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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.

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  • $\begingroup$ Hello, can you please give the text you are referring to? Seems we don't have the full reasoning. By NA, in a complete market, the price is invariant with respect to the measure you use. $\endgroup$ – siou0107 Jan 22 at 11:10
  • $\begingroup$ Hopefully this will suffice to illustate the point: (en.wikipedia.org/wiki/LIBOR_market_model): the LMM models a family of Forward Libors. As stated in the Wiki example, these follow the processes I described in my original question above, specifically: when the T-Forward measure corresponds with the Libor maturity, the drift is zero (case where Wiki says $j=p$). If the T-Forward measure is longer than the Libor maturity, that Libor then has a negative drift ($j<p$) and finally if the T-forward measure is shorter than the Libor maturity, the drift is positive ($j>p$). $\endgroup$ – Jan Stuller Jan 22 at 13:27
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    $\begingroup$ You use the LMM model to value multi cash-flow deals. When you price such a deal, you must use a single measure, namely you must choose a tenor $T_i\in\{T_i\}_i$ to define your measure. You cannot simply value each cash-flow separately under its own natural measure. $\endgroup$ – Daneel Olivaw Jan 22 at 22:44
  • $\begingroup$ @ Daneel Olivaw: thank you so much! So just to double check: if I wanna value a single caplet, I am free to use the T-forward measure that coincides with the maturity of the Libor that underlines the Caplet => I will get Black 76 formula for the Caplet price. If I start looking at multiple Libors under one measure, from your comment, I understand that the Libors that do NOT have the same maturity as the T-forward measure selected for valuation will NOT have the Black 76 formula as a solution to a Caplet valuation? Isn't this a paradox? $\endgroup$ – Jan Stuller Jan 23 at 9:57
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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 $P_{i-1}$.

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  • $\begingroup$ Gordon, thank you. Your formula $\frac{\sigma^2 \Delta_i L(t, T_{i-1}, T_i)}{1+\Delta_i L(t, T_{i-1}, T_i)}$ is my $r$. So under the $P_{i-1}$ measure, as you point out, the forward Libor has a drift term. The solution to that SDE is a Black-Scholes (with that drift term), not the Black 76 formula. That is what I was getting at basically but perhaps didn't explain it so well. My question is essentially: shouldn't the Forward Libor have the Black 76 formula solution under any measure? $\endgroup$ – Jan Stuller Jan 22 at 19:34
  • $\begingroup$ Given that the $r$ here is stochastic, you are not able to use the Black-Scholes, where the $r$ is assumed to be a constant. For such dynamics, you have to use Monte Carlo, or approximate the drift by a constant such as to replace $L(t, T_{i-1}, T_i)$ by $L(0, T_{i-1}, T_i)$, or to convert back to the $T_i$-forward measure. $\endgroup$ – Gordon Jan 22 at 20:12
  • $\begingroup$ I see, thank you. So in conclusion: when using the LMM, and choosing a Numeraire which doesn't coincide with the Libor maturity: the solution will NOT be the Black 76 formula. Correct? $\endgroup$ – Jan Stuller Jan 22 at 20:54
  • $\begingroup$ That is correct in a certain sense, that is, under a measure that is different from its natural forward measure, the Black 76 formula can not be used. However, under whatever measure, the final valuation results should be the same or have a close match. $\endgroup$ – Gordon Jan 22 at 21:01
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    $\begingroup$ Really sorry to keep asking, but I still find that confusing. Under LMM, we need to use one Numeraire for all Libors. One particular T-forward measure will only be "natural" for one one particular Libor within our range of forward Libors. That means that all forward Libors, except for one, will have a drift and will NOT have Black 76 formula as their solution. Since the choice of measure is arbitrary, I see a huge contradiction here, because each forward Libor can have the Black 76 solution under one specific measure, but will have a different solution under all other T-forward measures. $\endgroup$ – Jan Stuller Jan 22 at 21:29

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