Forward Measures and Consistent Caplet Pricing
Summary
The document addresses why a Libor forward rate can have zero drift under its own forward measure yet acquire drift under a different measure, while caplets and floorlets are commonly priced with Black’s formula. The response identifies an error in the proposed dynamics under the earlier forward measure. It derives the Radon–Nikodym density between the two forward measures using their zero-coupon bond numeraires, then applies the measure change to obtain the Libor dynamics under the new measure.
Under the new measure, the drift is state-dependent: it depends on the forward rate and accrual period, rather than being a constant short rate as proposed in the question. The example illustrates that a process’s drift depends on the chosen probability measure, while prices remain consistent when numeraire and measure changes are applied correctly. The discussion is a focused correction, not a full proof of pricing invariance or a general treatment of interest-rate models. Its displayed dynamics apply over the stated period and within the model assumptions used in the answer.
Key ideas
- A forward Libor has zero drift under its corresponding forward measure in the stated model.
- Changing numeraires changes the probability measure and generally changes the process drift.
- The proposed dynamics under the earlier forward measure omit the state-dependent drift created by the measure change.
- A density process derived from zero-coupon bond prices supports the measure-change calculation.
- Measure-dependent dynamics must be handled consistently when deriving derivative prices.
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# Intuition for consistent Derivative Prices under different Numeraires and Measures
# Intuition for consistent Derivative Prices under different Numeraires and Measures
This is essentially the Fundamental Theorem, however I am not asking for a thorough proof, I am more interested in the general intuition.
In words, it makes sense that whatever your unit of account (Numeraire), your derivative price should be the same. However, let's take the Libor Market Model as an example:
Under the $T_i$-forward measure (i.e. using a zero-coupon bond that matures at time $T_i$ as numeraire), the Libor $L(t,T_{i-1},T_i)$ is a martingale and has zero drift. We can write the process as: $$dL(t,T_{i-1},T_i) = \sigma L(t,T_{i−1},T_{i}) dW^{T_i}(t).$$ The solution is the Black 76 formula.
Shifting the measure to $T_{i-1}$-forward measure (i.e. using a zero-coupon bond that matures at time $T_{i-1}$, as numeraire), the Libor acquires a drift term and the process becomes: $$dL(t,T_{i-1},T_i) = rL(t,T_{i−1},T_{i})dt + \sigma L(t,T_{i−1},T_{i}) dW^{T_{i-1}}(t)$$ where $dW^{T_i}(t)$ under $T_{i-1}$ is a different process to the $dW^{T_{i-1}}(t)$ under the $T_i$ measure.
The solution to the second equation above is no longer the Black-76 formula, but a Black-Scholes formula with a drift term (the drift $r$ is a complicated term but we don't need to worry about it for the sake of this example).
Where is the logical mistake here? My understanding is that Caplets and Floorlets on forward Libors are always priced using the Black 76 formula, and so should always have no drift.
Thank you so much, J.
## Answer by Gordon (score 2, accepted)
https://quant.stackexchange.com/a/50798
Your dynamics under the $T_{i-1}$-forward measure is wrong.
Specifically, let $P_{i-1}$ and $P_i$ be, respectively, the $T_{i-1}$- and $T_i$-forward probability measures. Moreover, let $\Delta_i = T_i-T_{i-1}$. Then, for $0\le t \le T_{i-1}$, \begin{align*} \eta_t &\equiv \frac{dP_{i-1}}{dP_i}\big|_t \\ &= \frac{P_i(0, T_i)}{P_{i-1}(0, T_{i-1})}\frac{P_{i-1}(t, T_{i-1})}{P_i(t, T_i)}\\ &=\frac{1+\Delta_i L(t, T_{i-1}, T_i)}{1+\Delta_i L(0, T_{i-1}, T_i)}. \end{align*} Furthermore, \begin{align*} d\eta_t &= \frac{ \sigma\Delta_i L(t, T_{i-1}, T_i)}{1+\Delta_i L(0, T_{i-1}, T_i)}dW_t\\ &=\frac{\sigma \Delta_i L(t, T_{i-1}, T_i)}{1+\Delta_i L(t, T_{i-1}, T_i)} \eta_t dW_t. \end{align*} Consequently, under $P_{i-1}$, \begin{align*} dL(t, T_{i-1}, T_i) &= L(t, T_{i-1}, T_i)\left(\frac{\sigma^2 \Delta_i L(t, T_{i-1}, T_i)}{1+\Delta_i L(t, T_{i-1}, T_i)} dt + \sigma d\widehat{W}_t\right), \end{align*} where $\{\widehat{W}_t, 0\le t \le T_{i-1}\}$ is a standard Brownian motion under $P_{i-1}$.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
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