# option on bond future - any caplet representation out there ?

I'm trying to play with bond-future options. Bond future is a future contract on a basket of bonds. The short-side will deliver the so-called bond cheapest-to-deliver (CTD).

A bond-future option is therefore an option on this basket. Let's simplify things such that:

• the option is directly struck on the CTD;
• CTD is a zero-coupon bond;
• the option is European, $t < T_{opt} \leq T_{for} < T_{ctd}$ thus paying at option's expiration $T_{opt}$:

$$\left( P(T_{opt},T_{for},T_{ctd}) - K \right)^+$$

where: $T_{for}$ is the underlying forward maturity, $T_{ctd}$ the CTD bond maturity and $P(T_{opt},T_{for},T_{ctd})$ is the $T_{opt}$-value of the bond forward maturing in $T_{for}$.

If $T_{opt} \equiv T_{mat} = T$ then the bond-future option reduces to a standard option on the CTD bond, paying at $T$:

$$\left( P(T,T_{ctd}) - K \right)^+$$

where $P(T,T_{ctd})$ is the price at the future date $T$ of the CTD bond and I have applied the identity $P(T_{opt} = T,T_{for} = T,T_{ctd}) \equiv P(T,T_{ctd})$.

Now, it's known the caplet (or floorlet) representation for options on (zero-coupon) bond (see, for example, equation 2.26 of Brigo-Mercurio "Interest Rate Models - Theory and Practice: With Smile, Inflation and Credit").

My question is: does it exist any such representation for bond-future options in terms of options on the forward rates?

$$P(t,T_{for},T_{ctd}) = \frac{1}{1 + \tau(T_{for},T_{ctd}) F(t,T_{for},T_{ctd})}$$ where $F(t,T_{for},T_{ctd})$ denotes the time-$t$ value of the forward rate for accrual period $[T_{for};T_{ctd}]$.
There should be the analagous result for options on a forward zero coupon bond purchase, as follows. The payoffof a K-call, paid at $T_{for}$ (not $T_{opt}$) is $$[P(T_{opt},T_{for},T_{ctd})−K]^+$$ We also have that $$P(T_{opt},T_{for},T_{ctd})= 1/(1+\tau F(T_{opt},T_{for},T_{ctd}))$$ where $\tau$ is the accrual factor between $T_{for}$ and $T_{ctd}$, and $F$ is the forward interest rate observed at $T_{opt}$. Combining these expressions, we get that the payoff at $T_{for}$ is equal to $$[1-K(1+\tau F)]^+/(1+\tau F)$$By reinvesting this amount to $T_{ctd}$ at the market forward rate $F$, we see that the payoff at $T_{ctd}$ is simply $$[1-K(1+\tau F)]^+$$ which is equal to $$K\tau[(1-K)/K\tau-F]^+$$. This is the payoff of a floor on the forward rate, struck at $(1-K)/K\tau$. Likewise, a put on the forward ZCB is equivalent to a cap on the forward rate.
• Thanks @dm63. But I don't understand: the payment at $T_{for}$ (instead of $T_{opt}$) is an assumption you are making or it was my mistake saying that the option pays at $T_{opt}$? – Gabriele Pompa Feb 23 '18 at 8:27
• An option on a forward or future usually conveys the right to buy the underlying on $T_{for}$. For example, if the forward bond is trading at 99, and the strike is 98, then you have 1 point of intrinsic. This 1 point is paid on $T_{for}$, not on option expiration date. you could define an option that paid this on $T_{opt}$, but it wouldn't be natural. – dm63 Feb 23 '18 at 11:38