Put-Call Parity

A lot of relationships in finance are presented as a black box. Rather than understanding each of the steps between cause and effect, people are often aware only of the trigger and end result. This can lead to rather rigid, “if this then that” thinking and certainly obfuscates the interconnectedness within financial markets. Fortunately, there’s an entire idea built around examining the how and why of a certain financial relationship, namely, the price between European put and call options. In this article, we’re going to talk about the put-call parity model.

The assumptions

First and foremost, the put-call parity concept applies only to European options. This is a crucial prerequisite because these options can be of the same class, that is, they have not just the same underlying asset and strike price but also identical expiration dates. American options, unlike their European counterparts, can be exercised any time before the expiration date.?

The gist of the put-call parity is that simultaneously holding a?short put?(when an investor wants to profit from an increase in the stock's price by pocketing the premium they receive with selling the put) and a?long call?(when an investor wants to profit from an increase in the stock’s price by purchasing the stock at the option’s strike price and selling the stock at its, hopefully, higher market price) should provide the same return as one?forward contract?(an agreement to buy an asset at a specific time and price) with the same expiration date and forward price as that of the option’s.

Now let’s consider an example to better illustrate the situation. Say you:

• Buy a $10 call option with a strike price of $100 and a maturity date 6 months from now
• Sell a $10 put option with a strike price of $100 and a maturity date 6 months from now

According to the put-call parity, this is basically the same as buying the stock at its market price and borrowing an amount equal to the strike price discounted to today (i.e. in today’s money terms with inflation accounted for. More on this later).If the price goes up to $110 and you exercise the call option, you actually would’ve made, on net, $10. Remember, you had to pay $10 for the call option, and you still had to buy the stock at $100; the profit you make from selling it is canceled out so to speak.

But this is where the put option comes into play. You’ve sold it for $10, and because the underlying stock has increased in value, the buyer won’t exercise the put option, allowing you to pocket the $10.

Now imagine you had instead, as previously mentioned, bought the stock at $100 and borrowed $100 (the strike price). Then the stock price rises to $110 so you sell it and pay back the $100 you borrowed (assume, for the sake of simplicity, no interest). $110 - $100 equals a net profit of $10. Now here’s how this arrangement can be replicated as a forward contract. The strike price could be set to $100 and expiry date to the same as the option’s. That way, you’d be obligated to purchase the stock at $100 even if it’s market price is $110. And so you could sell the stock on the market and pocket the $10 difference.?

What this shows is that both portfolios return $10 – the first one where you buy the call option and sell the put option, and the second one where you buy the stock and borrow the strike price amount. Because both portfolios yield the same returns, the put-call parity holds.

The underlying math

Technically there is no single “put-call parity equation” simply because investors frequently rearrange the variables depending on what they’re looking for. Nevertheless, a common expression is as follows:

This says that a portfolio holding a call option and put option in the underlying asset equals a portfolio holding the underlying asset and the strike price amount. You’re subtracting the put option price (pt) from the call option price (ct) because, presumably, the latter is going to become out of the money and thus a liability for the portfolio. The?spot price?(St) is the current underlying asset’s price, and you’re subtracting the present value of the strike price discounted from the date of expiration, represented by X/(1 + r)^T. X is the strike price for both the put and call option, and “T” is the time, in years, until the option expires. Lastly, “r” is the risk-free interest rate (which can more or less be denoted with the European Central Bank bond yield rate).

One shouldn’t underestimate the power that comes with being able to rearrange this equation. Say an investor wanted to buy a stock but also protect themselves against potential losses. They may want a portfolio consisting of that stock along with a?protective put. This is a risk management strategy where puts would be used to set price floors – the premium paid for the put option – below which the investor won’t lose any further money even if the stock continues to fall.

As shown in the payoff diagram above, the maximum loss for buying that particular put option is the premium they paid for it, because they don’t have to exercise the option if the underlying stock price is above the strike price. Should an investor choose this strategy, all they would have to do is rearrange the equation from before like so:

This says that a portfolio holding a position in the stock (St) and a put option (pt) is equal to a portfolio holding the stock’s call option (ct) and the strike price amount.?

Arbitrage

So why care about the put-call parity in the first place? Because when the relationship described above doesn’t hold in real life, an arbitrage opportunity exists. In other words, any of the components must’ve been misvalued and investors can take advantage of the price difference to make essentially risk-free profit. And because these opportunities are rare and short-lived, savvy investors that act quick enough could, say, buy an undervalued call option, wait for its price to rise, and then sell it and pocket the difference.?

Econ IRL

When browsing any business-oriented news source, you’ll undoubtedly find something or another on the?yield curve. This may leave some wondering: why are people so obsessed with this one metric? Put simply, a country’s yield curve impacts the overall economic conditions, and this week’s?paper?demonstrates these effects by analyzing bank loans. The authors find that when the yield curve's slope steepens due to premium shocks, meaning investors suddenly require additional compensation for holding longer-term bonds over shorter-term bonds, banks increase lending, which stimulates economic growth.?

At the core of banks' business model, they transform short-term liabilities like deposits into long-term assets such as mortgages or business loans and profit from the discrepancy between short-term and long-term interest rates. Put simply, they give depositors less interest than they can collect from their loans and pocket the difference. Therefore, when term premiums are raised, banks expect greater profits from these maturity transformations – converting short-term deposits into long-term loans. The increased supply of loans might go towards consumption or business expansions, contributing to overall economic development.

‘Till next time,

SoBasically