Applying Put-Call Parity to Everyday Problems

While business is not rocket science, some would like to make it seem like it is in order to justify stratospheric salaries, particularly finance types. Marketers seem to acknowledge that they make too much money for what they actually do.

The Black-Scholes options pricing model was developed in 1973 by Fischer Black and Myron Scholes. Options are financial instruments called derivatives because their value is a function of another security, usually a stock. The duo’s breakthrough led to the following put-call parity to help students make sense of options pricing, as well as anything else in life. After all, parity is all about balance. Let's see how it applies in finance and then relate it to sports business and how VPs and General Managers go about deciding when to trade a player.

Put-call parity states that:

Call + Cash / (1+Rf)t = Put + Stock

A call is an option giving the holder the right, but not the obligation to buy the underlying security at a given time at a predetermined price (in the case of European options, with American options, the investor can buy anytime before maturity).

A put is an option giving the holder the right, but not the obligation to sell the underlying security at a given time at a predetermined price (in the case of European options, with American options, the investor can sell anytime before maturity).

The denominator of (1+Rf)t refers to discounting the present value to find what is needed today to eventually reach the strike price. The strike price (or exercise price) is the predetermined price.

Maturity is how long until the right expires.

Finally, the stock is a share of the company.

The Black Scholes pricing model is fairly sophisticated but put-call parity is fairly straightforward: imagine two investors, one is an optimist (Bull) and the other is a pessimist (Bear).

The Bull decides to buy the stock and ride the wave. But having taken a couple of finance classes, he realizes that he should probably hedge himself and buy the right to sell, locking in some protection in case something goes wrong.

The Bear figures that cash is king and decides to wait on the sidelines with good old fashion cash. But not wanting to miss out on the next Microsoft, he decides to invest in a call, giving himself the right to buy the security.

As you can see, these two investors have managed to take mirror positions. The Bull has the stock but can get rid of it. The Bear is hoarding cash but has given himself the right to buy. This much we agree, but why are these positions equal?

The value of the cash is obvious. The value of both the put and call are a function of the underlying security: the stock. If the stock rises, so will the right to buy (call). If the stock plunges, so will the call's value. Conversely, if the stock falls, the put's value rises but the call's value will sink. This mechanism must strike equilibrium; otherwise, there is the potential for a riskless profit, or arbitrage. However, there is no such thing as a riskless profit, the risk may be small, but it will still be present.

The final question is why is the cash discounted at the risk free rate, especially if there is no such thing as a riskless profit? Risk is synonymous with uncertainty. There is little uncertainty in cash, so that is one reason. Second, all of the uncertainty in the values of the put and the call are a direct function of the underlying security. This elimination of added risk means that a "common denominator" has been attained and all excess risk avoided. For this reason, the risk free rate can be used.

Now, how is this applied to the sports world? Well, consider the stock as an athlete. The call is the team's right to renew the athlete's contract, the put is their right to trade him away. The cash is the amount they will need to sign him when his contract expires (European option) or at anytime before (American option). They only need the present value of his signing amount since the cash will compound interest and reach the athlete's demanded salary (strike price). VPs and GMs use this formula implicitly and to a large extent, we all do too on a daily basis. So do sports agents.