# Put Call Parity and Arbitrage Opportunity

In this article, we will look at how we can seek arbitrage opportunities by using the put-call parity equation. As we know, the put-call parity equation is represented as follows:

c + PV(K) = p + s

If the prices of put and call options available in the market do not follow the above relationship then we have an arbitrage opportunity that can be used to make a risk-free profit. In the above equation the left side of the equation represents a fiduciary call and the right side of the equation is called a protective put. Depending on the asymmetry we can take our positions to earn a risk-free profit. We buy the underpriced side and sell the overpriced side. Let’s take an example to understand this.

Let’s say that we have we have the following information for a call and a put option on XYZ stock.

Exercise price: $100 Call option price:$7

Put option price: $5 Risk-free rate: 8% Current market price of XYZ:$98

Time to maturity: 0.5 years

Let’s plug these values in the put-call parity equation:

7 + 100/(1.08)^0.5 = 5 + 99

103.225 = 104

As we can see, the right hand side is greater than the left hand side by (104 – 103.225) = 0.775

To make use of this arbitrage opportunity, we will buy the fiduciary call and sell the protective put.

1. Sell the protective put: We sell a put option and receive the $5 premium. We also short sell the ABC stock and receive$99. The total cash inflow is $104. 2. Buy fiduciary call**:** We payout a total of$103.225 for the fiduciary call option. That is we pay $7 as premium for the call option and invest 96.225 in a bond for 6 months at 8%. 3. Net cash inflow: Our net cash inflow is (104 – 103.225) =$0.775

When the time to exercise these options comes, there are two scenarios, the actual spot price will either be above $100 or below$100, let’s see how our arbitrage will work out in each scenario:

 Underlying price is above $100 Underlying value is below$100 Receive $100 from the bondExercise call option, pay$100 and receive ABC stockDeliver the stock to cover the short saleThe put option expires without being exercisedNo net income or loss Receive $100 from the bondPut is in-the-money and is exercised. You pay$100 and take delivery of XYZ stockDeliver the stock to cover the short saleThe call option expires without being exercisedNo net income or loss

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