What is Put-Call Parity?
Put-call parity is defined as relationship between the prices of a European-style call option and a European-style put option, as long as they have the same strike and expiration
Call price − Put price = Stock price − Strike price (present valued from expiration).
An Example to know Put-Call Parity
Assume that Stock price is $98, and an European call option struck at $100 with an expiration of nine months has a value of $9.07. The nine-month, continuously compounded, interest rate is 4.5%. What is the value of the put option with the same strike and expiration? By rearranging the above expression we find Put price = 9.07 − 98 + 100 e−0.045×0.75 = 7.75. The put must therefore be worth $7.75.
Detailed approach about Put-Call Parity
The relationship,
C - P = S - K * e-r(T - t)
- C : European Calls
- P : Puts
- K : Same Strike
- T : Expiration
- t : time
If you buy a call option, at the same time write a put, and sell stock short, what will your payoff be at expiration? If the stock is above the strike at expiration you will have S − K from the call, 0 from the put and −S from the stock. A total of −K. If the stock is below the strike at expiration you will have 0 from the call, −S again from the stock, and −(K − S) from the short put. Again a total of −K. So, whatever the stock price is at expiration this portfolio will always be worth −K, a guaranteed amount. Because this amount is guaranteed, we can discount it to its present value. We must have,
C − P − S = −K * e−r(T −t)
This is put-call parity.
Alternative method for Understanding
In terms of implied volatility, put-call parity can also be understood. The implied volatility of calls and puts with the same strike and expiration must be equal.
Put-call parity is a model-independent relationship, which is its attractiveness. We require a model for the stock price, namely its volatility, in order to value a call on its own. The valuation of a put is no different. However, no model is required to value a portfolio that consists of a long call and a short put, or vice versa. There aren’t many model-independent interactions in the finance industry. There are two types of relationships: one is between forward and spot prices, and the other is between swaps and bonds.
In reality, there are two prices associated with options: an offer (or ask) and a bid. This indicates that if you are going short (long) on the options, you need to employ bid (offer) when searching for put-call parity violations. This adds a little amount of messiness to the calculations.
Put-call parity violations are significantly easier to identify when you consider implied volatility.
Non-Overlapping Implied Volatility Ranges
You need to search for implied volatility ranges that do not overlap. Assume, for instance, that the implied volatility terms of the bid/offer on a call is 22%/25% and that of a put (with the same strike and expiration) is 21%/23%. There is no opportunity for arbitrage because these two ranges (22–23%) overlap. On the other hand, there would be a breach of put-call parity and an obvious opportunity for arbitrage if the put prices were 19%/21%. However, in reality, don’t expect to locate many (if any) of these straightforward free-money possibilities.
When you initiate a deal, any arbitrage that you may have discovered usually vanishes. We must modify the calculation to account for dividends paid on the underlying stock during the options’ term. We now discover that
C − P = S − Present value of all dividends − E * e−r(T −t)
This, of course, assumes that we know what the dividends will be.
If interest rates fluctuate, simply use the value of a zero-coupon bond with a maturity that coincides with the option’s expiration to discount the strike back to the present. In a similar manner, dividends must be discounted.
American Options and the Breakdown of Put-Call Parity
Put-call parity does not hold when the options are American. This is due to the possibility that your short position could be used against you, exposing you to some degree to the stock price. As a result, you are unsure of your future value when you expire. Theoretically, it is never best to exercise an American call before it expires in the absence of dividends; nevertheless, if the stock drops sufficiently, an American put should be exercised.
Related Readings
- Arbitrage in Quantitative Finance: All You Need To Know
- Modelling Approaches in Quantitative Finance: All You Need To Know
- Central Limit Theorem In Finance: Power Your Portfolio Now
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