# Arbitrage strategy for a given put option price

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.

My attempt:

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.

• Found the answer if anyone is wondering At time t = 0, we short sell half unit of stock to gain \$2. Then, we sell a put option for \$1.21. Our cash position at this time is \$3.21. We keep \$0.01 aside for now. – Het Thakkar Jan 29 '19 at 15:59
• We invest this \$3.2 in bank to get \$4 at time t = 1(simple interest assumed). Now if the stock price increases to \$8, the holder will not exercise the option. We will buy half a stock at \$4 to replenish the short. We have \$0.01 still with us. If the price decreases to \$2, the holder will exercise the option and we will have to buy the stock for \$5. We will then sell it in the market for \$2. Leaving us with \$4(from bank)-\$3 = \$1. We will use this \$1 to buy half a stock and replenish the short leaving us with again \$0.01. We have thus proved an arbitrage opportunity – Het Thakkar Jan 29 '19 at 16:04
• Hypotheticals like this may work in the classroom but in the real world of options, your break even points are \$3.58 and \$6.42. Outside of that range, you're losing money, waiting for who knows how long for enough interest to accrue to offset the losses. Everything is predicated on being able to hold the position long enough for the \$3.21 to grow to \$4. That's not arbitrage (see Conversions, Reversals, Box Spreads, etc.). To add insult to injury, you haven't accounted for the carry cost of the short stock position or commissions. – Bob Baerker Jan 29 '19 at 16:57