Question

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.

Answer 1

Step-by-step explanation:

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.