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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. Under the risk-neutral measure \(\mathbb{Q}\), pricing requires computing \(\mathbb{E}^{\mathbb{Q}}[e^{-\int_t^T r_s\,ds}\,\Phi_T]\), where the stochastic discount factor and the payoff may be correlated. The forward measure absorbs this stochastic discount factor into the probability weighting, so that pricing reduces to a simple expectation \(P(t,T)\,\mathbb{E}^{\mathbb{Q}^T}[\Phi_T]\) with a deterministic prefactor. In this sense, the forward measure makes the payoff maturity \(T\) the natural reference point.

Definition

The Numéraire

The \(T\)-maturity zero-coupon bond has price process \(P(t,T)\) satisfying:

  • \(P(T,T) = 1\) (pays 1 at maturity)
  • \(P(t,T) > 0\) for \(t < T\)

The T-Forward Measure

The \(T\)-forward measure \(\mathbb{Q}^T\) is defined by the Radon-Nikodym derivative:

\[ \boxed{ \frac{d\mathbb{Q}^T}{d\mathbb{Q}}\bigg|_{\mathcal{F}_t} = \frac{P(t,T)}{P(0,T)B_t} } \]

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

1. Bond-Deflated Prices are Martingales

Under \(\mathbb{Q}^T\), for any traded asset \(S_t\):

\[ \frac{S_t}{P(t,T)} \text{ is a } \mathbb{Q}^T\text{-martingale} \]

2. Forward Prices are Martingales

The forward price of \(S\) for delivery at \(T\):

\[ F(t,T) = \frac{S_t}{P(t,T)} \]

is a \(\mathbb{Q}^T\)-martingale:

\[ F(t,T) = \mathbb{E}^{\mathbb{Q}^T}[S_T \mid \mathcal{F}_t] \]

3. Pricing Formula

For a claim with payoff \(\Phi_T\) at time \(T\):

\[ \boxed{ V_t = P(t,T) \cdot \mathbb{E}^{\mathbb{Q}^T}[\Phi_T \mid \mathcal{F}_t] } \]

No explicit discounting required—the bond price handles it.

Comparison: Risk-Neutral vs Forward Measure

Aspect Risk-Neutral \(\mathbb{Q}\) Forward \(\mathbb{Q}^T\)
Numéraire Money market \(B_t\) Bond \(P(t,T)\)
Martingale \(S_t/B_t\) \(S_t/P(t,T)\)
Pricing \(V_t = \mathbb{E}^{\mathbb{Q}}[e^{-\int_t^T r_s\,ds}\Phi_T]\) \(V_t = P(t,T)\mathbb{E}^{\mathbb{Q}^T}[\Phi_T]\)
Discount Stochastic Deterministic factor \(P(t,T)\)

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 \(P(t,T)\) induces a new measure under which bond-denominated price processes are martingales. The Radon-Nikodym derivative depends on the bond dynamics, and the corresponding Girsanov kernel is the bond volatility \(\sigma_P(t,T)\). Consequently:

\[ W_t^{\mathbb{Q}^T} = W_t^{\mathbb{Q}} - \int_0^t \sigma_P(s,T)\,ds \]

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

\[ \frac{dP(t,T)}{P(t,T)} = r_t\,dt + \sigma_P(t,T)\,dW_t^{\mathbb{Q}} \]

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

\[ \frac{dS_t}{S_t} = r_t\,dt + \sigma_S\,dW_t^{S,\mathbb{Q}} \]

to

\[ \frac{dS_t}{S_t} = \bigl(r_t + \sigma_S\,\sigma_P(t,T)\,\rho_{S,P}\bigr)\,dt + \sigma_S\,dW_t^{S,\mathbb{Q}^T} \]

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:

\[ 1 + \delta L(t;T,T+\delta) = \frac{P(t,T)}{P(t,T+\delta)} \]

Under the (T+δ)-Forward Measure

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

\[ L(t;T,T+\delta) = \mathbb{E}^{\mathbb{Q}^{T+\delta}}[L(T;T,T+\delta) \mid \mathcal{F}_t] \]

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

\[ V_t = P(t,T+\delta) \cdot \delta \cdot \mathbb{E}^{\mathbb{Q}^{T+\delta}}[(L_T - K)^+ \mid \mathcal{F}_t] \]

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

\[ V_t = P(t,T+\delta) \cdot \delta \cdot [L_t\Phi(d_1) - K\Phi(d_2)] \]

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

\[ V_t = \mathbb{E}^{\mathbb{Q}}\left[e^{-\int_t^T r_s\,ds}(S_T - K) \mid \mathcal{F}_t\right] \]

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

Under Forward Measure

\[ V_t = P(t,T)\mathbb{E}^{\mathbb{Q}^T}[S_T - K \mid \mathcal{F}_t] = P(t,T)(F(t,T) - K) \]

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

Bond Price Dynamics

In Vasicek:

\[ dr_t = \kappa(\bar{r} - r_t)\,dt + \sigma_r\,dW_t^{\mathbb{Q}} \]

The bond price is:

\[ P(t,T) = A(t,T)e^{-B(t,T)r_t} \]

with \(B(t,T) = \frac{1-e^{-\kappa(T-t)}}{\kappa}\).

Bond Volatility

\[ \sigma_P(t,T) = -B(t,T)\sigma_r \]

Interest Rate Under Q^T

\[ dr_t = [\kappa(\bar{r} - r_t) - \sigma_r^2 B(t,T)]\,dt + \sigma_r\,dW_t^{\mathbb{Q}^T} \]

The drift is modified by the "volatility of the drift" adjustment.

Multiple Forward Measures

For different maturities \(T_1 < T_2\):

\[ \frac{d\mathbb{Q}^{T_2}}{d\mathbb{Q}^{T_1}}\bigg|_{\mathcal{F}_t} = \frac{P(t,T_2)/P(0,T_2)}{P(t,T_1)/P(0,T_1)} \]

Each maturity has its own forward measure.

The Tower of Measures

\[ \mathbb{Q} \longleftrightarrow \mathbb{Q}^{T_1} \longleftrightarrow \mathbb{Q}^{T_2} \longleftrightarrow \cdots \]

All are equivalent measures connected by Radon-Nikodym derivatives.

When to Use the Forward Measure

Problem Use Forward Measure When
European options Payoff at single date \(T\)
Caps/Floors Separate caplet for each period
Bond options Option on \(P(T,S)\) at time \(T\)
Forward starting options Payoff depends on forward price

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

Summary

\[ \boxed{ V_t = P(t,T) \cdot \mathbb{E}^{\mathbb{Q}^T}[\Phi_T \mid \mathcal{F}_t] } \]
Property Statement
Numéraire Zero-coupon bond \(P(t,T)\)
Martingale Forward price \(F(t,T) = S_t/P(t,T)\)
Advantage Eliminates stochastic discounting
Use case Interest rate derivatives

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 \(\mathbb{E}^{\mathbb{Q}}[d\mathbb{Q}^T / d\mathbb{Q}|_{\mathcal{F}_T}] = 1\).

Solution to Exercise 1

By definition:

\[ \frac{d\mathbb{Q}^T}{d\mathbb{Q}}\bigg|_{\mathcal{F}_t} = \frac{P(t,T)}{P(0,T)B_t} \]

At \(t = T\): \(P(T,T) = 1\) (the bond pays 1 at maturity) and \(B_T = e^{\int_0^T r_s\,ds}\). Substituting:

\[ \frac{d\mathbb{Q}^T}{d\mathbb{Q}}\bigg|_{\mathcal{F}_T} = \frac{1}{P(0,T) \cdot e^{\int_0^T r_s\,ds}} = \frac{e^{-\int_0^T r_s\,ds}}{P(0,T)} \]

To verify \(\mathbb{E}^{\mathbb{Q}}[d\mathbb{Q}^T/d\mathbb{Q}|_{\mathcal{F}_T}] = 1\): Under \(\mathbb{Q}\), the discounted bond price \(P(t,T)/B_t\) is a martingale. Therefore:

\[ \mathbb{E}^{\mathbb{Q}}\!\left[\frac{P(T,T)}{B_T}\right] = \frac{P(0,T)}{B_0} = P(0,T) \]

since \(B_0 = 1\). Hence:

\[ \mathbb{E}^{\mathbb{Q}}\!\left[\frac{e^{-\int_0^T r_s\,ds}}{P(0,T)}\right] = \frac{1}{P(0,T)}\mathbb{E}^{\mathbb{Q}}\!\left[\frac{1}{B_T}\right] = \frac{P(0,T)}{P(0,T)} = 1 \]

This confirms the Radon-Nikodym derivative is properly normalized.

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.

Solution to Exercise 2

The forward contract pays \(\Phi_T = S_T - K\) at time \(T\). Under the \(T\)-forward measure:

\[ V_t = P(t,T) \cdot \mathbb{E}^{\mathbb{Q}^T}[S_T - K \mid \mathcal{F}_t] = P(t,T)\!\left(\mathbb{E}^{\mathbb{Q}^T}[S_T \mid \mathcal{F}_t] - K\right) \]

Since the forward price \(F(t,T) = S_t/P(t,T)\) is a \(\mathbb{Q}^T\)-martingale:

\[ \mathbb{E}^{\mathbb{Q}^T}[S_T \mid \mathcal{F}_t] = \mathbb{E}^{\mathbb{Q}^T}[F(T,T) \mid \mathcal{F}_t] = F(t,T) = \frac{S_t}{P(t,T)} \]

(using \(F(T,T) = S_T/P(T,T) = S_T\)). Therefore:

\[ V_t = P(t,T)\!\left(\frac{S_t}{P(t,T)} - K\right) = S_t - KP(t,T) = P(t,T)(F(t,T) - K) \]

The forward price \(K^*\) that makes the contract initially worth zero satisfies \(V_0 = 0\):

\[ P(0,T)(F(0,T) - K^*) = 0 \implies K^* = F(0,T) = \frac{S_0}{P(0,T)} \]

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 \(\mathbb{Q}^T\).

Solution to Exercise 3

With \(\kappa = 0.3\), \(\sigma_r = 0.02\), and \(T - t = 5\):

\[ B(t,T) = \frac{1 - e^{-\kappa(T-t)}}{\kappa} = \frac{1 - e^{-0.3 \cdot 5}}{0.3} = \frac{1 - e^{-1.5}}{0.3} = \frac{1 - 0.22313}{0.3} = \frac{0.77687}{0.3} = 2.58957 \]

The bond volatility is:

\[ \sigma_P(t,T) = -B(t,T)\sigma_r = -2.58957 \cdot 0.02 = -0.05179 \]

Under \(\mathbb{Q}^T\), the Brownian motion is \(W_t^{\mathbb{Q}^T} = W_t^{\mathbb{Q}} - \int_0^t \sigma_P(s,T)\,ds\), and the short rate dynamics become:

\[ dr_t = [\kappa(\bar{r} - r_t) - \sigma_r^2 B(t,T)]\,dt + \sigma_r\,dW_t^{\mathbb{Q}^T} \]

The drift adjustment is \(-\sigma_r^2 B(t,T) = -(0.02)^2 \cdot 2.58957 = -0.001036\) (evaluated at \(T - t = 5\)). This term shifts the short rate drift downward, reflecting the convexity adjustment arising from the correlation between the bond price and the short rate.

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 \(P(0, T+\delta) = 0.92\).

Solution to Exercise 4

Given: \(K = 0.05\), \(\delta = 0.25\), \(L_0 = L(0;T,T+\delta) = 0.048\), \(\sigma_L = 0.20\), \(P(0,T+\delta) = 0.92\), and maturity \(T\) (we need \(T\) to compute \(d_1, d_2\); we take \(T = 1\) as a typical assumption).

Under Black's formula for the caplet:

\[ V_0 = P(0,T+\delta) \cdot \delta \cdot [L_0\Phi(d_1) - K\Phi(d_2)] \]

where

\[ d_1 = \frac{\ln(L_0/K) + \frac{1}{2}\sigma_L^2 T}{\sigma_L\sqrt{T}} = \frac{\ln(0.048/0.05) + \frac{1}{2}(0.04)(1)}{0.20 \cdot 1} \]
\[ = \frac{\ln(0.96) + 0.02}{0.20} = \frac{-0.04082 + 0.02}{0.20} = \frac{-0.02082}{0.20} = -0.1041 \]
\[ d_2 = d_1 - \sigma_L\sqrt{T} = -0.1041 - 0.20 = -0.3041 \]

From normal distribution tables: \(\Phi(-0.1041) \approx 0.4585\) and \(\Phi(-0.3041) \approx 0.3806\).

\[ V_0 = 0.92 \cdot 0.25 \cdot [0.048 \cdot 0.4585 - 0.05 \cdot 0.3806] \]
\[ = 0.23 \cdot [0.02201 - 0.01903] = 0.23 \cdot 0.00298 \approx 0.000686 \]

The caplet price is approximately \(0.0686\%\) of notional, or about \(6.86\) basis points.

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 \(\mathbb{Q}\)?

Solution to Exercise 5

Under \(\mathbb{Q}\), the discounted price \(S_t/B_t\) is a martingale, where \(B_t = e^{\int_0^t r_s\,ds}\). The forward price is \(F(t,T) = S_t/P(t,T)\). Writing \(F(t,T) = (S_t/B_t) \cdot (B_t/P(t,T))\), note that \(S_t/B_t\) is a \(\mathbb{Q}\)-martingale but \(B_t/P(t,T)\) is a stochastic process (not constant), so their product \(F(t,T)\) is generally not a \(\mathbb{Q}\)-martingale.

To find the drift of \(F\) under \(\mathbb{Q}\), apply Itô's formula to \(F = S/P\). Under \(\mathbb{Q}\):

\[ \frac{dS_t}{S_t} = r_t\,dt + \sigma_S\,dW_t^{\mathbb{Q}}, \qquad \frac{dP(t,T)}{P(t,T)} = r_t\,dt + \sigma_P(t,T)\,dW_t^{\mathbb{Q}} \]

By the quotient rule (Itô):

\[ \frac{dF}{F} = \frac{dS}{S} - \frac{dP}{P} + \left(\frac{dP}{P}\right)^2 - \frac{dS}{S}\frac{dP}{P} \]
\[ = (r_t - r_t + \sigma_P^2 - \sigma_S\sigma_P)\,dt + (\sigma_S - \sigma_P)\,dW_t^{\mathbb{Q}} \]
\[ = \sigma_P(\sigma_P - \sigma_S)\,dt + (\sigma_S - \sigma_P)\,dW_t^{\mathbb{Q}} \]

The drift \(\sigma_P(\sigma_P - \sigma_S)\) is generally nonzero, confirming \(F\) is not a \(\mathbb{Q}\)-martingale. Under \(\mathbb{Q}^T\), this drift vanishes by construction (the Girsanov shift absorbs it), making \(F\) a \(\mathbb{Q}^T\)-martingale.

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?

Solution to Exercise 6

The Radon-Nikodym derivative between the two forward measures is:

\[ \frac{d\mathbb{Q}^{T_2}}{d\mathbb{Q}^{T_1}}\bigg|_{\mathcal{F}_t} = \frac{P(t,T_2)/P(0,T_2)}{P(t,T_1)/P(0,T_1)} \]

The measures differ because they use different bonds as numéraires, leading to different probability tilts. Under \(\mathbb{Q}^{T_1}\), the forward price \(S_t/P(t,T_1)\) is a martingale, while under \(\mathbb{Q}^{T_2}\), \(S_t/P(t,T_2)\) is a martingale. The Girsanov kernel connecting them involves the volatility difference \(\sigma_P(t,T_2) - \sigma_P(t,T_1)\).

The choice between these measures matters in applications such as:

  • Interest rate caps: Each caplet with reset at \(T_i\) is priced under \(\mathbb{Q}^{T_{i+1}}\), so different caplets in the same cap use different forward measures.
  • LIBOR Market Models (BGM): Forward LIBOR rates for different tenors are martingales under different forward measures, requiring careful measure changes when computing joint distributions.
  • Convexity adjustments: When a rate observed under one measure must be priced under another (e.g., CMS rates), the measure change introduces a convexity correction.

Exercise 7. Consider the exchange option with payoff \((S_T^1 - S_T^2)^+\). Using \(S_t^2\) as numeraire, derive Margrabe's formula. Explain why the interest rate \(r\) does not appear in the final formula, and identify the relevant volatility parameter \(\sigma\) in terms of \(\sigma_1\), \(\sigma_2\), and \(\rho\).

Solution to Exercise 7

The exchange option has payoff \((S_T^1 - S_T^2)^+\). Using \(N_t = S_t^2\) as numéraire, define the measure \(\mathbb{Q}^{S^2}\) under which \(S_t^1/S_t^2\) is a martingale.

The price is:

\[ V_t = S_t^2 \cdot \mathbb{E}^{\mathbb{Q}^{S^2}}\!\left[\left(\frac{S_T^1}{S_T^2} - 1\right)^+\;\middle|\;\mathcal{F}_t\right] \]

Let \(R_t = S_t^1/S_t^2\). Under \(\mathbb{Q}^{S^2}\), \(R_t\) is a martingale. If both assets follow geometric Brownian motions under \(\mathbb{Q}\) with volatilities \(\sigma_1, \sigma_2\) and correlation \(\rho\), then \(R_t\) is log-normal under \(\mathbb{Q}^{S^2}\) with volatility

\[ \sigma = \sqrt{\sigma_1^2 + \sigma_2^2 - 2\rho\sigma_1\sigma_2} \]

Since \(R_t\) is a martingale, its log-normal dynamics have zero drift (in the risk-neutral sense for this numéraire), and the problem reduces to a Black-Scholes call with underlying \(R_t\), strike 1, and volatility \(\sigma\). Multiplying by \(S_t^2\):

\[ V_t = S_t^1\Phi(d_1) - S_t^2\Phi(d_2) \]

where

\[ d_1 = \frac{\ln(S_t^1/S_t^2) + \frac{1}{2}\sigma^2(T-t)}{\sigma\sqrt{T-t}}, \qquad d_2 = d_1 - \sigma\sqrt{T-t} \]

The interest rate \(r\) does not appear because both \(S^1\) and \(S^2\) grow at rate \(r\) under \(\mathbb{Q}\), and the ratio \(S_t^1/S_t^2\) cancels this common growth. Equivalently, the option is self-financing when denominated in either asset, so no external financing (at rate \(r\)) is needed. The only relevant parameter is the volatility \(\sigma\) of the ratio process, which depends on \(\sigma_1\), \(\sigma_2\), and \(\rho\).