Question

Problem#7

Using the binominal model value three-year European put option with the periodically computed one-year interest rate as the underlying. Assume the notional amount of an option is $100,000, the strike rate is 2.5% of par, and the risk neutral (RN) probability of an up jump is 55%.

PROBLEMN.7 Consider the following three period interest rate lattice by year: 1 3 Maturity Rate 12,7 Maturity Rate 2,50% Maturity Rate 12,6 Maturity Maturity Rate 2,30% Rate 2,35 1 Maturity 1 2 Rate ,15% Maturity 1 Rate 2,15 Maturity 1 | Rate 2 Maturity Rate 11,7 Using the binominal model value three-year European put option with the periodically computed one-year interest rate as the underlying. Assume the notional amount of an option is $100,000, the strike rate is 2.5% of par, and the risk neutral (RN) probability of an up jump is 55%.

Answer 1

European options are only eligible to be exercised at maturity. given is the coupon rate, interest rate in the rate box.

given the up probability, 55%, down probability 45%

notations used are p for put option price, p+ value of put option when interest rate value goes up, p- value of put option when interest rate value goes down and p++ value of put option when interest rate value goes up after it went up once, and so on till last nodes, pv is value of present value of the interest rate.

for example, at year 2, to calculate p++, the formula used is p++ = pv*((up probability*p+++) +(down probability*p++-)) = 0.953*((55%*0.024)+(45%*0) = 0.012

pv at the same node is calculated with the formula, (2%/(1+7%)+(1/(1+7%)) = 0.953

similarly all values at nodes are calculated. finally the pv value at year 0 is calculated and multiplied with notional amount to obtain the price of put option = 206.38

year 3 Maturity rate strike price notion strike rate 3% 0.97560976 100000 2.50% 55% 45% 39 year 2 OV down Maturity +++ 0.02439 2% year 1 0.953271028 0.012787782 Maturity Maturity rate 2%. 6% 29 50% 0.68 0.00478263 pv0 .962264 year o ++- Maturity rate Maturity rate 2% 30% pv 0.755555556 0.784615 0.002064 Maturity Maturity 1 Trate 1 2% 15% 0.886956522 rate 2% 15% .886957 price of european put 206.3889 pv pv0 Maturity Maturity rate pv 0.943925