We develop a theoretical arbitrage-free and complete model to price options on the stocks of companies involved in a merger or acquisition deal allowing for the possibility that the deal might be called off at an intermediate time possibly creating discontinuous impacts on the stock prices. Our model is intended to be a valuable tool for market makers to quote prices for options on stocks involved in merger or acquisition deals and also for risk arbitrageurs and options traders to gauge the fair value for such options. Although we specifically consider stock-for-stock and cash acquisition deals, our basic framework is applicable to the investigation of much more general deals.
We model the stock price processes as jump diffusions with the jumps representing the price impacts of the deal being called off. We show that fundamental economic considerations imply specific functional forms for the price impact functions that allow us to obtain analytical expressions for the option prices. We demonstrate the completeness of our model when there are marketed securities that represent the fundamental values of the stocks involved in the deal, i.e the values in the absence of the synergies associated with the deal. In the situation where such securities do not exist, we derive the optimal risk-minimizing strategies in the underlying stocks and a risk-free bond for any option on either stock. These strategies show how one may hedge the risks associated with merger or acquisition deals using traded options on the stocks involved. We also show how one may use the model to infer the probabilities of success of deals from observed option price data.
Finally, we test our model on real option price data. We investigate several merger deals and show that the model is able to explain observed option prices on stocks involved in such deals remarkably well. In the process, we also conclusively demonstrate the inadequacy of the Black-Scholes framework in explaining observed option prices.