Question

: A stock price is currently $60. It is known that at the end of eight months it will be either $65 or $55. The risk-free interest rate is 8% per annum with continuous compounding. (1) What is the value of an eight-month European put option with a strike price of $60? (2) Verify that no-arbitrage arguments and risk-neutral valuation arguments give the same answers.

Answer 1

At the end of 8 months, the value of the put option will be either $5 (if the stock price is $$55) or $0 (if the stock price is $65).

Let us consider a portfolio :
- Δ : shares
+1 : option

The Δ of a put option is negative

The value of the portfolio is either -55Δ + 5 or -65Δ

-55Δ + 5 = -65Δ

=> Δ = -0.5

The value of the portfolio is certain to be -55*(-0.5) + 5 = 32.5. For this value of Δ, the portfolio is hence riskless

The current value of the portfolio is -60Δ + f

where f is the value of the option. Since the portfolio must earn the risk-free rate of interest,

=> (60*0.5 + f)e^0.08*8/12 = 32.5

=> f = 0.81

Hence, the value of the put option is $0.81

Calculating using risk neutral valuation. Let p be the probability of upward stock price movement.

Hence, 65p + 55(1-p) = 60*e^0.08*8/12

=> 10p + 55 = 63.29

=> p = 0.829

The expected value of the option in risk neutral world is

0*0.829 + 5(1-0.829) = 0.855

This has a present value of 0.855 / e^0.08*8/12 = 0.81

This is consistent with the no arbitrage option