We propose a formal definition of approximate arbitrage which can be used to extend the applicability of theories based on the absence of arbitrage. Our definition is based on the radio of gain to loss, where gain (loss) is the expectation of the positive (negative) part of the excess payoff. Arbitrage is characterized by infinite gain-loss ratio, an approximate arbitrage by gain-loss ratio close to infinity. Our definition of approximate arbitrage has a use dual interpretation in terms of pricing kernels. This allows us to compare the pricing kernel restriction implied by a limit on the maximum gain-loss ration to the other in the literature. We show theoretically that only the gain-loss restriction is consistent with the absence of arbitrage and approximate arbitrage opportunities. We demonstrate the practical differences of these alternative pricing kernel restrictions by examining their implications for the prices of call options on an asset that does not trade.

This paper offers a new approach for pricing options on assets with stochastic volatility. We start by taking as given the prices of a few simple, liquid European options. More specifically, we take as given the “surface” of Black-Scholes implied volatilities for European options with varying strike prices and maturities. We show that the Black-Scholes implied volatilities of at-the-money options converge to the underlying asset’s instantaneous (stochastic) volatility as the time to maturity goes to zero. We model the stochastic processes followed by the implied volatilities of options of all maturities and strike prices as a joint diffusion with the stock price. In order for no arbitrage opportunities to exist in trading the stock and these options, the drift of the processes followed by the implied volatilities is constrained in such a way that it is fully characterized by the volatilities of the implied volatilities. Finally, we suggest how to use the arbitrage-free joint process for the stock price and its volatility to price other derivatives, such as standard but illiquid options as well as exotic options, using numerical methods. Our approach simply requires as inputs the stock price and the implied volatilities at the time the exotic option is to be priced, as well as estimates of the volatilities of the implied volatilities.

We develop an estimation method for the Diagonal Multivariate GARCH model. For a vector of size N unidimensional GARCH processes for the diagonal elements of the conditional covariance matrix, and N(N-1)/2 bivariate GARCH processes for the off-diagonal elements of the conditional covariance matrix. The coefficient matrices are then transformed in such a way that ensures the positive semi-definiteness of the conditional covariance matrix. Under a technical assumption, the estimator has the same asymptotic properties as the univariate and bivariate maximum likelihood GARCH estimators. The method is computationally feasible for large problems, of size N=100 or larger. We do not need to impose any particular simplifying structure on the coefficient matrices. The conditional covariance is ensured to be stationary, and is, in general, well conditioned. We provide an empirical application in the context of international stock markets and offer Monte Carlo evidence of the good small sample properties of the model.

We study a rational expectation model of bubbles and crashes. The model has two components: (1) our key assumption is that a crash may be caused by local self-reinforcing imitation between noise traders. If the tendency for noise traders to imitate their nearest neighbors increases up to a certain point called the “critical” point, all noise traders may place the same order (sell) at the same time, thus causing a crash. The interplay between the progressive strengthening of imitation and the ubiquity of noise is characterized by the hazard rate, i.e. the probability per unit time that the crash will happen in the next instant if it has not happened yet. (2) Since the crash is not a certain deterministic out come of the bubble, it remains rational for traders to remain invested provided they are compensated by a higher rate of growth of the bubble for taking the risk of a crash. Our model distinguishes between the end of the bubble and the time of the crash: the rational expectation constraint has the specific implication that the date of the crash must be random. The theoretical death of the bubble is not the time of the crash because the crash could happen at any time before, even though this is not very likely. The death of the bubble is the most probable time for the crash. There also exists a finite probability of the attaining the end of the bubble without crash. Our model has specific predictions about the presence of certain critical log-periodic patterns in pre-crash prices, associated with the deterministic components of the bubble mechanism. We provide empirical evidence showing that these patterns were indeed present before the crashes of 1929, 1962 and 1987 on Wall Street and the 1997 crash on the Hong Kong Stock Exchange. These results are compared with the statistical tests on synthetic data.