Pricing Equity Derivatives Under the Convenience Yield Measure

To capture the extra returns embedded in the tails of the market distributions, the literature focused on adding stochastic processes to the diffusion coefficient of the asset prices or even jumps to the asset prices as the drift was forced to match the risk-free rate. However, if we assume that part of the information contained in the implied volatility surface is incomplete and results from matching supply and demand, we loose the notion of risk-neutral measure and can modify the drift to match the market risk premium. Similarly to the commodity markets where the holder of the spot is compensated for holding one unit of inventory in case of shortage, we are going to compensate the holder of the spot in the equity market for the risk of a large downward jump. Assuming a stochastic convenience yield, we create a self-financing portfolio involving the delivery of one unit of the stock at maturity of the forward contract, and use it to compute the model dependent forward price. Specifying an Ornstein-Uhlenbeck process for the convenience yield, we derive the dynamics of the forward price, define the convenience yield measure and apply it to compute analytically the price of a call option when the volatility of the stock price is a deterministic function of time. We then extend the approach to the case where the instantaneous volatility is a deterministic function of time as well as the diffused underlying, quantify the extra bias in the local volatility with stochastic convenience yield, and solve it semi-analytically. Finally, we consider the case where the instantaneous volatility is a stochastic process external to the stock price, compute European option prices by deriving the characteristic function of the logarithm of the forward price both analytically and numerically, and use Malliavin calculus to derive approximation to these prices