What is put-call parity?

By , CEO of Volcube.

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Put-call parity refers to the idea that simple put options and call options of the same strike and same expiry date are essentially the same instrument. The only difference between a simple put option on an underlying product, say with a strike of 100 and expiring in December, and a simple call option on the same underlying product with the same strike and expiration date, is that the put allows its owner to sell the underlying whereas the call allows its owner to buy the underlying. But this difference is actually (surprisingly perhaps) trivial. With a simple hedge using the underlying, a call option can be transformed into a synthetic put option. And the reverse is also true; a put option can be hedged with the underlying product to be transformed into a synthetic call option. This is the essence of put-call parity. Puts and calls are pretty much the same thing!

The easiest way to see how this works is to go back to the definition of the options, to look at their basic payoffs at expiration and compare this to trades in the underlying. Let’s work through a simple example. Imagine the underlying product is trading at $100.00. Now let’s suppose the $100 strike call options are available to purchase for 50 cents. Suppose we buy 100 lots of these options and let’s assume each option let’s us buy 1 lot of the underlying (also known as the spot product) if we exercise the option.

Now, at expiration what is our profit and loss from having bought these calls for 50 cents? With the spot trading below $100, we won’t be exercising our calls (who wants to buy the spot product for $100 when it is trading below this price?). So below $100, we just lose our 50 cents. If the spot is above $100 at expiration, the calls are worth exercising. We will make 1 cent in profit for every cent above $100 that the spot trades up to. But we still lose our 50 cents no matter what. So we will break even if the spot trades up to $100.50. At that price, our calls make 50 cents in profit which covers the cost to us of buying them. Above $100.50, we make a penny of real profit for every penny the spot product rallies. So if the spot rallies to $102, we will make $1.50 in profit.

Let’s look at the $100 strike puts. Imagine these too are trading at 50 cents. If we buy these, what is our payoff profile? It’s the mirror image of the calls we just looked at. The puts give us the option to sell the underlying at $100; clearly we are not going to exercise this option if the spot is trading higher than $100. So above that price, we just lose our 50 cents paid for the puts. Below $100, our puts are worth exercising. We make a penny for every penny below $100 that the spot falls to. At $99.50, we have made back the amount we paid for the puts, so this is a break-even point. Again, this is just the opposite of the situation with the $100 calls. Below $99.50, we make a penny of net profit for every penny the spot price falls. At $98, we have $1.50 of net profit, exactly as we had with the calls $2 above $100.

So we can see how similar these two profiles are. The are essentially the same but with a minus sign added! Now to make these payoffs look exactly the same, we can just trade the underlying product at the same time. Let’s suppose that when we buy the $100 calls for 50 cents, we also sell 100 lots of the spot product. In other words, we are going to hedge our calls 100%; we sell short 1 lot of the underlying for every 1 lot of the calls we own. What does this do to the payoff profile? Recall that our call options lost 50 cents no matter what if the spot was trading below $100 when the options expired. But now we are short the spot product from a price of $100. So our short spot position will make a profit below $100. At $99.50, our call options still lose their usual 50 cents, but the short spot position makes 50 cents back. We break-even. Below $99.50, the short spot position makes a penny for every penny the spot falls. The calls just lose their 50 cents. Notice something about this long call-short spot profile? It’s identical to the long put position. Let’s check above the $100 strike to be sure. Remember above $100, the long put option position just loses 50 cents. How about the long call-short spot? The long calls make a penny for every penny the spot rises above $100. But the short spot position cancels this out exactly; it loses a penny for every penny the spot rises above $100. Plus, we still lose the 50 cents we paid for the call. Net result? The long call-short spot position is synthetically identical to a long put position. And that is put-call parity!

Try working through another example like this. Maybe look at a short put position; work out the payoffs at different spot prices. Then work out the payoff for a short call-long spot position. Due to put-call parity, the results should be the same.

In a later article, we will look at some exemptions to this rule (that typically involve interest rate considerations or dividend payments on stock options). We’ll also look at how this situation applies to options that aren’t hedged 100%. But from this article, the point to take away is that put-call parity means that in the simplest cases, puts and calls are really the same thing.

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