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# Options Pricing

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Most people want to forget their experience with high school calculus, that nebulous math whose underlying theory few really tried to understand. The trick was to memorize the rules of calculation and pass the tests. Once accomplished, those memories were quickly discarded as useless in the real world. But if you live in the world of options, it is the real world. In this case, however, it is much more important to understand the theory behind calculus than to know how to do the calculations. There are models for that. The good news is that the theory is actually much easier to master than the mechanics.

Calculus is a methodology that measures the rate of change of one thing relative to another. In pricing an option, that is all you really need to know: The higher the rate of change in a stock (volatility), the more valuable an option is, regardless of whether it is a put or a call. This makes intuitive sense: The more you think a stock can move over a specified period of time, the more you will be willing to pay for an option. In my earlier column, you only have to look at the graphs to see that when buying an option, the real payoff comes when a stock moves a lot. The bottom line is that for stocks that move a lot, that have a higher volatility, their options, both puts and calls, will be priced higher.

The first and most important step then in pricing an option is to quantify a stock's volatility. We will look at two types of volatility: 1) normal day-to-day changes and 2) the likelihood of an abnormal large change on any given day.

The following graph is the first step in analyzing normal day-to-day volatility. The stock price is measured horizontally and the number of observations vertically to create the bar graph. We look back any number of days, perhaps 220 trading days (one year) and record the number of closing prices at each price, represented by a bar. For example, the light blue bar shows that most closing prices occurred at 0% away from the stock's current price. The red bars represent the number of closing prices below the current stock price, the dark blue above.

It is to be expected that few closing prices occur 45% above and below the current price, more 30% above and below, and even more right around the current stock price.

If we fit a curve over these bar graphs it will look like the graph below. The more observations we include, the smoother the curve. This represents the distribution of stock prices weighted by the frequency of observations. This distribution, referred to as a "normal distribution," is a valuable statistical tool and can tell us a great deal about the volatility of the stock and consequently, the pricing of its options.

Without going into statistical calculations (which are a little more refined than represented here), we can now define several terms that are the main drivers in pricing options. The graph is labeled 1 STD around a 30% range over and under the current stock price and 2 STD around a 45% range. STD (standard deviation) indicates a level of certainty or confidence interval: 1 STD incorporates 66% of all observations and 2 STD incorporates 95%. Basically this means that we would expect the stock to be up or down 30% in one year's time with a 66% certainty; or 45% with a 95% certainty (based on historical data). This is what is used when referring to the volatility of a stock. The 30% (1 STD is usually used) or 45% number is plugged directly into option pricing models to determine a theoretical price. The higher this volatility number, the higher the theoretical value of an option.

This volatility number tells us something about the normal or day-to-day volatility of a stock. A 30% annualized volatility number translates into a 2% move per day. This is calculated by taking 30% and dividing it by the square root of 220, the number of trading days per year.

These volatility numbers are where most people stop, but they tell only half of the story. Much can be learned additionally, however, from the shape of the distribution.

The following graph depicts the distribution of two stocks. Notice that the blue stock relative to the red stock has a higher frequency of observations around the current stock price. If we stop there it would seem that the volatility number for the blue stock would be lower than that of the red stock; the options of the blue stock should be priced lower. But further notice that the blue stock has a higher frequency of trades far away from the current stock price (tail observations). It is possible then that these two stocks could have the same volatility number, or the blue stock could even have a higher number. The blue stock has a lower day to day volatility, but a higher "event" volatility or tail risk.

So which stock is riskier and which options should be priced higher? As it turns out in a case like this, the options of different strikes (above and below the current stock price) would be priced to reflect these distributions. It is likely that the options for the blue stock would be relatively cheaper for strike prices near the current stock price and more expensive for strikes away from the stock price than for the red stock.

Relative option prices are compared by backing out the volatility number. We can use this as a normalized number to compare options of different strikes, options of different stocks, and various index options to stock options. The following chart will introduce this concept and several derivative numbers useful to analyze options.

The red dashed line depicts the return profile of a 40 strike call purchased at \$3.65 with the stock price at \$40 per share. We decide to pay this price because the past historical volatility number is 40%. If the next day the rest of the world decides that the future volatility of the stock will come down (perhaps an earnings report overnight calms investors' fears) to say 35%, then the option will cheapen and we will experience a loss depicted by the blue dashed line. In each case we are deciding to pay a certain price because we plugged our volatility assumption into a statistical model (like the Black-Scholes). This is called the "implied volatility:" the option price implies a certain volatility number. This is the way to compare one option price to another; we compare the implied volatilities rather than the prices themselves.

The option pricing model will also generate numbers we call "partials" that will explain how the option behaves relative to the stock price. These partials are first and second derivatives of the underlying stock price. The "delta" is the amount the option price will move for a \$1 move in the stock price.

In this case, if the stock is up or down \$1, the option price will move about 63 cents. This number is also the probability the option will expire in the money, in this case 63%. This delta, however, will change as the stock price changes. For a call, as the stock price goes up, the delta will also go up: the probability that the stock expires in the money increases and the option acts more and more like stock (it is the opposite for a put). Interestingly, there is a confluence of variables that affect how much the delta will change. For example, the delta will increase more for a short-term call than for a call with a longer expiration. The rate of change of the delta is measured by the second derivative and is called "gamma." In this case, the delta will change by .054 for a \$1 change in the stock price (if the stock is up \$1, the delta will rise to .63 + .054 = .684). The "vega" measures the sensitivity to changes in implied volatility. In this case, if the implied volatility number goes from 40% to 41%, the option price will increase by 7 cents to \$3.72. Vega will be larger for longer term options than shorter term options. The "theta" measures the decay of the option: because the option has an expiration date, it is a wasting asset. In this case each day the option will decay by \$ .018. The rate of decay will accelerate over time; most of the decay will occur near expiration. Because of decay, the return profile of the option in the above graph will move from the original red dashed line to the black dashed line over time and eventually to the solid blue line at expiration. The "rho" measures the sensitivity of the option price to changes in interest rates: as interest rates rise, the price of a call will increase and the price of a put will decrease.

It is evident that an option is a dynamic asset: Its price will change based on several variables. The greatest influence by far is the price of the underlying stock. The second greatest influence, and the one to concentrate on, is the implied volatility. This is where the art come into play when pricing an option. All the other variables (interest rates, dividends, and time to expiration) are given.

There are limitations in using statistical analysis in deciding on an appropriate implied volatility number when pricing options. It is linear in nature: it uses historical data and extrapolates this into the future. This assumption is generally incorrect. There is a subjective side to pricing options that comes with experience and even intuition, using statistical analysis as a guide.

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No positions in stocks mentioned.

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