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.

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.
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%.
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 bond Exercise call option, pay $100 and receive ABC stock Deliver the stock to cover the short sale The put option expires without being exercised No net income or loss	Receive $100 from the bond Put is in-the-money and is exercised. You pay $100 and take delivery of XYZ stock Deliver the stock to cover the short sale The call option expires without being exercised No net income or loss

As you can see, in both the scenarios, there is no net inflow or outflow at maturity. Our risk-less profit is $0.775 that we made in the beginning. This example used European options. If we had to form a similar strategy with American options, it would be much more complicated.

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