Compute the initial price of a swaption that matures at time t=5 and has a strik

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.)

Underlying Swap: initial value of a forward-starting swap that begins at t=1, with maturity t=10 and a fixed rate of 4.5%. (The first payment then takes place at t=2 and the final payment takes place at t=11 as we are assuming, as usual, that payments take place in arrears.) You should assume a swap notional of 1 million and assume that you receive floating and pay fixed.) is equal to 33,374

Explanation / Answer

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(0.045f10,11,j)=1,000,000(0.045r10,j).S10,j=1,000,000(0.045f10,11,j)=1,000,000(0.045r10,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(0.045f9,10,j)=1,000,000(0.045r9,j).Q10,j=1,000,000(0.045f9,10,j)=1,000,000(0.045r9,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.

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