Defining Options Pricing and Implied Volatility

The fundamental question of any investment decision is the price. Is it a bargain, or, is it too expensive? In options trading, inevitably you will turn to price-to-earnings ratios, book values, earnings growth rates, and many other measures to determine a stock’s price. Back in 1973, Fisher Black and Myron Scholes published a theoretical model to price equity options. Later, Robert Merton built upon that original paper with his own work, and was the first to refer to the pricing equation as the “Black-Scholes” options pricing model. This equation enabled investors to calculate a quantitative measure of an option’s value  and facilitated the beginning of an options trading on a large public scale.

The Black-Scholes Pricing Model

The Black-Scholes options pricing model is quite complex, but the computation will be done for you on your brokerage platform or in the options analysis software. Here’s Black-Scholes pricing model to calculate a call option:

Pc = SØ(d1) – XE-rtØ(d2)

where:

Pc = the calculated, or theoretical price of the call optionoptions pricing and implied volatility

d1 = [log(s/x) + (r + σ2 /2)t]/[σ√t]

d2 = d1 – σ√t

s = the stock price

x = the strike price

r = the interest rate

t = time in years until options expiration

σ = historical volatility

Ø = the normal cumulative probability distribution function

As you can see the option price depends on several variables: the stock price, the strike price, the interest rate, time to expiration, and the stock’s historical volatility. Among these variables, the interest rate is the least important for options with reasonably short-term expirations. But, the interest rate will become more important when calculating the value of Long-Term Equity Anticipation Securities (LEAPS) options, with expiation up to two years. For the most part, the seldom and small changes in interest rates together with the low dependence of the Black-Scholes equation on the interest rate prompts investors to generally discard interest rates when computing theoretical option prices.

Options Pricing – The Greeks:

The Greeks are quantitative measures of the sensitivity of the option’s theoretical price to the several variables in the Black-Scholes model. Differential calculus allows you to determine the sensitivity of your calculated option price to changes in individual variables of the equation, such as the stock price, while holding all other variables constant.

Let’s examine the relationship of the variables in the Black-Scholes equation in broad qualitative terms. The larger the separation is between the current stock price and the strike price, the smaller the option’s price will become. There is a smaller probability of a price’s occurrence as you move away from the peak of the distribution. The Greek delta (Δ) represents in quantitative terms the relationship of the option price and the stock price.

You can expect the price of an option to be higher, as you have more time available for your prediction to come true. Generally speaking, longer-term options will be more expensive. The Greek theta (θ) represents the relationship of time to the price of the option. Theta determines the change in the option price due to the passage of one day of time while all other variables remain constant. Theta values for individual options are always negative.

The Greek vega(ν) represents the relationship of volatility to the option price. Highly volatile stocks have more expensive options.

The sensitivity of the option price to changes in the interest rate is measured by the Greek Rho (ρ). Commonly, changes in interest rates have a negligible effect on the commonly traded options of a few weeks or months in duration.

Gamma (γ) is unique among the Greeks in that it determines the change in one of the Greeks with a change in the stock price; gamma computes the change in delta with a $1 change in the stock price. Gamma is a measure of sensitivity for delta.

Implied Volatility

If you enter all the appropriate data into the Black-Scholes equation and compute the option price, you will see a discrepancy with the actual market price for that option. The market price may be higher or lower than the calculated or theoretical option price, but more often it is higher. The only variable in the Black-Scholes equation that might be responsible for this discrepancy is the historical volatility. Thus, if you were to enter the market price of the option into the Black-Scholes equation and compute volatility, the result will be volatility “implied” by the market price, or implied volatility. If the market price was higher than the theoretical price, then implied volatility was higher than the historical volatility. The market has priced this option higher because they expected the price volatility of this stock to be higher than it has been in the past. Therefore, implied volatility is a measure of the market’s consensus estimate of future volatility for this stock.

Option prices may vary widely due to swings in implied volatility. This is why many traders follow a small number of stocks and track their historical fluctuations in implied volatility. They will buy options when implied volatility is low and sell options when implied volatility is high. As implied volatility returns to its norm, these options positions will tend to appreciate in value.

Options Pricing – Building the Price of an Option:

The market price of any option consists of its intrinsic value, the time remaining to expiration, and implied volatility. The price of an option increases as the option has more intrinsic value, more time to expiration, and higher implied volatility.

Consider this example. The intrinsic value of an option is the value of that option if you were to exercise it today. For instance, a $200 call option for ABC Construction is selling $211 because you can exercise your call and buy 100 shares of ABC Constructions at $200 per share, sell it for $211 per share, and have a profit of $11 per share. It’s quite possible, though, that the market price of that call option is much higher than $11 because the market price also accounts for how much time is left until expiration as well as the implied volatility of ABC Construction at this time.

The same call option with 12 days to expiration may be selling for $12.50, but the call option with 48 days to expiration may be selling for $18.50. Assuming the implied volatility of the two options for ABC Construction is identical, the difference of $6 is due to the additional 36 days time.

Options Pricing – How to Use the Greeks:

You will use the Greeks in two ways:

1. To evaluate and compare several trade candidates;

2. To manage the ongoing trade and determine appropriate adjustments when necessary.

Your web site’s brokerage software will calculate the position Greeks for you. Here are some ideas for analysis:

-if one trade has a much larger delta value than another, then that trade is much more bullish if the delta is positive, or much more bearish if the position delta is negative;

-large vega values tell you that the position is sensitive to changes in implied volatility. Large positive vega trades will gain in value with increasing implied volatility and lose value as implied volatility decreases;

-positions with large negative values of vega lose value when implied volatility increases;

-positions with negative theta will slowly lose value with the passage of time if nothing else changes.

Once a trade position has been established, you will look to the Greeks of the position to assess your ongoing risk. The Greek values will trigger your decision to close the trade and/or make adjustments to the position.

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Filed under: Options Trading

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