Buoy Don’t Float

Two economists, Black and Scholes, came up with the idea of modeling options and eventually reaching a closed-form solution to pricing options. Rumor has it that they made up some models and then went across the hall to some mathematicians seeking help to work out the solutions. In the end, they won a nobel prize. Was their discovery that big of a deal?

What are options? Options are–like the name implies–the option to buy or sell something, generally for a predetermined amount and after a certain delivery date.

An option contract can be structured many different ways but typically includes the following:

Option price – the price to buy the option

Underlying price – the current value of the stock that the option is pinned to. If we were dealing with commodities, an option on natural gas would be pinned to the current price of natural gas.

Strike price – the price to invoke the option after the expiration date

Expiration date – the date at which the option expires

European / American options – European options allow you to exercise the option only on the expiration date, whereas American options allow you to exercise the option at any point in time up until the expiration date.

Put / Call – Call is the option to buy at the strike price; put is the option to sell at the strike price.

Some option speak:

A put option would be an at-the-money put option if the strike price = underlying price. (There is no gain/loss if the option were exercised at that instant.)

I would exercise a call option if it was in-the-money (and the expiration date allowed me to do so). For example, if the current price of one share at Bui, INC was $25 and I exercised a call option that had a strike price of $20 then I would essentially make $5 (if I bought it at $20 and immediately sold it for $25). Alternatively, if I had a call option with a strike price of $30 then the contract would be out-of-the-money since I would rather buy one share at the market rate of $25 than to exercise the option and buy a share for $30.

The Black-Scholes equation is an indecipherable beast to most, but the important idea is that you can essentially break it down to this relationship:

option price = f(strike price, interest rate, underlying price, expiration date, volatility)

*reading between the lines, options are like trading volatility since all four other parameters are either set or known at the time an option contract is traded.

Other ways to think of Black-Scholes:

closed-form solution – this is to say, the option contract can be accurately valued and is *not* an arbitrary price.

the arbitrage-free pricing of an option contract – that is to say, if you took the premium paid to you for an option and actively managed it (bought/sold equivalent forward contracts and essentially made the right investment decisions) you would have the same overall outcome.

Keep in mind that exchanges handle all of these trades. Especially with options, there is a lot of leverage at play. For example, who is to prevent you from selling 3,000 call options tomorrow and running off with the premiums? It is the exchange’s job to do that by margin calls / ensuring that you can cover your position. In doing this, I believe they essentially evaluate your position vs. the settle price at the end of each trade daty. Also keep in mind that most option contracts are cash-settled in that they usually don’t involve the changing of hands of the underlying. Instead, the parties pay the difference in the contract instead of physically trading a share of Bui, INC.