A stock price is currently $40. It is known that at the end of three months it will be either $45 or $35. The risk-free rate of interest with quarterly compounding is 8% per annum. Calculate the value of a three-month European put option on the stock with an exercise price of $40. Verify that no-arbitrage arguments and risk-neutral valuation arguments give the same answers.

Consider a portfolio consisting of: shares1option-  + (Note: The delta, , of a put option is negative. We have constructed the portfolio so that it is + 1 option and -  shares rather than 1-option and +  shares so that the initial investment is positive.) The value of the portfolio is either 355- +or 45-. If: 35545- + = -  i. e., 0 5 = -  the value of the portfolio is certain to be 22.5. For this value of  the portfolio is therefore riskless. The current value of the portfolio is 40f- + where fis the value of the option. Since the portfolio must earn the risk-free rate of interest (400 5) 1 0222 5f  +   = Hence 2 06f= i. e., the value of the option is $2.06. This can also be calculated using risk-neutral valuation. Suppose that pis the probability of an upward stock price movement in a risk-neutral world. We must have 4535 (1) 40 1 02pp+-=  i. e., 105 8p= or: 0 58p= The expected value of the option in a risk-neutral world is: 00 5850 422 10  +  = This has a present value of 2 102 061 02 =  This is consistent with the no-arbitrage answer.