The Normal Distribution Curve

Per the suggestion of a reader (Thanks JB), today’s blog discusses a mathematical topic – in simple, layman’s terms, and how it relates to the world of option trading.

Hello Mark,

You might want to include “normal curve” and a brief description vis-a-vis its usage in the options world.

JB

In our world, random variation tends to conform to a specific probability distribution – the ‘normal distribution. It’s known as the Gaussian distribution in the scientific community.

For readers interested in more details, try this entry from Wikipedia.

The shape of the graph resembles a bell. Thus, it’s often referred to as a bell curve. 

Although there are many normal distribution curves, they all share one important property:

The 68/95/99.7% Rule:

All normal density curves satisfy the following property (the Empirical Rule).

68% of the observations fall within 1 standard deviation (sd) of the mean*

95% of the observations fall within 2 standard deviations of the mean

99.7% of the observations fall within 3 standard deviations of the mean

Thus, for a normal distribution, almost all values lie within 3 standard deviations of the mean.

* Below, I’ve assumed the ‘mean’ is zero.

OK, what does all this have to do with options?

When buying or selling options, the investor must be concerned with how far the asset that underlies the option is likely move (in either direction) during the lifetime of the option.  To determine that number, we must calculate the standard deviation.  The math can be done using a simple calculator:

sd = S * V * SQRT (t/252)

Where:

S = Stock price in $/share

V = Implied volatility of underlying (as a decimal; i.e., 0.25 for a volatility of 25)

t = number of days until expiration arrives

NOTE: I prefer to use 252 as the number of trading days in one year.  Others use 365, and that makes a significant (20%) difference in the sd.

The normal distribution tells us:

·        Approximately 2/3 of the time, that asset moves less than 1.0 sd by the time the option expires

·        The asset moves more than 2 sd only 5% of the times

·        A huge move of 3 sd is expected only 3 times per 1,000 events. For a monthly stock or index move, that’s once every 28 years

This normal distribution tells us that events such as the massacre of 1987 should never occur [a decline of 20 sd (find the phrase ‘1987’ on the linked page)].  Some observers (Talib, Mandelbrot) believe that the normal curve does not truly represent events as we observe them, and explain that the ‘tails of the curve’ occur much more frequently than predicted by the normal distribution curve.  The occurrence of these so-called ‘black swan’ events is important for those of us who trade options.

It tells us that selling far out of the money options is not as safe as predicted by the normal distribution curve and that the reward for selling such inexpensive options is not worth the risk.  It also suggests that buying such options as insurance for a portfolio exposed to a large loss if that black swan event occurs, is probably worthwhile over the long-term.

66

9 Responses to The Normal Distribution Curve

  1. sync 08/23/2008 at 8:59 AM #

    Does the SD give you any more useful information than what you get from Delta?

  2. Mark 08/23/2008 at 11:05 AM #

    It gives you different information.
    The delta is an approximation of the probability that the option will be in the money when expiration arrives. And that’s true no matter how far in the money or out of the money that option is when its delta is determined.
    When you calculate a one standard deviation move, you are measuring a specific price change for the underlying. By definition, the likelihood that the stock will move that far, or less, by expiration is 68%.
    Mark

  3. JB 08/23/2008 at 4:58 PM #

    “not as safe as predicted by the normal distribution curve and that the reward for selling such inexpensive options is not worth the risk.” <<-- This doesn't bode well for iron condors and similar strategies. Thinking out loud --this is not favorable to any strategy that takes in a credit at the open--True? Thanks. JB. PS if one uses a 5 day week in the 252 day year does that make the 365 vs 252 moot? Obviously, if you're trading out 1 day there is a big difference but for the more usual 4 - 12 week time to expiration.

  4. JB 08/23/2008 at 5:18 PM #

    Mark,
    I may be getting hung up in some of the theory and math here. If I buy a call option 1 sd above the current price and price movement follows the normal curve wouldn’t there be a 16% chance of it being in the money? That is, I only care about 1/2 of the curve so 1 -(50%+0.5*.68) = .16
    Thanks.
    JB

  5. sync 08/23/2008 at 10:10 PM #

    >>It tells us that selling far out of the money options is not as safe as predicted by the normal distribution curve
    I did some research today and found that current thinking is that stocks follow a lognormal curve. This means that most of time stocks move sideways more than is indicated by the normal curve. But extreme moves are also more common.

  6. Mark 08/24/2008 at 7:12 PM #

    JB,
    1) I think you’ve gone too far in stating that “This doesn’t bode well for iron condors and similar strategies.”
    You don’t receive fair compensation for taking the risk of selling far OTM options because the premium is not only very small in terms of cash, but that cash is less than it should be. In other words, FOTM options tend to be undervalued because ‘black swan’ events – unexpected events – occur more often than predicted by the normal distribution model.
    Investors who sell options that bet against black swan events lose more often than they ‘should,’ Thus, it’s a losing strategy.
    Buying iron condors with options that are closer to the money results in fair compensation for the risk taken. Sure, they are exposed to a black swan event also, but they collect a premium sufficient to allow long-term profitability (assuming good risk management techniques.
    If you disagree, then this strategy is one you should avoid because it cannot fit into your comfort zone.
    2) Regarding the number of days in one year, I’ve been thinking about this problem for many years and have not been satisfied with any solution. I use 252 because that’s the number of trading days. I am not certain that it’s correct.
    3) If you buy an option 1 sd OTM, I agree that there is approximately a 16% chance it FINISHES ITM. 32% of the time it moves more than 1 sd – and there’s no reason why one direction should be preferred to another.
    But, don’t ignore the additional times that the option moves ITM and then OTM by expiration. For an option owner, that’s important because the option can (and should be, IMHO) sold before expiration arrives.
    Mark

  7. Mark 08/24/2008 at 7:17 PM #

    Sync,
    If stocks truly move sideways more than expected, then buying iron condors is a winning strategy, even if they occasionally incur a maximum loss when the black swan event occurs.
    And because those BS events are more common than expected, the overall iron condor strategy can be improved by ownership of a small number of cheap, near-term, FOTM options in as insurance for the iron condor portfolio.
    I own a few extra cheapies most of the time.
    Mark

  8. JB 08/25/2008 at 2:22 PM #

    Hi Mark,
    The “This doesn’t bode well for iron condors and similar strategies.” was more of question than a statement. Poor wording on my part. In any case thanks for the response.
    When you talk about “selling far OTM options” , in general, how many sd’s is far OTM?
    Thanks

  9. Mark 08/25/2008 at 2:42 PM #

    This is one of those questions that doesn’t have a good answer.
    Two sd is always FOTM if you go strictly by definition because a move that far occurs only 5% of the time.
    If that 95% confidence level makes you uncomfortable, then it’s not for you. If you go too far OTM, the premium you collect shrinks to a level that makes the whole trade undesirable.
    To my comfort zone, FOTM refers to the option premium. If I can buy an iron condor that generates my minimum cash requirements for a given expiration month, I will buy that iron condor – but only if I feel comfortable selling the puts and calls.
    I won’t be limited by how many sd OTM my shorts are. I’ll be concerned if I feel ok about being short options with those strike prices.
    Right now, I like selling RUT 610/620 put spread in Nov. The 630/640 just doesn’t feel good enough. That’s how I decide. I’m not suggesting you do that, but I do not worry about being a bit less than 1 sd out of the money.
    I’m also afraid of the upside and will not sell a call below 810, and I’d prefer to sell 830s. Just my comfort zone and difficult to explain.
    I have a friend who prefers to sell options much closer to the money. In fact, he sells RUT options that are only 10 or 20 points OTM. That suits his comfort zone because he can sell fewer iron condors and the total dollars at risk relatively small. I don’t like that method, but he makes it work for him.
    Mark