1
$\begingroup$

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?

Thanks in advance. gab


Addendum: if it helps the relation between bond and forward rate is (should be ;) ):

$$ P(t,T_{for},T_{ctd}) = \frac{1}{1 + \tau(T_{for},T_{ctd}) F(t,T_{for},T_{ctd})} $$ where $F(t,T_{for},T_{ctd})$ denotes the time-$t$ value of the forward rate for accrual period $[T_{for};T_{ctd}]$.

$\endgroup$
1
$\begingroup$

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.

$\endgroup$
  • $\begingroup$ 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 $T_{opt}$? $\endgroup$ – Gabriele Pompa Feb 23 '18 at 8:27
  • $\begingroup$ 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. $\endgroup$ – dm63 Feb 23 '18 at 11:38

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.