The concept of "forward measure" is a fundamental tool in quantitative finance, specifically in the pricing of financial derivatives. It is closely related to the concept of "risk-neutral measure," but while the risk-neutral measure deals with the valuation of derivatives at the present time, the forward measure focuses on the valuation of derivatives at some future time.
A forward measure is a probability measure under which the price of an underlying asset, such as a stock or a bond, is a martingale when discounted by the price of another traded asset, typically a zero-coupon bond. The forward measure is used to calculate expected payoffs of financial derivatives in a manner that is consistent with the no-arbitrage principle. It simplifies the valuation process and allows us to compute option prices without finding the optimal hedging strategy explicitly.
To provide a more detailed explanation, let us start with the concept of a risk-neutral measure.
1. Risk-neutral measure (β): In a complete market, the risk-neutral measure is a probability measure that is equivalent to the real-world measure (i.e., has the same null sets) under which the discounted price process of a traded asset (e.g., stock) becomes a martingale. This plays a central role in pricing financial derivatives as it allows us to price a derivative using the expected payoff under the risk-neutral measure, discounted at the risk-free rate.
Suppose we have a stock price process St. Under the risk-neutral measure β, the discounted stock price process $M_t = \frac{S_t}{B_t}$, where Bt is the risk-free money market account, is a martingale. Mathematically,
Eβ[Mtβ
+β
s|β±t]β=βMt,
where Eβ represents the expectation under the risk-neutral measure, β±t is the filtration (information set) up to time t, and sβ>β0.
Now, let us move on to the concept of forward measure.
2. Forward measure (βT): A forward measure is a probability measure under which the price of an asset, when discounted by the price of another traded asset, typically a zero-coupon bond maturing at time T, is a martingale. Let P(t,βT) be the price of a zero-coupon bond at time t, which pays 1 at time T. The forward measure βT is defined such that $\widetilde{M}_t = \frac{S_t}{P(t, T)}$ is a martingale. Formally,
EβT[MΜtβ
+β
s|β±t]β=βMΜt,
where EβT represents the expectation under the forward measure.
Now let us discuss how the forward measure is used in the pricing of financial derivatives.
Pricing a derivative under the forward measure involves two main steps: (1) change the pricing measure from the real-world measure to the forward measure, and (2) compute the expected value of the derivativeβs payoff under the forward measure, discounted by the zero-coupon bond price. Mathematically, the price of a derivative is given by
V(t)β=βP(t,βT)EβT[VT|β±t],
where V(t) is the value of the derivative at time t, and VT is its payoff at time T.
To illustrate with an example, let us consider the pricing of a European call option with strike price K and maturity T on a stock with a price process St. The payoff of the call option is given by
VTβ=βmaxβ(STβ
ββ
K,β0).
Using the forward measure βT, we can price the option as follows:
$$\begin{aligned}
V(t) &= P(t, T) \operatorname{E}^{\mathbb{Q}^T}[\max(S_T - K, 0) | \mathcal{F}_t] \\
&= P(t, T) \left( \operatorname{E}^{\mathbb{Q}^T}[S_T | \mathcal{F}_t] - K\operatorname{Pr}^{\mathbb{Q}^T}(S_T \ge K | \mathcal{F}_t) \right),\end{aligned}$$
where we have used the fact that $\widetilde{M}_T = \frac{S_T}{P(T, T)} = S_T$, and $\widetilde{M}_t = \frac{S_t}{P(t, T)}$ are martingales under the forward measure.
In conclusion, the concept of forward measure simplifies the pricing of financial derivatives by transforming the prices of assets, such as stocks or bonds, into martingales when discounted by another traded asset like a zero-coupon bond. This allows us to price derivatives as the expected value of their payoffs under the forward measure, discounted by the zero-coupon bond price, ultimately ensuring consistency with the no-arbitrage principle.