Question

:A stock price is currently $50. It is known that at the end of six months it will be either $52 or $48. The risk-free interest rate is 5% per annum with continuous compounding.

1. What is the value of a six-month European call option with a strike price of $50?
2. Verify that no-arbitrage arguments and risk-neutral valuation arguments give the same answers.

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Answer 1

At the end of 6 months, the value of the call option will be either $2 (if the stock price is $52) or $0 (if the stock price is $48).

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

The value of the portfolio is either 52Δ - 2 or 48Δ

48Δ = 52Δ - 2

=> Δ = 0.5

The value of the portfolio is certain to be 52*(0.5) - 2 = 24. For this value of Δ, the portfolio is hence riskless

The current value of the portfolio is 50Δ - f

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

=> (50*0.5 - f)e^0.05*6/12 = 24

=> f = 1.59

Hence, the value of the call option is $1.59

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

Hence, 52p + 48(1-p) = 50*e^0.05*6/12

=> 4p + 48 = 51.27

=> p = 0.818

The expected value of the option in risk neutral world is

2*0.818 + 0(1-0.818) = 1.636

This has a present value of 1.636 / e^0.05*6/12 = 1.59

This is consistent with the no arbitrage option