In the last post we went over how calls and puts work. Specifically, we discussed what calls and puts are worth when they expire. Options contracts are written in terms of the expiration date, so their value is easy to figure out at that point.

It's also important to know what options are worth before they expire. Traders usually don't buy and hold options until expiration; even when they do, they need to know what price is fair to pay upfront.

What's a call worth a year before it expires?

### Options Pricing

The fair price of a financial instrument is equal to its *expected value*, or the average price of the instrument taking all outcomes into consideration. How does this work in practice?

Let's say someone tells you that they're going to flip a fair coin. If it comes up heads they'll give you $100, and if it comes up tails you get nothing. What's this deal worth to you? Half of the time you make $100, and half of the time you make $0. On average you make $50, which is also the flip's fair price. [1]

The expected value of an option is harder to calculate. First, we need to treat the underlying stock price as a random process. This is usually done by assuming that stocks follow a specific type of random movement: Geometric Brownian motion. [2]

Under this assumption, here's a 50-call's value a year before expiration:

(You can follow along with this post in Excel with the Options_Intro script, available here or on Github. You'll need to install DataNitro, which lets you run Python scripts in Excel.)

### Volatility

Let's break down this model. The most important property of the underlying stock, for valuing an option, is its volatility. A stock's volatility measures how far from its current price you can expect the stock to move over a certain period of time (usually a year).

If a stock has no volatility, options on it would be useless - no one wants a call option on a $50 bill (unless its strike is under $50). If a stock is really volatile, then an option on it is worth a lot - even there's only a 50% chance of going in the right direction, we're likely to make a lot of money when it does move our way. [3]

The graph above shows an option on a stock with 30% volatility. Here's one with 5% volatility:

and one with 150%:

At 5%, the call's value looks almost like a call at expiration. That's because the stock is unlikely to move far from its current price in one month. On the other hand, at 150% volatility, the call looks almost like a stock - it's worth some money even close to 0, and gains about a dollar in price for each dollar the underlying moves up. This is because even if the stock is at $10, it has a reasonable chance of moving to $50 or higher within the year; and for every dollar it moves up, we expect it to close one dollar higher.

### Time

After volatility, the most important attribute of an option is time to expiration. In fact, time and volatility are largely interchangeable - an option with a lot of time left but not a lot of volatility is very similar to an option with a lot of volatility but not much time remaining. This is because we're interested in how far the stock moves before expiration - we don't care if it moves a lot because it moves quickly, or because it can move for a long time. [4]

Here's are two graphs of 50-calls with 30% volatility. This is the call with 5 days to expiration:

and with 5 years:

Because time and volatility are so closely related, the first graph looks like the call with 5% volatility, and the second looks like the call with 150% volatility.

There are two other values that affect an option's price: the interest rate and the dividends on a stock. If you own a call, a high interest rate is good for you, and a high dividend rate is bad for you. [5] However, both interest rates and dividends rarely change much in the short term, so they're relatively unimportant for option pricing. [6]

### Black-Scholes

The standard way to price options is by the Black-Scholes formula. [7] This formula combines volatility, time, and interest and dividend rates (as well as the stock price and option's strike price):

S and K are the stock price and strike price; σ and τ are the volatility of the stock and the time to expiration; r and q are the annual interest rate and dividend rate; and N is the standard normal cumulative distribution function.

What does this formula say? It's easier to understand if we ignore interest and dividends (setting r and q to 0):

You can think of N(d1) as the chance that the stock will be at or above the strike price when the option expires, so N(d1)S is the expected value of getting the stock in the future at the price S. N(d2)K is similar - it's the expected value of having to pay the strike price for the stock. So, together, these two terms say that the value of a Call is the expected value of being able to get the stock at the price S, minus the expected value of having to pay the strike K. The value of a Put is the expected value of being able to sell the stock at a price K, minus the expected value of having to buy it at a price S. [8]

The full formula is the same, but takes interest and dividends into account in the following way: the future stock price, S, is decreased by the proportion of dividends paid out between now and expiration (since you won't be receiving those dividends by holding the call). In finance, this is called *discounting* by the dividend rate. Similarly, the strike price, K, is discounted by the interest rate, to account for the different in the value of dollars at expiration as compared to dollars now.

Footnotes:

[1] Why is this the fair price? If you could buy this contract for $49, you would make - on average - $1 for every coin flip. (Half of the time you would make $51 after the flip, and the other half you would lose $48. In the long run, the wins and losses even out, and you're left with $2 for every two flips, or $1 for each flip.) On the other hand, if you could sell the contract for $51, you'd also make an average of $1 for every flip.

If you're buying the contract for $49, there's someone selling it for $49 - which is a losing bet for them. A seller would lose money at less than $50, and a buyer would lose money at more than $50, so $50 is the only price they can agree on. In practice, people do sometimes trade at a loss, especially since the price of options (and stocks) is much harder to figure out than in this example. If someone consistently places trades at a loss, though, they'll run out of money and leave the market very quickly.

[2] If a stock does move randomly, what is its expected value?

[3] As we saw last time, you can combine different options to make money no matter what direction the stock moves in.

[4] There are, however, important practical differences in trading options with different times to expiration. Trading options on the day they expire is a whole topic in and of itself.

[5] Specifically, a high interest rate encourages people to keep money in the bank and earn interest. Normally, you'd buy a stock if you think it's going to increase in value; but if interest rates are particularly high, you can instead keep most of your money in the bank to earn interest, and buy calls instead to capture most of the stock's upside. On the other hand, if a stock pays high dividends, you only earn those dividends when you hold the stock, not if you just own a call on it.

[6] Last time, we mentioned that different options have different rules about expiration. If you're trading in a liquid market, these rules only make a difference when a stock has high dividends - for example, if you have a call and a particularly high dividend has been announced, you can exercise it early to capture the dividend.

If you're trading in an illiquid market (one where there aren't a lot of buyers and sellers), you can also exercise a stock early to help close a position.

[7] If you have a background in physics, you might notice that the Black-Scholes formula looks like a thermodynamic equation. This is because we're modeling the stock with Brownian motion, a basic thermodynamic process.

[8] This is a rough explanation. More accurately, N(d1) and N(d2) are risk-adjusted probabilities, which are different from real probabilities. In particular, they're different from each other, and take into account that you'll exercise the option if and only if it's profitable.

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