Fixed Income Derivatives: Caplets and Floorlets

In this lesson, we learned about caplets and floorlets.  A caplet is similar to a European call option on the short rate interest rate, r(t).  It is usually settled after maturity but can also be settled in advance.  The maturity it thought of in time (tau).  The strike price is c.    The payoff of a caplet with maturity tau and with strike c and settled at time tau is:

The caplet can be thought of as a call option on the short rate prevailing at time (tau) -1, but is settled at time tau.

A floorlet is the caplet except the payoff is:

A cap consists of a sequence of caplets with the same strike.
A floor consists of a sequence of floorlets with the same strike.


Following is the short-rate lattice we have been using for the past few lessons:

u = 1.25 and d = 0.90.  We will now price a caplet using this model.

We assume an expiration of t=6 and a strike of 2%.
Remember that the payment is actually made in time tau = 6, but the payment is made on the rate of (tau) - 1 = 5.  Therefore we will build our lattice based on t=5 instead of t=6.  Also, since we are building our lattice for t=5 instead of t=6, we must discount the payments accordingly.

For example:
On the node N(5,0), the rate is 3.54%.  The caplet is worth: (r(5) - c)/(1+r(5)) = (0.0354-0.02)/(1+0.0354) = 0.15.  Below is the full lattice:

We can then move backwards in the lattice as we usually do using risk-neutral probabilities equation:
S(t) = 1/(1+r(t)) * E[S(t+1)].

The value of the caplet at time t=0 is 0.042.  This is the value of the security that pays off max[0,r(5) - 2%] at time t=6.  This is in the notional value of $1, whereas in practice you would actually buy or sell many thousand of these securities.