Valuing a Libor Cap as a Bond Put Option
Summary
The document distinguishes a caplet written on a forward Libor rate from one whose payoff uses the spot Libor rate observed at the option’s reset date. The forward-rate caplet can be expressed under the relevant forward measure as an option on a martingale forward rate, which supports the standard Black-style valuation framework when its modeling assumptions apply.
For a payoff on spot Libor, the answer derives a different expression by conditioning at the reset date and using the relationship between Libor and the discount bond. This identifies the payoff’s value with a put option on a zero-coupon bond, expiring at the reset date and maturing at the later payment date. The proposed simpler forward-measure transformation in the question is challenged: it omits a discounting factor, so it is not generally the stated bond-put price. The answer says a specified short-rate model, such as Hull–White, can provide an analytical valuation. The result depends on the rate model; Black’s formula does not follow simply from treating the bond price as lognormal under the selected measure.
Key ideas
- A forward Libor caplet can be valued as an option on a forward rate under the corresponding forward measure.
- A caplet on spot Libor at reset has a payoff that can be rewritten using the price of a zero-coupon bond.
- The spot-Libor caplet is equivalent to a bond put expiring at reset and paying based on a later-maturing bond.
- The corrected valuation retains a discounting factor that the proposed transformation leaves out.
- A short-rate model such as Hull–White can be used to price the bond option analytically.
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Full text
# Cap option on Libor
# Cap option on Libor
We denote `discount factor` $D(t),$ `zero coupon bond` $B(t,T),$ $E_t[X] = E[X|\mathcal{F}(t)]$ and $T$-forward measure $E_t^{T}[\ ].$
First, let me fix the `Libor` and `Forward Libor` to avoid ambiguity
`Libor` $L(t,T):$ $$B(t, T)\cdot \Big(1 + (T-t) L(t, T)\Big) = 1.$$ `Forward Libor` $F(t,T-\delta,T):$ $$\Big(1 + (T-t)F(t,T-\delta,T)\Big)B(t,T) = B(t,T-\delta)$$
Now we see the `cap` $$C(t;T,L^*) = \dfrac{1}{D(t)}E_t\left[D(T)\delta\Big(F(t,T-\delta,T) - L^*\Big)^+\right]$$ We can change into forward measure $$C(t;T,L^*) = \delta B(t,T)E^T_t\left[\Big(F(t,T-\delta,T) - L^*\Big)^+\right]$$ and $F(t,T-\delta,T)$ is $T$-forward martingale, the above formula become the standard `Black-Scholes.`
But if we choose $$C(t;T,L^*) = \dfrac{1}{D(t)}E_t\left[D(T)\delta\Big(L(T-\delta,T) - L^*\Big)^+\right]$$ then we can transform into $$C(t;T,L^*) = (1+\delta L^*)\cdot E^{T}_{t}\left[\left(\dfrac{1}{1+\delta L^*} - B(T-\delta,T)\right)^+\right]$$ it become a bond put option expiring at time $T - \delta$ maturing at time $T.$
But $B(t,T)$ is impossible `log-normal` under $T$-forward measure, then we can't use `Black-Scholes.` So how to deal with for this case?
## Answer by Gordon (score 4, accepted)
https://quant.stackexchange.com/a/34480
Note that \begin{align*} &\ \dfrac{1}{D(t)}E_t\left(D(T)\delta\Big(L(T-\delta,T) - L^*\Big)^+\right)\\ =&\ \dfrac{1}{D(t)}E\left(D(T-\delta) E\left(\frac{D(T)}{D(T-\delta)}\delta\Big(L(T-\delta,T) - L^*\Big)^+\mid\mathcal{F}_{T-\delta}\right) \mid \mathcal{F}_t\right)\\ =&\ \dfrac{1}{D(t)}E\left(D(T-\delta) B(T-\delta, T)\delta\Big(L(T-\delta,T) - L^*\Big)^+\mid\mathcal{F}_t\right)\\ =&\ (1+\delta L^*)\dfrac{1}{D(t)}E\left(D(T-\delta)\left(\dfrac{1}{1+\delta L^*} - B(T-\delta,T) \right)^+\mid\mathcal{F}_t\right)\tag{1}\\ =&\ (1+\delta L^*)B(t, T)E^T_t\left(\frac{D(T-\delta)}{D(T)}\left(\dfrac{1}{1+\delta L^*} - B(T-\delta,T) \right)^+\right). \end{align*} Your transformation from $$C(t;T,L^*) = \dfrac{1}{D(t)}E_t\left(D(T)\delta\Big(L(T-\delta,T) - L^*\Big)^+\right)$$ to $$C(t;T,L^*) = (1+\delta L^*)\cdot E^{T}_{t}\left(\left(\dfrac{1}{1+\delta L^*} - B(T-\delta,T)\right)^+\right)$$ does not appear correct.
We also note that $(1)$ is indeed the value of a put bond option with maturity $T-\delta$. Based on a certain short rate model such as the Hull-White model, this value can be computed analytically.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.