Pricing Delayed-Payment Libor Options and Digital Payoffs

# Equality under T-forward measure for convexity adjustment


# Equality under T-forward measure for convexity adjustment












I've been working with the convexity adjustment for interest rates that arises when changing from one measure $Q_{T_p}$ with a numéraire $N_p=P(t,T_p)$ to a measure $Q_{T_e}$ with a numéraire $N_e=P(t,T_e)$ where $T_p$ is the time of payment and $T_e$ is the time where the interest of the forward rate ends.

So I have the forward rate:

\begin{align*} L(t, T_s, T_e) = \frac{1}{\Delta_s^e}\bigg(\frac{P(t, T_s)}{P(t, T_e)}-1 \bigg), \end{align*}

And the expecation of the payment with the change of measure:

\begin{align*} &\ P(t_0, T_p)E^{T_p}\big(L(T_s, T_s, T_e) \mid \mathcal{F}_{t_0}\big) \\ =&\ P(t_0, T_p)E^{T_e}\Big(\frac{\eta_{T_p}}{\eta_{t_0}}L(T_s, T_s, T_e) \mid \mathcal{F}_{t_0}\Big)\\ =&\ P(t_0, T_p)E^{T_e}\Big(\frac{P(t_0, T_e)}{P(t_0, T_p)P(T_p, T_e)} L(T_s, T_s, T_e)\mid \mathcal{F}_{t_0}\Big)\\ =&\ P(t_0, T_e)E^{T_e}\Big(\frac{1}{P(T_p, T_e)} L(T_s, T_s, T_e)\mid \mathcal{F}_{t_0}\Big)\\ =&\ P(t_0, T_e)E^{T_e}\Big(\big(1+ \Delta_p^e L(T_p, T_p, T_e) \big) L(T_s, T_s, T_e)\mid \mathcal{F}_{t_0}\Big) \end{align*}

Now I believe, this last equation is the tricky one since I have two forward rates observed in times $T_p$ and $T_s$ I'd like to confirm my thinking that since we are under the $Q_{T_e}$ measure and both $L(T_p, T_p, T_e)$ and $L(T_s, T_p, T_e)$ are martingales under it, I can use this equality:

\begin{align*} P(t_0, T_e)E^{T_e}\Big(\big(1+ \Delta_p^e L(T_p, T_p, T_e) \big) L(T_s, T_s, T_e)\mid \mathcal{F}_{t_0}\Big)= P(t_0, T_e)E^{T_e}\Big(\big(1+ \Delta_p^e L(T_s, T_p, T_e) \big) L(T_s, T_s, T_e)\mid \mathcal{F}_{t_0}\Big)\tag{1} \end{align*}

Does this equality holds because that reason? Now from there I've seen two possible solutions:

- Assume a model (e.g. log-normal) for both $L(t, T_p, T_e)$ and $L(t, T_p, T_e)$ since now we'll be working with both values observed in $T_s$ when I use Ito's lemma to compute the product and then integrate I'll do it from $t_0$ to $T_s$ and with that solution can solve the expectation.

- Use the linear model proposed here in page 19,Section 4.2. where essentially the calculations are made with respect to $(1+ \Delta_s^e L(T_s, T_s, T_e)$ but with a specific value to the $\Delta_s^e$ being multiplied in order to account for the payment being in $T_p$ instead of in $T_e$.

I'm working on this to price an option, specifically a digital option (which I think should be almost the same). Have anyone here used any of this results of would have a preference over one of these options?

Much help appreciated

## Answer by Antoine Conze (score 1, accepted)

https://quant.stackexchange.com/a/37483

The usual approach is akin to your 2. and is known as replication pricing for unnatural payment date Libor, also sometimes called delayed libor. You can find it in any book on interest rate derivatives pricing, for instance Andersen & Piterbarg Interest Rate Modeling.

Also the standard approach for digital options is to price it by differentiating the regular call or put with respect to strike, so as to be consistent with the volatility smile.

So in your particular case, to price a digital option that pays $\text{Indicator}(L(T_s, T_s, T_e) > K)$ on $T_p$, you first apply replication to price options with payoff $(L(T_s, T_s, T_e) - K)^+$ that pay on $T_p$, and you then differentiate (numerically) with respect to $K$.