Review of Quantitative Finance and Accounting, 19: 261–272, 2002
2002 Kluwer Academic Publishers. Manufactured in The Netherlands.
A Note on Forward Price and Forward Measure
FOM/SOB-NB, Rutgers University, Levin Bldg., Rockafeller Rd., Piscataway, NJ 08854
E-mail: email@example.com; Tel.: (732) 445-4236; Fax: (732) 445-2333.
Penn State University, Smeal College of Business, University Park, PA 16802
Abstract. The forward measure is convenient in calculating various contingent claim prices under stochastic
interest rates. We demonstrate that caution needs to be drawn when the forward measure is used to price contin-
gent claims that involve multiple cash ﬂows. We also derive partial different equations for the forward price to
demonstrate how forward contracts can be used for dynamic hedging and how hedges can be conducted if the
payoff of a contingent claim depends on the forward price.
Key words: forward measure, forward price, stochastic interest rates
JEL Classiﬁcation: G13
The forward measure pricing methodology (Jamshidian, 1987, 1989; and Geman, Karoui
and Rochet, 1995) has been widely used in pricing securities when interest rates are stochas-
tic. This technique provides great ease in deriving closed form solutions for various deriva-
tive contracts with European-style payoffs. However, for many ﬁnancial contracts whose
payoffs are dependent upon forward prices, the forward measure needs to be used with cau-
tion. In this paper, we demonstrate, using two popular interest rate contracts, how roll-over
expectations under different forward measures should be used. We also derive an alternative
partial differential equation (PDE) to the one in Jamshidian (1987) to demonstrate how the
forward price can be interpreted alternatively. This alternative PDE is consistent with the
interpretation that the forward price is an asset price, as opposed to the interpretation that
the forward price is an index. In doing so, we effectively transfer an “index forward price”
into an “forward asset price” that satisﬁes the Capital Asset Pricing Model, as argued in
Cox, Ingersoll and Ross (1981, hereafter CIR).
The theoretical behavior of the forward price was ﬁrst derived and summarized by CIR.
They also derived closed form solutions for forward and futures prices under a single factor