Binomial Option pricing model question
In the following question, S0 = stock price at time t = 0, u = up factor, d = down factor. r = risk free interest rate. Stock price at time t = 1 is either S0*u or S0*d.
Consider the one period binomial model with S0 = 4, u = 2, d = 1/2, r = 1/4. Consider a put option with strike price K = $5. Show how an arbitrage can be done if the price of the put option is $1.21 by starting with zero capital and building a portfolio to make a risk free proﬁt.
We can first try to value the option through portfolio replication. For this we assume a position of having x stocks and y money at time t = 0. Now at t = 1
8x + y(1+1/4) = 0 (as put option will not be exercised)
2x + y(1+1/4) = 3 (profit K-S0d)
On solving, we get x = -0.5, and y = 3.2. So the option value is $1.2(S0*x + y)
As the option price is more than what we valued it at, we can sell the option at time t = 0 for $1.21. I can't figure out what we can do after this. The stocks we need to have at time t = 0 for replication came out negative.