A number of futures contracts conveys to the short position various delivery options regarding the quality and exact timing of delivery. Moreover, the compensation to the long position is not solely determined by the market value of the delivered asset at the time of delivery. Sometimes, the long position can hedge this delivery risk by holding an appropriate portfolio of the underlying asset. It often has been stated that whenever the long position can form a dynamic hedge against the delivery risk, the delivery option has a zero value. This paper demonstrates the implication of such erroneous intuition to the pricing of options. It is shown that the root of the issue is the property of diffusion processes whereas, within a given time interval, a random variable either will never cross a given boundary or else, cross it an infinite number of times.

Alan J. Marcus, David M. Modest, The Valuation of a Random Number of Put Options: An Application to Agricultural Price Supports, The Journal of Financial and Quantitative Analysis, Vol. 21, No. 1 (Mar., 1986), pp. 73-86

Equilibrium in the market for real assets requires that the price of those assets be bid up to reflect the tax shields they can offer to levered firms.Thus there must be an equality between the market values of real assets and the values of optimally levered firms.The standard measure of the advantage to leverage compares the values of levered and unlevered assets, and can be misleading and difficult to interpret.We show that a meaningful measure of the advantage to debt is the extra rate of return, net of a market premium for bankruptcy risk, earned by a levered firm relative to an otherwise-identical unlevered firm.We construct an option valuation model to calculate such a measure and present extensive simulation results.We use this model to compute optimal debt maturities, show how this approach can be used for capital budgeting, and discuss its implications for the comparison of bankruptcy costs versus tax shields.