Derivative pricing based on local utility maximization

Abstract. This paper discusses a new approach to contingent claim valuation in general incomplete market models. We determine the neutral derivative price which occurs if investors maximize their local utility and if derivative demand and supply are balanced. We also introduce the sensitivity process of a contingent claim. This process quantifies the reliability of the neutral derivative price and it can be used to construct price bounds. Moreover, it allows to calibrate market models in order to be consistent with initially observed derivative quotations.

[1]  S. Pliska,et al.  Mathematics of Derivative Securities , 1998 .

[2]  W. Schachermayer.  Optimal investment in incomplete markets when wealth may become negative , 2001 .

[3]  Jan Kallsen,et al.  Utility-Based Derivative Pricing in Incomplete Markets , 2002 .

[4]  S. Shreve,et al.  Martingale and duality methods for utility maximization in a incomplete market , 1991 .

[5]  Murad S. Taqqu,et al.  Option Pricing in ARCH‐type Models , 1998 .

[6]  J. Duan.  THE GARCH OPTION PRICING MODEL , 1995 .

[7]  F. Delbaen,et al.  Exponential Hedging and Entropic Penalties , 2002 .

[8]  M. Schweizer.  Option hedging for semimartingales , 1991 .

[9]  H. Föllmer,et al.  Hedging of contingent claims under incomplete in-formation , 1991 .

[10]  P. Grandits.  On Martingale Measures for Stochastic Processes with Independent Increments , 2000 .

[11]  Jan Kallsen.  A utility maximization approach to hedging in incomplete markets , 1999, Math. Methods Oper. Res..

[12]  Neil D. Pearson,et al.  Consumption and portfolio policies with incomplete markets and short-sale constraints: The infinite dimensional case , 1991 .

[13]  J. Jacod,et al.  Integrales stochastiques par rapport a une semimartingale vectorielle et changements de filtration , 1980 .

[14]  Y. Miyahara.  Minimal Entropy Martingale Measures of Jump Type Price Processes in Incomplete Assets Markets , 1999 .

[15]  Ludger Rüschendorf,et al.  Minimax and minimal distance martingale measures and their relationship to portfolio optimization , 2001, Finance Stochastics.

[16]  Aleš Černý,et al.  The Theory of Good-Deal Pricing in Financial Markets , 1998 .

[17]  Jan Kallsen,et al.  Optimal portfolios for logarithmic utility , 2000 .

[18]  A. Shiryaev,et al.  Limit Theorems for Stochastic Processes , 1987 .

[19]  W. Schachermayer,et al.  The asymptotic elasticity of utility functions and optimal investment in incomplete markets , 1999 .

[20]  M. Schweizer.  Approximation pricing and the variance-optimal martingale measure , 1996 .

[21]  Ole E. Barndorff-Nielsen,et al.  Processes of normal inverse Gaussian type , 1997, Finance Stochastics.

[22]  S. Shreve,et al.  Optimal portfolio and consumption decisions for a “small investor” on a finite horizon , 1987 .

[23]  Jan Kallsen,et al.  The cumulant process and Esscher's change of measure , 2002, Finance Stochastics.

[24]  Jan Kallsen,et al.  Optimal portfolios for exponential Lévy processes , 2000, Math. Methods Oper. Res..

[25]  J. Cochrane,et al.  Beyond Arbitrage: 'Good Deal' Asset Price Bounds in Incomplete Markets , 1996 .

[26]  R. Udiger Frey.  Derivative Asset Analysis in Models with Level-dependent and Stochastic Volatility , 1997 .

[27]  Neil D. Pearson,et al.  Consumption and Portfolio Policies With Incomplete Markets and Short‐Sale Constraints: the Finite‐Dimensional Case , 1991 .

[28]  E. Eberlein,et al.  Hyperbolic distributions in finance , 1995 .

[29]  J. Jacod.  Calcul stochastique et problèmes de martingales , 1979 .

[30]  E. Eberlein,et al.  New Insights into Smile, Mispricing, and Value at Risk: The Hyperbolic Model , 1998 .

[31]  F. Black,et al.  The Pricing of Options and Corporate Liabilities , 1973, Journal of Political Economy.

[32]  Marco Frittelli,et al.  On the Existence of Minimax Martingale Measures , 2002 .

[33]  J. Harrison,et al.  Martingales and stochastic integrals in the theory of continuous trading , 1981 .

[34]  N. Karoui,et al.  Dynamic Programming and Pricing of Contingent Claims in an Incomplete Market , 1995 .

[35]  Jakša Cvitanić,et al.  Convex Duality in Constrained Portfolio Optimization , 1992 .

[36]  E. Ziirich.  HEDGING BY SEQUENTIAL REGRESSION: AN INTRODUCTION TO THE MATHEMATICS OF OPTION TRADING , 2000 .

[37]  I. Karatzas,et al.  On the pricing of contingent claims under constraints , 1996 .

[38]  R. Frey.  Derivative Asset Analysis in Models with Level-Dependent and Stochastic Volatility , 1997 .

[39]  M. Frittelli.  The Minimal Entropy Martingale Measure and the Valuation Problem in Incomplete Markets , 2000 .

[40]  Jean-Michel Bismut,et al.  Growth and optimal intertemporal allocation of risks , 1975 .

[41]  Peter Grandits,et al.  The p-optimal martingale measure and its asymptotic relation with the minimal-entropy martingale measure , 1999 .

[42]  J. Cox,et al.  Optimal consumption and portfolio policies when asset prices follow a diffusion process , 1989 .

[43]  Vasant Naik,et al.  General Equilibrium Pricing of Options on the Market Portfolio with Discontinuous Returns , 1990 .

[44]  S. Shreve,et al.  Stochastic differential equations , 1955, Mathematical Proceedings of the Cambridge Philosophical Society.

[45]  Stanley R. Pliska,et al.  A Stochastic Calculus Model of Continuous Trading: Optimal Portfolios , 1986, Math. Oper. Res..

[46]  S. Ross,et al.  Option pricing: A simplified approach☆ , 1979 .

[47]  Hui Wang,et al.  Utility maximization in incomplete markets with random endowment , 2001, Finance Stochastics.

[48]  T. Chan.  Pricing contingent claims on stocks driven by Lévy processes , 1999 .