How the Pricing Measure Affects a Forward Price's Drift
Summary
The document distinguishes the drift of a forward price from the drift of the value of a forward contract. In the stated simplest setting, with a constant risk-free rate, the underlying follows a diffusion whose drift under the risk-neutral measure is that rate. The forward price for delivery at a fixed date is then described as having zero drift under that measure, even though the contract's value has risk-free-rate drift.
The response also states that the forward price is a martingale under the forward measure associated with its delivery date, so its drift there is zero. Under other measures, the forward price need not have zero drift; its drift depends on the chosen measure. The explanation is brief and provides no derivation beyond the constant-rate example. It does not cover stochastic interest rates or spell out drift expressions under the real-world measure, so those cases require additional assumptions and analysis.
Key ideas
- The drift of a forward price depends on the probability measure used to describe it.
- Under the relevant forward measure, the forward price is described as a martingale with zero drift.
- With a constant risk-free rate, the example gives the forward price zero drift under the risk-neutral measure.
- The value of a forward contract is distinct from its forward price and has risk-free-rate drift in the stated setting.
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Full text
# Does a forward price have a drift component in any measure? # Does a forward price have a drift component in any measure? Going by intuition, a forward price should already take into account the drift in the underlying price process. Further, assuming interest rates are deterministic, the stochasticity in the forward price process comes solely from the underlying price process. That much I can intuitively accept. But what about the drift component of the forward price process? Is it non-zero and does the choice of measure (risk-neutral, forward, real-world) influence its presence and magnitude? Another issue I cannot reconcile is that I've read that all tradable securities under the risk-neutral measure must have a drift component the same as the risk-free interest rate. A forward contract is surely a tradable security? Please excuse my confusion and possible bludgeoning of different concepts. ## Answer by zsljulius (score 1) https://quant.stackexchange.com/a/17027 I will formalize my answer later. But one thing you will have to know is that the price of a forward contract will be a martingale under t forward measure, meaning the drift term is 0. This is not true under other measures. So the drift of the process depends on the measure you use to price the contract. ## Answer by Joern (score 1) https://quant.stackexchange.com/a/17151 It the value of the forward contract and not the forward price that has drift r under the risk-neutral measure. In fact, in the simplest case where the risk-free interest rate is a constant r, then the forward price process f(t,T) has zero drift under the risk-neutral measure: If the spot price process satisfies dS(t)=S(t)(rdt+bdW(t)), then dF(t,T)=bF(t,T)dW(t).
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