OTC), between financial institutions and their clients. For a detailed account of the fundamental features of spot (i.e., cash) and futures financial markets the reader is referred, for instance, to Cox and Rubinstein (1985), Ritchken (1987), Chance (1989), Duffie (1989), Merrick (1990), Kolb (1991), Dubofsky (1992), Edwards and Ma (1992), Sutcliffe (1993), Hull (1994, 1996) or Redhead (1996).