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|
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.