Question

Compute the initial price of a swaption that matures at time t=5 and has a strike of 0. The underlying swap is the same swap as described in the previous question with a notional of 1 million. To be clear, you should assume that if the swaption is exercised at t=5 then the owner of the swaption will receive all cash-flows from the underlying swap from times t=6 to t=11 inclusive. (The swaption strike of 0 should also not be confused with the fixed rate of 4.5% on the underlying swap.)

(should be answered by building an n =10-period binomial model for the short-rate, ri,j​. The lattice parameters are: r0,0​=5%, u = 1.1, d = 0.9 and q = 1-q = 1/2.)

Answer 1

Start at time i=10i=10 .

For each node (10,j)(10,j) with j = 0,1,…,100,1,…,10 , the forward price of the swap (ex payments received at time 10) is a discounted, expected value:

S10,j=(1+r10,j)−1(12S11,j+12S11,j+1),S10,j=(1+r10,j)−1(12S11,j+12S11,j+1),

where, for a receive fixed / pay float swap,

S10,j=1,000,000(0.045−f10,11,j)=1,000,000(0.045−r10,j).S10,j=1,000,000(0.045−f10,11,j)=1,000,000(0.045−r10,j).

Note that the forward rate f10,11,jf10,11,j equals r10,jr10,j on a tree where the time spacing between the nodes matches the period for floating-rate resets and fixed rate payments.

Now find the forward swap price at each node (9,j)(9,j) with j = 0,1,…,90,1,…,9 :

S9,j=(1+r9,j)−1(12S10,j+12S10,j+1+Q10,j),S9,j=(1+r9,j)−1(12S10,j+12S10,j+1+Q10,j),

where the net payment received at time i=10i=10 is

Q10,j=1,000,000(0.045−f9,10,j)=1,000,000(0.045−r9,j).Q10,j=1,000,000(0.045−f9,10,j)=1,000,000(0.045−r9,j).

Work your way back on the tree until you find the current swap price S0,0S0,0 . Since this is a forward starting swap beginning at time i=1i=1 , do not include any net payments Q1,jQ1,j .

To price the swaption, set the terminal values at expiry i=5i=5 and j=0,1,…,5j=0,1,…,5 to

C5,j=max(S5,j,0).C5,j=max(S5,j,0).

Then work backwards from i=5i=5 , calculating discounted expected values at each node until you arrive at the current price C0,0C0,0 .