Forward versus Spot Price Modeling

It is possible to base an equity derivatives pricing model on a stochastic driving process for the share price (spot), or for the equity forward. While the former is the classical approach pioneered by Black and Scholes [9], the latter approach separates the modeling of exogenous random price fluctuations from the cost-of-carry modeling of the stock, probably the first and most prominent example of this technique being the paper by Black [8]. While the Black—Scholes spot price approach and Black’s forward approach are equivalent, the present note demonstrates that the introduction of local volatility and/ or level-dependent default intensity into the stochastic driving process destroys this equivalence, if applied carelessly. The advantages and disadvantages of both approaches are discussed.