Question

Consider six months as one period. Assume that the six-month and one-year spot rates are 2% and 2.5%, respectively. Assume that six months from now, the six-month spot rate will be either 1.8% or 2.2% with equal probability. What is the price of a security that pays $100 six months from now if the six-month spot rate then is 1.8% and pays $20 otherwise? Semi-annual compounding

Answer 1

Sol:

Six month spot rate = 2%

One year spot rate = 2.5%

Equal probability of spot rate six month from now will be 1.8% or 2.2%

Let the spot rate for the period t to T be r(t,T) (Semiannual compounding) Assuming it to be continuous compounding rate.

Payoff will be = 100 if r(6,12) = 1.8% and 20 if r(6,12) = 2.2%

Now price of payoff is equal to the discounted value of its expected payoff.

Therefore Price of payoff = e^-r(0,6) x expected payoff

= e^-2% x (∑ payoff x probability)

= e^-0.02 x [100 x P (r(6,12) = 1.8%) + 20 x P (r(6,12) = 2.2%)]

= e^-0.02 x (100 x 0.5) + (20 x 0.5)

= e^-0.02 x 60 = 58.81

Therefore price of payoff = 58.81