• Pennsylvania State University

• MATH 448

A stock price is currently $40. It is known that at the end of one month it will be either $42 or $38. The risk-free interest rate is 8% per annum with continuous compounding. What the value ofa one-month European call option with a strike price of $39? Solution: Consider a portfolio consisting of -1: Call option +A: Shares If the stock price rises to $42, the portfolio is worth 42-3. If the stock price falls to $38, it is worth 38Д. These are the same when 43-3 = 38Δ or ▲ = 0.75. The value of the portfolio in one month is 28.5 for both stock prices. Its value today must be the present value of 28.5, or 28.5e-0.08-0.083333 = 28.31. This means that -f+40A = 28.31 where f is the call price. Because ▲ = 0.75, the call price is 40. 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: so that 42p +38(1-p) = 40€0.08-0.08333 4p =40e0.08-0.08333- 38 or p = 0.5669. The value of the option is then its expected payoff discounted at the risk-free rate