The Forward Measure¶

The \(T\)-forward measure is a probability measure that uses the zero-coupon bond \(P(t,T)\) as numéraire. It is particularly useful for pricing interest rate derivatives where the payoff occurs at a specific future date \(T\).

The central advantage of the forward measure is that it removes the randomness of discounting: pricing reduces to \(P(t,T)\,\mathbb{E}^{\mathbb{Q}^T}[\Phi_T]\) with a deterministic prefactor, avoiding the correlation between stochastic discount factor and payoff that complicates the risk-neutral valuation formula.

Definition¶

The forward measure is the numéraire framework applied to the \(T\)-maturity zero-coupon bond \(P(t,T)\) (with \(P(T,T) = 1\) and \(P(t,T) > 0\)).

By the change-of-numéraire formula, the \(T\)-forward measure \(\mathbb{Q}^T\) has Radon–Nikodym derivative:

where \(B_t = e^{\int_0^t r_s\,ds}\) is the money market account and \(\mathbb{Q}\) is the standard risk-neutral measure.

Key Properties¶

Recall (see § The Fundamental Theorem for Numéraires): for any traded asset \(S_t\), the bond-deflated price \(S_t / P(t,T)\) is a \(\mathbb{Q}^T\)-martingale. Specialised to \(S_t\), this is the forward price

since \(F(T,T) = S_T\). For a claim with payoff \(\Phi_T\) at time \(T\):

No explicit discounting is required — the bond price handles it.

Comparison: Risk-Neutral vs Forward Measure¶

Dynamics Under the Forward Measure¶

Brownian Motion Change¶

Under \(\mathbb{Q}\): \(W_t^{\mathbb{Q}}\) is Brownian motion.

The change of numéraire from \(B_t\) to \(P(t,T)\) shifts the Brownian motion by the bond volatility \(\sigma_P(t,T)\):

is Brownian motion, where \(\sigma_P(t,T)\) is defined through the bond dynamics:

Asset Dynamics¶

If under \(\mathbb{Q}\) the asset and bond are driven by (possibly correlated) diffusions with volatilities \(\sigma_S\) and \(\sigma_P(t,T)\) and instantaneous correlation \(\rho_{S,P}\), then the asset dynamics change from

to

The drift acquires the extra term \(\sigma_S\,\sigma_P(t,T)\,\rho_{S,P}\), which is precisely the instantaneous covariance between the asset return \(dS/S\) and the bond return \(dP/P\). This covariance correction arises because the Girsanov shift from \(\mathbb{Q}\) to \(\mathbb{Q}^T\) is driven by the bond volatility \(\sigma_P(t,T)\): the change of numéraire tilts probabilities in proportion to the bond's own diffusion, and every correlated diffusion picks up a corresponding drift adjustment.

Example: LIBOR Forward Rate¶

Definition¶

The forward LIBOR rate \(L(t;T,T+\delta)\) for period \([T, T+\delta]\) is defined by:

Under the (T+δ)-Forward Measure¶

The forward LIBOR rate is a \(\mathbb{Q}^{T+\delta}\)-martingale:

At time \(T\), the spot LIBOR fixes: \(L(T;T,T+\delta) = L_T\).

Caplet Pricing¶

A caplet with strike \(K\) pays \(\delta(L_T - K)^+\) at time \(T+\delta\).

Under \(\mathbb{Q}^{T+\delta}\):

If \(L(t;T,T+\delta)\) is log-normal under \(\mathbb{Q}^{T+\delta}\) with volatility \(\sigma_L\):

This is Black's formula for caplets.

Example: Forward Contract¶

Setup¶

Forward contract to buy asset \(S\) at time \(T\) for price \(K\).

Payoff at \(T\): \(\Phi_T = S_T - K\).

Under Risk-Neutral Measure¶

Requires knowing the joint distribution of \(r\) and \(S\).

Under Forward Measure¶

Since \(F(t,T)\) is a \(\mathbb{Q}^T\)-martingale: \(\mathbb{E}^{\mathbb{Q}^T}[S_T] = F(t,T)\).

Much simpler! No need to model \(r\) and \(S\) jointly.

The Forward Measure in Vasicek Model¶

Recall (see § Vasicek Example): under \(\mathbb{Q}\), \(dr_t = \kappa(\bar r - r_t)\,dt + \sigma_r\,dW_t^{\mathbb{Q}}\) and the bond price is \(P(t,T)=A(t,T)e^{-B(t,T)r_t}\) with \(B(t,T)=(1-e^{-\kappa(T-t)})/\kappa\). The bond volatility is \(\sigma_P(t,T) = -B(t,T)\sigma_r\), and substituting into the Girsanov shift \(W^{\mathbb{Q}^T}_t = W^{\mathbb{Q}}_t - \int_0^t \sigma_P(s,T)\,ds\) gives the short-rate dynamics under the \(T\)-forward measure:

The extra drift \(-\sigma_r^2 B(t,T)\) is the convexity adjustment.

Multiple Forward Measures¶

For different maturities \(T_1 < T_2\):

Each maturity has its own forward measure.

The Tower of Measures¶

All are equivalent measures connected by Radon–Nikodym derivatives.

When to Use the Forward Measure¶

Avoid forward measure for:

• Path-dependent options (use \(\mathbb{Q}\))
• American options (early exercise)
• Options with multiple payment dates (use swap measure)

Summary¶

All forward measures \(\mathbb{Q}^{T_1}, \mathbb{Q}^{T_2}, \ldots\) and the risk-neutral measure \(\mathbb{Q}\) are equivalent --- they agree on which events are possible and differ only in how they weight outcomes. The choice among them is a matter of computational convenience, not of economic content.

The forward measure transforms the problem of stochastic discounting into a problem of computing a simple expectation, making it indispensable for interest rate modeling.

Exercises¶

Exercise 1. Write the Radon–Nikodym derivative \(d\mathbb{Q}^T / d\mathbb{Q}|_{\mathcal{F}_t}\) in terms of \(P(t,T)\), \(P(0,T)\), and \(B_t\). Verify that at \(t = T\), this expression simplifies to \(e^{-\int_0^T r_s\,ds} / P(0,T)\). Explain why \(\mathbb{E}^{\mathbb{Q}}[d\mathbb{Q}^T / d\mathbb{Q}|_{\mathcal{F}_T}] = 1\).

Exercise 2. A forward contract on a stock \(S\) for delivery at \(T\) has payoff \(S_T - K\) at maturity. Using the forward measure, show that the value at time \(t\) is \(V_t = P(t,T)(F(t,T) - K)\) where \(F(t,T) = S_t / P(t,T)\). Determine the forward price \(K^*\) that makes the contract initially worth zero.

Exercise 3. In the Vasicek model with \(\kappa = 0.3\), \(\bar{r} = 0.05\), \(\sigma_r = 0.02\), and \(B(t,T) = (1 - e^{-\kappa(T-t)})/\kappa\), compute the bond volatility \(\sigma_P(t,T) = -B(t,T)\sigma_r\) for \(T - t = 5\). Write the drift adjustment for the short rate under \(\mathbb{Q}^T\).

Exercise 4. A caplet with strike \(K = 0.05\) on the LIBOR rate \(L(T; T, T+\delta)\) with \(\delta = 0.25\) pays \(\delta(L_T - K)^+\) at \(T + \delta\). If \(L(0; T, T+\delta) = 0.048\) and the forward LIBOR volatility is \(\sigma_L = 0.20\), use Black's formula to price the caplet under \(\mathbb{Q}^{T+\delta}\). Assume \(P(0, T+\delta) = 0.92\).

Exercise 5. Explain why the forward price \(F(t,T) = S_t / P(t,T)\) is a \(\mathbb{Q}^T\)-martingale but not a \(\mathbb{Q}\)-martingale in general. What is the drift of \(F(t,T)\) under the standard risk-neutral measure \(\mathbb{Q}\)?

Exercise 6. For two different maturities \(T_1 < T_2\), write the Radon–Nikodym derivative \(d\mathbb{Q}^{T_2}/d\mathbb{Q}^{T_1}|_{\mathcal{F}_t}\) and explain why the measures \(\mathbb{Q}^{T_1}\) and \(\mathbb{Q}^{T_2}\) differ. In which financial applications does the choice between these measures matter?

Exercise 7. Recall (see § Numéraire, Example: Exchange Option) that Margrabe's formula uses \(S^2\) as numéraire. In contrast, when the payoff \((S_T - K)^+\) on a single asset is dated at \(T\), the \(T\)-forward measure is preferred over the money-market numéraire. State the precise advantage of \(\mathbb{Q}^T\) over \(\mathbb{Q}\) when interest rates are stochastic and explain why this advantage disappears when \(r\) is constant.