The Black-Scholes Model

Options are a financial instrument giving the holder the right to buy or sell an underlying stock or commodity (underling asset ) at a future point in time, at an agreed upon price. The Black-Scholes model, for which Fischer Black, Myron Scholes and Robert Merton were awarded the Nobel Prize in Economics, is a tool for pricing equity options. Prior to its development there was no standard way to price options; in a very real sense, the Black-Scholes model marks the beginning of the modern era of financial derivatives.

The Black-Scholes model is used to calculate a theoretical call price (ignoring dividends paid during the life of the option) using the five key determinants of an option’s price: stock price, strike price, volatility, time to expiration, and short-term (risk free) interest rate.

Options

Stock options can be priced using almost the exact same factors that your insurance company uses to determine the premium of your auto policy. Both depend chiefly on (a) the value of the asset (the price of the car/stock) and (b) the risk (your driving record/the stock’s average price changes).

Here’s a quick comparison:

Insurance Agent…Options Trader
Value of car…Current stock price
Deductible…Strike price
Time span of policy…Time until expiration
Interest Rates…Cost of money
Risk…Volatility

High Risk Equals High Premium

Example #1

Take a look at both Drivers, A and B.

Let’s say Driver A is a 17-year old male high school senior. Currently, he has two speeding tickets in the last year. In addition, his parents purchased him a Corvette with a bumper sticker that reads, “Born to Speed!”

On the other hand, Driver B is a 42-year old stay-at-home mother. She has had no traffic violations in the last ten years and drives a minivan.

It’s obvious that Driver A is going to have a higher risk and, therefore, have a higher insurance premium.

Example #2

Let’s now make the same comparison to someone trying to price an option on a $20 tire company stock versus a $200 dot.com stock.

As we already know, the larger the deductible, the lower the premium, and vice versa. In fact, the insurance deductible can be compared to the option strike price.

Components of Option Pricing

There are different components which are used to determine the theoretical value of an option:

• The price of the underlying stock: The higher the stock price, the more (less) a call (put) is worth.
• The strike price of the option: The higher the strike price, the less a call (the more a put) is worth. The right to purchase the stock for a cheaper price is worth more than the right to buy it at a higher price.
• The time until the option expires: The more time left before the option expires, the more any option is worth.
• The cost of money

Interest Rate (The higher the interest rate, the more the call (less the put) is worth. This is because a call buyer uses less cash to buy the call than he would use to buy stock, and the difference can be invested to earn interest. The more interest earned, the more a call buyer is willing to pay for the option.)

Dividend (An option holder is not entitled to cash dividends, and dividends reduce the price of the stock (when the stock goes ex-dividend, the stock price is decreased by the amount of the dividend). The higher the dividend, the less a call is worth, and the more a put is worth.

The volatility of the underlying stock: This is the only one of the factors that is not known. (Of course dividends, interest rates, stock prices and time to expiration are constantly changing, but they are known at the time the option transaction is made.) The volatility used in the model is an estimate of the potential price movement that will occur during the life of the option. The higher the volatility, the more any option is worth because a high volatility increases the probability that the option will make a large favorable move for the holder of the option (Only a favorable move is considered, because if the move is unfavorable, the option holder loses nothing extra. Loss is limited to the price paid for the option.)

If we take all these components and plug them into the Black-Scholes formula, the model will calculate an option’s theoretical value.

The Options Pricing Calculator

Luckily, you don’t have to become a theoretical mathematician to be able to use the Black-Scholes formula. There’s a plethora of financial companies and individuals who have constructed calculators to aid you in determining an option’s theoretical value.

The CBOE (Chicago Board Options Exchange) offers free downloading of their Options Toolbox. It is a self-directed tutorial on options trading. Within this toolbox is an option pricing calculator, comparable to the ones used by traders on the floor of the CBOE.

Here is the link to get the CBOE Options Toolbox.

In-, At-, and Out-of-the-Money Options

An important concept to be knowledgeable of is the designation of options as In-the-money, At-the-money, and Out-of-the-Money.

In-the Money: This option has a strike price which, for Calls, is below the present market price and, for Puts, is above the current market price.

Out-of-the-money: This option has a strike which, for Calls, is above the present market price and, for Puts, is below the current market price.

At-the-money: This option has a strike whether Call or Put which is equal to or near equal to the present price of the underling asset.

Intrinsic Value and Time Value

The distinctions of whether an option is In-, At- or Out-of-the-Money are also pertinent to understand because they help to illustrate the concepts of Intrinsic Value and Time Value.

Earlier, we covered the aspects that go into pricing an option. Well, once you have an option price, you can break that figure down into two parts; its “Intrinsic Value,” or the In-the-Money amount of an option’s price, and its “Time Value,” or the opportunity value that the option may become more valuable in the future.

As you can see, the only options that have Intrinsic Value are In-the-Money options. For example, a 50 Call on a $55 stock is intrinsically worth at least $5, since you could exercise the option, buy the stock at $50 and sell it at $55, earning $5.

This is where Time Value enters. Most certainly option will trade at more than that. Why? Because all options, In-, At- or Out-of-the-Money have Time Value. Time Value is the added premium the option has due to the possibility for additional price movements in the underling asset.
If the 50 Call premium is $6, and the stock is trading at $55, the Intrinsic or In-the-Money amount is $5. The remaining $1 is the Time Value. Therefore, this option is valued at $6 even though, intrinsically, it is only worth $5. The additional 1 dollar exists because of the stock’s volatility (the possibility that it may move more than where it is right now).

On the other hand, the 55 Call is trading for $3, and the 60 Call is trading for $1. With the stock trading at $55, neither of these options have any Intrinsic Value, but they have Time Value because of the possibility that the stock may move that way.

Note: Options, Intrinsic Values, and Time Values can never be negative.

Put/Call Parity

Put Call Parity is an option pricing concept that requires the extrinsic values (premium/time value) of call and put options to be in equilibrium so as to prevent arbitrage.

Arbitrage is the simultaneous purchase and sale of financial instruments in order to benefit from price discrepancies. Option traders frequently look for price discrepancies of the same option contract between different option exchanges, thereby benefiting from a risk free trade.

Time Premium (Time Decay)

Time Decay is the ratio of the change in an option price to the decrease in time to expiration. Since options are a wasting asset, their value declines over time. As an option approaches its expiration date without being in-the-money, its time value declines since the probability of that option being profitable (in-the-money) is reduced.

Time decay of an option begins to accelerate in the last 30 to 60 days before expiration, providing the option is not in-the-money. The greater the certainty about an option’s expiration value, the lower the time value. Conversely, the greater the uncertainty about an option’s expiration value, the greater the time value.

Volatility

It’s well-known in the marketplace that a stock will vary in price throughout the trading day. With each successive trade, the price can go up or down, or remain unchanged.

If you review a stock’s daily closing prices over a period of time, you can observe these net changes more precisely (called returns). These fluctuating trading prices represent a stock’s volatility. The degree to which a stock fluctuates varies on whether a stock’s price trend is advancing (bullish), declining (bearish), or remains steady.

When using options, it is important to familiarize yourself with changes in time decay and volatility changes. Volatilities are always stated in percentages. For example, if XYZ stock, which is currently at $50, has a volatility of 16%, which entails that it is expected to trade in the range up or down of $42-$58. (To calculate the range, use the current stock price as the mean.)

Be cognizant of the fact that lower volatilities imply less movement in the stock price, and higher volatilities imply more movement in the stock price.