Question

VI.

n = 10
r0,0 = 5%
u = 1.1
d = 0.9
q = 1 - q = ½

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 = 6t=6 to t = 11t=11 inclusive. The swaption strike of 0 should also not be confused with the fixed rate of 4.5% on the underlying swap.

Answer 1

Starting swap that begins at t =1

maturity t =10

fixed rate = 4.5%

the first payment then takes place at t=2

final payment takes place at t=11

for each node (10,j) j=0,2,3,4,5,6,7,8,9,10 the forward price of the swap is a discounted

expected value;

(1+r10,j)-1 (1/2 S11,j + 1/2 S11,j+1)

where, for a receive fixed/ pa float swap,

S10,j = 1000000(0.045 - f10,11,j)

= 1000000(0.045-r10,j)

note that the forward rate f10,11,j equals r10,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) with j=0,1,....,9

=(1 + r9,j)-1 (1/2 S10,j + 1/2 S10,j+1 + Q10,j)

where the net payment received at time i= 10 is

Q10,j= 1000000(0.045 - f9,10,j)

= 1000000(0.045 - r9,j)

work your way back on the tree until you find the current swap price S0,0. Since this is a forward starting swap beginnig at time 1

to price the swaption, set the terminal values at 5

   C5,j = max(S5,j,0)