Fixed Income Derivatives: Swaps and Swaptions

In this lesson, we learned about pricing swaps and swaptions.  We learned about swaps a few weeks ago.  A swaption is simply an option on a swap contract.

We want to price a swap based on our short-rate, r, lattice.  Once again, here is our lattice:

We want to price an interest rate swap with a fixed strike rate of 5% that expires at time t=6.  The first payment will be made at time t=1 and the final payment will be made at time t=6.  The payment will be (r(i,j) - K) if you are long, and will be -(r(i,j) - K) if you are short.  It will be made in arrears, so this payment will be made at time t=i+1.



To price the swap, we will want to, once again, start from the ending (time t=6), then work backwards.  However, since the payment at time t=6 is based on the short-rate at time t=5, we will simply start at the values for t=5 (discount by 1 period) and work backwards.  The formula for pricing the swap prices at times t=5 is as follows:

(r(5,j) - K) at time t=6 is worth (r(5,j)-K)/(1+r(5,j)) at time t=5 where 1/(1+r(5,j)) is the discounting factor for the difference between periods 5 and 6.

The next step in pricing the swap will be working backwards in the lattice.  In this regard it will differ from what we have done previously since a swap will contain intermediate coupon payments.  So we must use the formula we previously had for risk-neutral pricing with intermediate coupon payments.  This formula is:
S(t) = E[(S(t+1) + C(t+1))/(1+r(t))]
where C(t+1) is simply (r(t,j) - K) at the node in which it is being calculated. An example is provided below the lattice:

And here is the example for node N(2,2):

We then moved on to pricing swaptions.  A swaption is simply an option on a swap.  We priced a swaption on the swap we just developed.  We are going to assume that the option strike is 0% (this is not to be confused with the strike of 5%, or fixed rate, on the underlying swap) and the swaption expiration is at t=3.
Therefore at time t=3, the owner of the swaption has the right to exercise and have ownership of the underlying swap for a strike value of 0.  So the payoff of the swaption is: max(0,S(3)) where S(3) is the underlying swap price.

In order to price this swaption, we will take max(0,S(3)) for time t=3 of the underlying swap, then simply work backwards in our lattice using risk-neutral pricing.  However, we will not be factoring in the intermediate cash flows for this lattice since the holder of the swaption will not get the cash flows until the exercise at time t=3.  Below is the swaption lattice with the values of the swaption replacing the values of the swap in blue: