Joint stock/option dynamics and application to option trading strategies.

Summary This thesis explores theoretically and empirically the implications of the joint stock/option dynamics on various issues related to options trading. First, we study the joint dynamics between an option on a stock and an option on the market index. The CAPM model provides an adequate mathematical framework for this study because it allows to model the joint dynamics of a stock and its market index. Moving to option prices, we show that beta and idiosyncratic volatility, parameters of the model, allow us to characterize the relationship between the implied volatility surfaces of the stock and the index. We then turn to the estimation of the beta parameter under the risk-neutral probability using option prices. This measure, called implied beta, represents the information contained in the option prices about the realization of the beta parameter in the future.For this reason, we try to see, if implied beta has any predictive power of the future beta.By conducting an empirical study, we conclude that implied beta does not improve the predictive ability compared to the historical beta which is computed through the linear regression of the stock returns on the index returns. Better yet, we note that the oscillation of the implied beta around the future beta can lead to arbitrage opportunities, and we propose an arbitrage strategy that allows to monetize this gap. On the other hand, we show that the implied beta estimator could be used to hedge options on the stock using index instruments, this hedging concerns notably the volatility risk and also the delta risk. In the second part of our work, we are interested in the problem of market making on options. In this study, we assume that the model of the underlying's dynamics under the risk-neutral probability could be misspecified, which reflects a mismatch between the implied distribution of the underlying and its historical distribution.First, we consider the case of a risk-neutral market maker who aims to maximize the expectation of his future wealth. Using a stochastic optimal control approach, we determine the optimal call and put prices on the option and interpret the effect of price inefficiency on the optimal strategy. In a second step, we consider that the market maker is risk averse and therefore tries to reduce the uncertainty associated with his inventory. By solving an optimization problem based on a mean-variance criterion, we obtain analytical approximations of the optimal buying and selling prices. We also show the effects of inventory and price inefficiency on the optimal strategy. We then turn to the market making of options in a higher dimension. Thus, following the same reasoning, we present a framework for the market making of two options with different underlyings with the constraint of variance reduction related to the inventory risk held by the market maker. In the last part of our work, we study the joint dynamics between the implied volatility at the currency and the underlying, and we try to establish the link between these joint dynamics and the implied skew. We are interested in an indicator called "Skew Stickiness Ratio" which has been introduced in the recent literature. This indicator measures the sensitivity of the implied volatility of the currency to the movements of the underlying. We propose a method that allows us to estimate the value of this indicator under the risk-neutral probability without the need to admit assumptions on the dynamics of the underlying. [.].