Contingent Claims On Assets With Conversion Costs

Contingent claim contracts on certain natural resources may involve underlying assets which exist in either developed (S) or undeveloped (S-c) states. While the positive prices of the developed asset may be modelled by the standard geometric Brownian motion over the specified term of the contract, the lower priced undeveloped asset may become negative due to a conversion cost (c) given in the contracts. In particular pricing such contracts involves natural arbitrage opportunities on undeveloped assets in the event that developed prices drop below c and then rise before expiry T. Thus we consider the effect on contract price of requiring contract settlement (with possible rebates) at time τc = inf{t ≤ T : S(t) = c}. In particular, we obtain the final-value pricing equation (with knock-out boundary) ∂π ∂t + rs ∂π ∂s + 1 2 σs ∂2π ∂s2 − rπ = rc ∂s where r is the risk free interest rate, σ the volatility in developed asset. Mathematically, the success of this approach is based on martingale localization of the classic theory of Harrision-Pliska (1981) to provide a simple and precise meaning of (localized) arbitrage-free pricing in this context.