Hull.11.01
Category:Hull -> Chapter 11
Question:
A stock price is currently $40. It is know that at the end of 1 month it will be either $42 or $38. The risk-free interest rate is 8% per annum with continuous compounding. What is the value of a 1-month European call option with a strike price of $39?
Answer:
Consider a portfolio consisting of:
-1: call option
+Δ: shares
If the stock price rises to $42, the portfolio is worth 42Δ – 3. If the stock price falls to $38, it is worth 38Δ. These are the same when
42Δ – 3 = 38Δ
or Δ = 0.75. The value of the portfolio in one month is 28.5 for both stock prices. It’s value today must be present value of 28.5, 28.5e^-0.08x0.08333 = 28.31. This means that
-f + 40Δ = 28.31
where f is the call price. Because Δ = 0.75, call price is 40 x 0.75 – 28.31 = $1.69. As an alternative approach, we can calculate the probability, p, of an up movement in a risk-neutral world. This must satisfy:
42p + 38(l – p) + 40e^0.08x0.08333
so that
4p = 40e^0.08x0.08333 – 38
or p = 0.5669. The value of the option is then its expected payoff discounted at the risk-free rate:
[3 x 0.5669 + 0 x 0.4331]e^-0.08x0.08333 = 1.69
or $1.69. This agrees with the previous calculation.