On the pricing of forward starting options in Heston’s model on stochastic volatility

We consider the problem of pricing European forward starting options in the presence of stochastic volatility. By performing a change of measure using the asset price at the time of strike determination as a numeraire, we derive a closed-form solution within Hestons stochastic volatility framework applying distribution properties of the volatility process. In this paper we develop a new and more suitable formula for pricing forward starting options. This formula allows to cover the smile effects observed in a Black-Scholes environment, in which the extreme exposure of forward starting options to volatility changes is ignored.Received: July 2004, Mathematics Subject Classification (2000): 91B28, 60G44, 60H30, 60E10JEL Classification: G13It is a pleasure to thank the anonymous referee for his valuable comments and suggestions on this paper. Furthermore, we would like to thank Holger Kraft, University of Kaiserslautern, and Alexander Giese, HypoVereinsbank AG Munich, for fruitful discussions and suggestions.     

Wealth-path dependent utility maximization in incomplete markets

Motivated by an optimal investment problem under time horizon uncertainty and when default may occur, we study a general structure for an incomplete semimartingale model extending the classical terminal wealth utility maximization problem. This modelling leads to the formulation of a wealth-path dependent utility maximization problem. Our main result is an extension of the well-known dual formulation to this context. In contrast with the usual duality approach, we work directly on the primal problem. Sufficient conditions for characterizing the optimal solution are also provided in the case of complete markets, and are illustrated by examples.Received: December 2003, Mathematics Subject Classification (2000): 91B28, 91B16, 49N15, 49N30JEL Classification: G11The authors would like to thank the anonymous referees for their remarks and suggestions which greatly improved this paper. We also thank participants at the Oberwolfach workshop in 2003 for comments and discussions.     

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.     

Fitting prices with a complete model

The aim of this paper is to introduce some methodologies for parameter estimation in Hobson and Rogers stochastic volatility model (1998). We pay a specific attention to the so-called feedback parameter, which is shown to be crucial for the model to fit correctly the smile curve of implied volatility and we introduce different procedures for the estimation of the volatility parameters. We finally test the pricing capability of the model on market options prices on the FTSE100 and the S&P500 Indexes, according to the estimation methodologies introduced.     

From the Minimal Entropy Martingale Measures to the Optimal Strategies for the Exponential Utility Maximization: the Case of Geometric Lévy Processes

In this article, we will consider a multi-dimensional geometric L'evy process as a financial market model. We will first determine the minimal entropy martingale measure (MEMM); we will next derive the optimal strategy for the exponential utility maximization of terminal wealth concretely from the representation of the MEMM. JEL Classification: D46, D52, G12 AMS (2000) Subject Classification: 60G44, 60G51, 60G52,60H20, 60J75, 91B16, 91B28, 94A17     

Valuation of credit default swaps and swaptions

This paper presents a conceptual and general framework for valuation of single-name credit derivatives. The general subfiltration approach of [J-R] to modelling default risk, which includes the Cox-process setting of [L], is integrated with a numeraire invariant approach. Several known results are reformulated and extended in this framework. New concepts and results are presented for change of numeraire in presence of default and valuation of credit swaptions. A new formula on fractional recovery of pre-default value is derived, generalizing that of [D-S]. A Black-Scholes formula for credit default swaptions due to [S] is shown to serve as a least-squares approximation to the general case.Received: 1 November 2003, Mathematics Subject Classification: 91B28, 60G44, 60G40JEL Classification: E43, G13I would like to thank the editor and two anonymous referees for their valuable comments and suggestions.     

Implications of state-dependent pricing for dynamic macroeconomic models

State-dependent pricing (SDP) models treat the timing of price changes as a profit-maximizing choice, symmetrically with other decisions of firms. Using quantitative general equilibrium models which incorporate a “generalized (S,s) approach,” we investigate the implications of SDP for topics in two major areas of macroeconomic research, the early 1990s SDP literature and more recent work on persistence mechanisms. First, we show that state-dependent pricing leads to unusual macroeconomic dynamics, which occur because of the timing of price adjustments chosen by firms as in the earlier literature. In particular, we display an example in which output responses peak at about a year, while inflation responses peak at about 2 years after the shock. Second, we examine whether the persistence-enhancing effects of two New Keynesian model features, namely specific factor markets and variable elasticity demand curves, depend importantly on whether pricing is state dependent. In an SDP setting, we provide examples in which specific factor markets perversely work to lower persistence, while variable elasticity demand raises it.     

Asymmetric information and imperfect competition in a continuous time multivariate security model

This paper deals with the problem of price formation in a market with asymmetric information and several risky assets. We then extend the multivariate security model of Caballé and Krishnan (1994) to a continuous time framework, and general utility function. Our model enables us to observe some results which are specific to multi security markets such as Giffen effect. An application of the main result will be the non trivial generalizations of the models of Back (1992) and Cho (1997).Mathematics Subject Classification (1991): 49L10, 60G44, 90A15JEL Classification: G11, G12The author would like to thank his supervisor H. Pham, K. Back and an anonymous referee for useful comments and discussions.     

Additional utility of insiders with imperfect dynamical information

In this paper we consider a market driven by a Wiener process where there is an insider and a regular trader. The insider has privileged information which has been deformed by an independent noise vanishing as the revelation time approaches. At this time, the information of every trader is the same. We obtain the semimartingale decomposition of the original Wiener process under dynamical enlargement of the filtration, and we prove that if the rate at which the additional noise in the insiders information vanishes is slow enough then there is no arbitrage and the additional utility of the insider is finite.Received: 1 October 2003, Mathematics Subject Classification: 60G48, 90A09, 60H07, 90A60JEL Classification: D82, G11, G14     

Liquidity risk and arbitrage pricing theory

Classical theories of financial markets assume an infinitely liquid market and that all traders act as price takers. This theory is a good approximation for highly liquid stocks, although even there it does not apply well for large traders or for modelling transaction costs. We extend the classical approach by formulating a new model that takes into account illiquidities. Our approach hypothesizes a stochastic supply curve for a securitys price as a function of trade size. This leads to a new definition of a self-financing trading strategy, additional restrictions on hedging strategies, and some interesting mathematical issues.Received: 1 November 2003, Mathematics Subject Classification: 60G44, 60H05, 90A09JEL Classification: G11, G12, G13Umut Çetin: This work was performed while Dr. Çetin was at the Center for Applied Mathematics, Cornell UniversityPhilip Protter: Supported in part by NSF grant DMS-0202958 and NSA grant MDA-904-03-1-0092 The authors wish to thank M. Warachka and Kiseop Lee for helpful comments, as well as the anonymous referee and Associate Editor for numerous helpful suggestions, which have made this a much improved paper.     

On the use of measure-valued strategies in bond markets

We propose here a theory of cylindrical stochastic integration, recently developed by Mikulevicius and Rozovskii, as mathematical background to the theory of bond markets. In this theory, since there is a continuum of securities, it seems natural to define a portfolio as a measure on maturities. However, it turns out that this set of strategies is not complete, and the theory of cylindrical integration allows one to overcome this difficulty. Our approach generalizes the measure-valued strategies: this explains some known results, such as approximate completeness, but at the same time it also shows that either the optimal strategy is based on a finite number of bonds or it is not necessarily a measure-valued process.Received: November 2002, Mathematics Subject Classification: 60H05, 60G60, 90A09JEL Classification: G10, E43The first author gratefully acknowledges financial support from the CNR Strategic Project Modellizzazione matematica di fenomeni economici. We thank professors A. Bagchi, R. Douady and J. Zabczyk for helpful discussions. A special thanks goes to professors T. Björk, Y. Kabanov and W. Schachermayer for comments and suggestions which contributed to improve the final version of this paper.     

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.     

Understanding the Implied Volatility Surface for Options on a Diversified Index

This paper describes a two-factor model for a diversified index that attempts to explain both the leverage effect and the implied volatility skews that are characteristic of index options. Our formulation is based on an analysis of the growth optimal portfolio and a corresponding random market activity time where the discounted growth optimal portfolio is expressed as a time transformed squared Bessel process of dimension four. It turns out that for this index model an equivalent risk neutral martingale measure does not exist because the corresponding Radon-Nikodym derivative process is a strict local martingale. However, a consistent pricing and hedging framework is established by using the benchmark approach. The proposed model, which includes a random initial condition for market activity, generates implied volatility surfaces for European call and put options that are typically observed in real markets. The paper also examines the price differences of binary options for the proposed model and their Black-Scholes counterparts. Mathematics Subject Classification: primary 90A12; secondary 60G30; 62P20 JEL Classification: G10, G13