Mathematical Finance, Vol. 16, No. 3 (July 2006), 519–547
MODEL UNCERTAINTY AND ITS IMPACT ON THE PRICING OF DERIVATIVE INSTRUMENTS
RAMA CONT Centre de Math´ematiques Appliqu´ees, Ecole Polytechnique, Palaiseau, France
Uncertainty on the choice of an option pricing model can lead to “model risk” in the valuation of portfolios of options. After discussing some properties which a quan- titative measure of model uncertainty should verify in order to be useful and relevant in the context of risk management of derivative instruments, we introduce a quantita- tive framework for measuring model uncertainty in the context of derivative pricing. Two methods are proposed: the first method is based on a coherent risk measure com- patible with market prices of derivatives, while the second method is based on a convex risk measure. Our measures of model risk lead to a premium for model uncertainty which is comparable to other risk measures and compatible with observations of mar- ket prices of a set of benchmark derivatives. Finally, we discuss some implications for the management of “model risk.”
KEY WORDS: model uncertainty, Knightian uncertainty, option pricing, incomplete markets, volatility, ambiguity, coherent risk measures, convex risk measures
1. INTRODUCTION In March 1997, Bank of Tokyo/Mitsubishi announced that its New York-based deriva- tives unit had suffered a $83 million loss because their internal pricing model overvalued a portfolio of swaps and options on U.S. interest rates. A few weeks later, NatWest Capital Markets announced a £50 million loss because of a mispriced portfolio of German and U.K. interest rate options and swaptions run by a single-derivatives trader in London. According to observers having followed these events, many “of the situations [...] that led to (recent) derivatives losses were attributable to model risk” (Elliott 1997). With the dissemination of quantitative methods in risk management and advent of complex derivative products, mathematical models have come to play an increasingly important role in financial decision making, especially in the context of pricing and hedging of derivative instruments. While the use of models has undeniably led to a better
This project was supported by a research grant from the Europlace Institute of Finance. Part of this work was done in the framework of a research project on model uncertainty at HSBC-CCF, Division of Market and Model Risk. Previous versions of this work were presented at the Hermes Center (Nicosia), 3rd METU Economics Research Conference (2003), the European Conference on Arbitrage Theory and Applications (Paris, 2003), and the Workshop on Advanced Mathematical Methods in Finance (Munich, 2004). We thank Sana BenHamida, Joel¨ Bessis, Jean-Franc¸ois Boulier, Remi´ Bourrette, Nicole El Karoui, Larry Epstein, Daniel Gabay, Jocelyne Nadal, Walter Schachermayer, Franck Viollet, and an anonymous referee for helpful comments and Emmanuelle Hammerer for research assistance. Manuscript received June 2004; final revision received March 2005. Address correspondence to Rama Cont, Centre de Mathematiques´ Appliquees,´ Ecole Polytechnique, F-91128 Palaiseau, France; e-mail: [email protected].