1. Assume that the strike price is $100. The time to maturity Tis 2 months. Risk-free rate is 0.1, u= 1.21, d = 0.82. Price the European and American binary calls if the spot price differs from the strike by 0.01 using the 2-step binomial tree. Use monthly compounding. (9 pts)
Answers
Step-by-step explanation:
1. A stock price is currently $ 100. Over the next two six-month periods it is expected
to go up by 10% or go down by 10%. The risk-free interest rate is 8% per annum
with continuous compounding.
(i) What is the value of a one-year European call option with a strike price of $
100.
(ii) What is the value of a one-year European put option with a strike price of $
100.
(iii) Verify that the European call and the European put satisfy put-call parity.
Solution:
Parameters are u = 0.1, d = −0.1, 1 + r = e0.5×0.08. So the risk-neutral probability is
p∗ = 0.7. After evaluation of the options at the terminal nodes we use the risk-neutral
valuation to get (i)
πC(0) = e−2(0.5×0.08) £0.72 × 21 + 2 × 0.7(1 − 0.7) × 0 + (1 − 0.7)2 × 0¤
= 9.61
and (ii)
πP (0) = e−2(0.5×0.08) £0.72 × 0 + 2 × 0.7(1 − 0.7) × 1 + (1 − 0.7)2 × 19¤
= 1.92
(iii) For put-call parity one has to verify S − πC + πP = Ke−r
, here :
100 − 9.61 + 1.92 = 100e−0.08
.