Real world pricing and affine representation for forward contracts

Abstract

AbstractThis paper makes use of an integrated benchmark modeling framework that allows us to derive a term structure for the forward contract under real world probability measure. The benchmark or numeraire is chosen to be the growth optimal portfolio (GOP). In general forward prices are not martingales under the real world probability measures. To overcome this problem, we introduce the benchmarked forward measure under which we recover the martingale property for forward prices. Under the benchmarked forward measure we establish an actuarial pricing formula for contingent claims that are not necessarily independent of the GOP. Thereafter, the issue of relating state variables in forward price models to market observed quantities is addressed. The paper links state variables directly, and explicitly, to forward prices, and establishes that models are, in fact, affine with respect to a finite number of forward prices. Applying the methodology to Oil forward models we remark that the forward price is independent of fluctuations of the zero‐coupon bond price and therefore of those of the interest rate. The paper shows also that under the benchmark setting the contribution of the market price of risk is naturally included in the modeling of the zero‐coupon bond and the forward price, which is not systematic in the risk neutral setting. Copyright © 2010 Wilmott Magazine Ltd.