MODEL UNCERTAINTY AND ITS IMPACT ON THE PRICING OF DERIVATIVE INSTRUMENTS

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 quantitative 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 quantitative 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 compatible 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 market prices of a set of benchmark derivatives. Finally, we discuss some implications for the management of "model risk."     

A NEW METHOD OF PRICING LOOKBACK OPTIONS

A new method for pricing lookback options (a.k.a. hindsight options) is presented, which simplifies the derivation of analytical formulas for this class of exotics in the Black-Scholes framework. Underlying the method is the observation that a lookback option can be considered as an integrated form of a related barrier option. The integrations with respect to the barrier price are evaluated at the expiry date to derive the payoff of an equivalent portfolio of European-type binary options. The arbitrage-free price of the lookback option can then be evaluated by static replication as the present value of this portfolio. We illustrate the method by deriving expressions for generic, standard floating-, fixed-, and reverse-strike lookbacks, and then show how the method can be used to price the more complex partial-price and partial-time lookback options. The method is in principle applicable to frameworks with alternative asset-price dynamics to the Black-Scholes world.     

A GENERAL EQUILIBRIUM MODEL OF A MULTIFIRM MORAL‐HAZARD ECONOMY WITH FINANCIAL MARKETS

We present a general equilibrium model of a moral‐hazard economy with many firms and financial markets, where stocks and bonds are traded. Contrary to the principal‐agent literature, we argue that optimal contracting in an infinite economy is not about a tradeoff between risk sharing and incentives, but it is all about incentives. Even when the economy is finite, optimal contracts do not depend on principals’ risk aversion, but on market prices of risks. We also show that optimal contracting does not require relative performance evaluation, that the second best risk‐free interest rate is lower than that of the first best, and that the second‐best equity premium can be higher or lower than that of the first best. Moral hazard can contribute to the resolution of the risk‐free rate puzzle. Its potential to explain the equity premium puzzle is examined.     

PRICING A CLASS OF EXOTIC OPTIONS VIA MOMENTS AND SDP RELAXATIONS

We present a new methodology for the numerical pricing of a class of exotic derivatives such as Asian or barrier options when the underlying asset price dynamics are modeled by a geometric Brownian motion or a number of mean-reverting processes of interest. This methodology identifies derivative prices with infinite-dimensional linear programming problems involving the moments of appropriate measures, and then develops suitable finite-dimensional relaxations that take the form of semidefinite programs (SDP) indexed by the number of moments involved. By maximizing or minimizing appropriate criteria, monotone sequences of both upper and lower bounds are obtained. Numerical investigation shows that very good results are obtained with only a small number of moments. Theoretical convergence results are also established.     

EXISTENCE OF A NONNEGATIVE EQUILIBRIUM PRICE VECTOR IN THE MEAN-VARIANCE CAPITAL MARKET

We derive a necessary and sufficient condition for the existence of a nonnegative equilibrium price vector under which the total demand and supply of each asset balances in the standard mean-variance capital market. Also, we give an explicit formula for such a price vector. This formula shows that the price of assets is an increasing function of , the weighted average of the requested rate of return of individual investors, which tends to infinity as approaches the expected rate of return on the market portfolio. Further, we construct a macroeconomic index which gives information about the soundness of the capital market.