Article

# Arbitrage Opportunities in Misspecified Stochastic volatility Models

arXiv.org, Quantitative Finance Papers 02/2010; DOI: 10.1137/100786678

Source: RePEc

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**ABSTRACT:**In an incomplete market framework, contingent claims are of particular interest since they improve the market efficiency. This paper addresses the problem of market completeness when trading in contingent claims is allowed. We extend recent results by Bajeux and Rochet (1996) in a stochastic volatility model to the case where the asset price and its volatility variations are correlated. We also relate the ability of a given contingent claim to complete the market to the convexity of its price function in the current asset price. This allows us to state our results for general contingent claims by examining the convexity of their “admissible arbitrage prices.”Mathematical Finance 09/1997; 7(4):399 - 412. · 1.25 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Derman and Kani ~1994!, Dupire ~1994!, and Rubinstein ~1994! hypothesize that asset return volatility is a deterministic function of asset price and time, and develop a deterministic volatility function ~DVF! option valuation model that has the potential of fitting the observed cross section of option prices exactly. Using S&P 500 options from June 1988 through December 1993, we examine the predictive and hedging performance of the DVF option valuation model and find it is no better than an ad hoc procedure that merely smooths Black--Scholes ~1973! implied volatilities across exercise prices and times to expiration. EXPECTED FUTURE VOLATILITY PLAYS a central role in finance theory. Consequently, accurately estimating this parameter is crucial to meaningful financial decision making. Finance researchers generally rely on the past behavior of asset prices to develop expectations about volatility, documenting movements in volatility as they relate to prior volatility and0or variables *Bernard Dumas is Professor of Finance, Hautes tudes Commerciales ~HEC! School of Management, Research Professor of Finance, Fuqua School of Business, Duke University, National Bureau of Economic Research ~NBER! Research Associate, and Center for Economic Policy Research ~CEPR! Research Fellow; Jeff Fleming is Assistant Professor of Administrative Science, Jones Graduate School of Administration, Rice University; and, Robert E. Whaley is T. Austin Finch Foundation Professor of Business Administration, Fuqua School of Business, Duke University. This research was supported by the HEC School of Management and the Futures and Options Research Center at the Fuqua School of Business, Duke University. We gratefully acknowledge discussions with Jens Jackwerth and Mark Rubinstein and comments and sugge...The Journal of Finance 07/2001; · 4.22 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**The main purpose of the paper is to provide a mathematical background for the theory of bond markets similar to that available for stock markets. We suggest two constructions of stochastic integrals with respect to processes taking values in a space of continuous functions. Such integrals are used to define the evolution of the value of a portfolio of bonds corresponding to a trading strategy which is a measure- valued predictable process. The existence of an equivalent martingale measure is discussed and HJM-type conditions are derived for a jump-diffusion model. The question of market completeness is considered as a problem of the range of a certain integral operator. We introduce a concept of approximate market completeness and show that a market is approximately complete if an equivalent martingale measure is unique.Finance and Stochastics 01/1997; · 1.21 Impact Factor

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## Peter Tankov |