A geometric approach to portfolio optimization in models with transaction costs

We consider a continuous-time stochastic optimization problem with infinite horizon, linear dynamics, and cone constraints which includes as a particular case portfolio selection problems under transaction costs for models of stock and currency markets. Using an appropriate geometric formalism we show that the Bellman function is the unique viscosity solution of a HJB equation.Mathematics Subject Classification (1991): 60G44JEL Classification: G13, G11This research was done at Munich University of Technology supported by a Mercator Guest Professorship of the German Science Foundation (Deutsche Forschungsgemeinschaft). The authors also express their thanks to Mark Davis, Steve Shreve, and Michael Taksar for useful discussions concerning the principle of dynamic programming.     

Completion of a Lévy market by power-jump assets

Except for the geometric Brownian model and the geometric Poissonian model, the general geometric Lévy market models are incomplete models and there are many equivalent martingale measures. In this paper we suggest to enlarge the market by a series of very special assets (power-jump assets) related to the suitably compensated power-jump processes of the underlying Lévy process. By doing this we show that the market can be completed. The very particular choice of the compensators needed to make these processes tradable is delicate. The question in general is related to the moment problem.Received: June 2004, Mathematics Subject Classification (2000): 91B28, 91B26, 91B16, 91B70JEL Classification: C61The work of José Manuel Corcuera and David Nualart is partially supported by the MCyT grant no. BFM200304294. W. Schoutens is a Postdoctoral Fellow of the Fund for Scientific Research - Flanders (Belgium) (F.W.O. - Vlaanderen).     

Hazard rate for credit risk and hedging defaultable contingent claims

We provide a concise exposition of theoretical results that appear in modeling default time as a random time, we study in details the invariance martingale property and we establish a representation theorem which leads, in a complete market setting, to the hedging portfolio of a vulnerable claim. Our main result is that, to hedge a defaultable claim one has to invest the value of this contingent claim in the defaultable zero-coupon.Received: April 2003Mathematics Subject Classification: 91B24, 91B29, 60G46JEL Classification: G10The authors would like to thank D. Becherer and J.N. Hugonnier for interesting discussions and the anonymous referee whose pertinent questions on the first version of this paper help them to clarify the proofs. All remaining errors are ours.     

Optimizing the terminal wealth under partial information: The drift process as a continuous time Markov chain

We consider a multi-stock market model where prices satisfy a stochastic differential equation with instantaneous rates of return modeled as a continuous time Markov chain with finitely many states. Partial observation means that only the prices are observable. For the investors objective of maximizing the expected utility of the terminal wealth we derive an explicit representation of the optimal trading strategy in terms of the unnormalized filter of the drift process, using HMM filtering results and Malliavin calculus. The optimal strategy can be determined numerically and parameters can be estimated using the EM algorithm. The results are applied to historical prices.Received: March 2004, Mathematics Subject Classification (2000): 91B28, 60G44JEL Classification: G11Supported by NSERC under research grant 88051 and NCE grant 30354.     

Diversified Portfolios with Jumps in a Benchmark Framework

This paper considers diversified portfolios in a sequence of jump diffusion market models. Conditions for the approximation of the growth optimal portfolio (GOP) by diversified portfolios are provided. Under realistic assumptions, it is shown that diversified portfolios approximate the GOP without requiring any major model specifications. This provides a basis for systematic use of diversified stock indices as proxies for the GOP in derivative pricing, risk management and portfolio optimization. 1991 Mathematics Subject Classification: primary 90A12; secondary 60G30; 62P20 JEL Classification: G10, G13     

Maturity cycles in implied volatility

The skew effect in market implied volatility can be reproduced by option pricing theory based on stochastic volatility models for the price of the underlying asset. Here we study the performance of the calibration of the S&P 500 implied volatility surface using the asymptotic pricing theory under fast mean-reverting stochastic volatility described in [8]. The time-variation of the fitted skew-slope parameter shows a periodic behaviour that depends on the option maturity dates in the future, which are known in advance. By extending the mathematical analysis to incorporate model parameters which are time-varying, we show this behaviour can be explained in a manner consistent with a large model class for the underlying price dynamics with time-periodic volatility coefficients.Received: December 2003, Mathematics Subject Classification (2000): 91B70, 60F05, 60H30JEL Classification: C13, G13Jean-Pierre Fouque: Work partially supported by NSF grant DMS-0071744.Ronnie Sircar: Work supported by NSF grant DMS-0090067. We are grateful to Peter Thurston for research assistance.We thank a referee for his/her comments which improved the paper.     

A Benchmark Approach to Filtering in Finance

The paper proposes the use of the growth optimal portfolio for pricing and hedging in incomplete markets when there are unobserved factors that have to be filtered. The proposed filtering framework is applicable also in cases when there does not exist an equivalent risk neutral martingale measure. The reduction of the variance of derivative prices for increasing degrees of available information is measured. 1991 Mathematics Subject Classification: primary 90A09; secondary 60G99; 62P20 JEL Classification: G10, G13     

Optimal portfolios when stock prices follow an exponential Lévy process

We investigate some portfolio problems that consist of maximizing expected terminal wealth under the constraint of an upper bound for the risk, where we measure risk by the variance, but also by the Capital-at-Risk (CaR). The solution of the mean-variance problem has the same structure for any price process which follows an exponential Lévy process. The CaR involves a quantile of the corresponding wealth process of the portfolio. We derive a weak limit law for its approximation by a simpler Lévy process, often the sum of a drift term, a Brownian motion and a compound Poisson process. Certain relations between a Lévy process and its stochastic exponential are investigated.Received: January 2003Mathematics Subject Classification: Primary: 60F05, 60G51, 60H30, 91B28; secondary: 60E07, 91B70JEL Classification: C22, G11, D81We would like to thank Jan Kallsen and Ralf Korn for discussions and valuable remarks on a previous version of our paper. The second author would like to thank the participants of the Conference on Lévy Processes at Aarhus University in January 2002 for stimulating remarks. In particular, a discussion with Jan Rosinski on gamma processes has provided more insight into the approximation of the variance gamma model.