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Put-Call Parity: Definition, Examples, Working & Formula

What is Put-Call Parity? 

A fundamental concept in financial mathematics and options pricing theory, put call parity represents the relationship between the put and call options. Hans Stoll, an economist, proposed the put call parity concept in a research paper titled “The relationship between put and call option prices” published in 1969 in “The Journal of Finance.” The put call parity holds true for European options and the relationship is between the prices of put options and call options of the same underlying asset, expiry date, and the exercise price (or strike price). It is based on the underlying principle that the addition of the call option value and the present value of the strike price is equal to the addition of the put option value and current value of the underlying asset. You can see a detailed article on put and call options.

Essentially, the put call parity theory asserts that the value of the call option represents a particular fair value of a corresponding put option at the same exercise price and expiry date, and vice versa. If the markets do not follow this relationship, then it implies a mispricing that smart traders are aware of how to make profits. The modern markets rarely feature such instances of deviations, though, and usually, the relationship is followed.

The fundamental equation that governs the put call parity is as follows:

A0 + P0 = C0 + E(1 + r)-T

Here, 

A0 = Underlying asset’s price

P0 = Premium on put option price

C0 = Premium on call option price

E = Exercise price

r = risk-free rate

T = Time to expire

The concept of put call parity assumes no arbitrage and efficient markets. Though the above put call parity equation was derived for European put options, it can also be applied to American options by accounting for interest rates and dividends. When the dividend surges, the value of the put options expiring after the date of ex-dividend will also increase but the value of the call options will decrease by a similar magnitude. The effect of interest rates are quite opposite when compared to dividends. When interest rates shoot up, the call option values increase while the put option values decrease. Let us understand put call parity through some examples along with illustrations of arbitrage opportunities.  

Practical Examples of Put-Call Parity

Suppose there is an investor who wants to benefit from put call parity through a rise in the underlying value of the asset and also hedge the investment against a reduction in value. Let us consider two portfolios to understand put call parity practically. 

1st Portfolio

When time is zero, the investor purchases a premium call option at a price C0 with the exercise price E for an underlying asset along with a redeemable bond at the same exercise price after time T. Let us assume that the call option expires at time T. Therefore, the cost of this investment strategy will be C0 + E(1 + r)-T.

So, in this portfolio, the purchases a call option that will result in a positive payoff if the value of the underlying asset surpasses the exercise price. The investor also invests his surplus cash in a risk-free bond. The condition for profit in this portfolio can be represented by AT > E.

2nd Portfolio

Assume that the investor purchases an asset with an underlying price A0 at time t equals to zero and also a put option at a price P0 with an exercise price of E after time T. So, the resultant cost for this investment strategy will be A0 + P0.

For both the above portfolios, the investor can gain from an increase in the underlying value without being exposed to a fall below the exercise price. So, even for these two portfolios, the put call parity equation holds true. 

Now that we have seen the put call parity strategies for two investment portfolios, let us see an example and calculate the value for a term in the equation. 

Example

For an exercise price of $45 that expires in 3 months, the underlying asset value is $55 and there are no cash payments during the options. If the selling price of the put option is $4.2 and the risk-free rate is 4.8%, calculate the value of the call option.

Solution

The put call parity equation condition specifies that A0 + P0 = C0 + E(1 + r)-T.

Now,  C0 =  A0 + P0 - E(1 + r)-T, i.e., 55 + 4.2 - 45(1.048)-0.25 = $14.74

Arbitrage Opportunities Arising from Put-Call Parity

The arbitrage strategies resulting from open-arbitrage are referred to as synthetic positions. The condition for using synthetic positions is that the exercise price and the expiry date of all the puts and the calls must be the same.Synthetic positions can be utilized to gain from put call parity through two common arbitrage strategies - conversion and reversal.

  1. Conversion strategy

    In a conversion strategy, an investor purchases the underlying asset along with concurrently purchasing a put option and offloading a call option. This is also referred to as a synthetic short stock position.

  2. Reversal strategy

    Quite opposite to the conversion strategy, the reversal strategy involves short-selling the underlying asset along with concurrently offloading a put option and purchasing a call option. This is also referred to as the synthetic long position. See this link for a detailed note on short-selling.

As you can see, you can implement multiple investment strategies through the put call parity relationship.

Conclusion

Put call parity is one of the foundational guiding principles in the area of options pricing. When the condition is defied, an investor can use arbitrage strategies to benefit from such opportunities. The put call parity therefore has several implications in risk management and options trading. As an investor, you need to be mindful of the fact that this relationship assumes an ideal market. You may have to incorporate the effects of dividends, interest rates, transaction costs and other factors in order to make this a viable strategy for earning profits.

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