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Chris Conlan
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I trade: Mid-term equities, Short term equities options, Tick by tick Forex arbitrage I am especially knowledgable in: Clothing/Retail stocks, Options valuation/modeling, The Yen and its currency pairs
  • Black-Scholes: Simplified 0 comments
    Sep 1, 2013 5:56 AM

    As a self-taught options trader myself, I always found it frustrating to hear anything along the lines of "The Black-Scholes options valuation model is based on highly advanced niche calculus and extremely complex probability theory, so it is best that you just appreciate and use the formula as opposed to understand its derivation." I felt that I could not confidently apply the formula until I understood the ideas behind it.

    After my arduous journey through academia, I was disappointed to conclude that I did not need to learn the minutia of the derivation but needed it explained differently. The goal of this article is to demonstrate the logical procession of the Black-Scholes without using esoteric terms and techniques from Stochastic Calculus and Probability Theory. Instead I will use only elementary Statistics. In a later article, I will explain how to use this knowledge to help trade.

    Keep in mind that the Black-Scholes model is highly theoretical, so some of the situations assumed in this argument are known to be unrealistic by certain counts.

    To start, stocks under the model's formulation are assumed to be a Brownian Motion. This means that over any time period, the change (from close to close) of a stock's price is a random number from a normal distribution. Each price move is also independent from one another, so each move is a new random value from the same bell curve. It doesn't depend on what past periods did.

    en.wikipedia.org/wiki/Normal_distribution

    Obviously not every stock has the same distribution. They all have a mean of zero, (theoretically) but they have different volatilities and can be measured in different time frequencies (daily, hourly, weekly). A Perfect Normal Brownian Motion (normal dist. with standard deviation of 1) is always multiplied by the stock's standard deviation to make sure it is the right size. For example, PCLN with a price of about $960/share, will have a higher volatility than DELL with a price of about $13.40. Also, say we measure changes daily, the distribution for the net change over 2, 3, or 20 days is a normal distribution with 2, 3 or 20 times as much variance. Std Dev = Square root of Variance. So, a distribution for net change of a stock with Std Dev of 4 over 20 days is a normal distribution with Std Dev of (4 * sqrt(20)).

    Every stock possesses a Brownian Motion model that reflects the probability it will change a certain amount, and that model can be converted back to a Perfect Normal Brownian Motion if we divide it by Std Dev of the stock times Sqrt(# of periods past).

    (1)

    Now that we established that, we can begin to construct to Black-Scholes probability measure.

    Say I want the probability that a stock's change over a period of time starting at t = 0 will be greater than a certain value, say the strike of a call, K, minus the current stock price, S(0).

    (2)

    Mind that S(0) is the current stock price and S(NYSE:T) is the price at expiry of the option.

    By standardizing both sides with the denominator, we can say that the Probability that S(T) will be above K is the Cumulative Dist. Function of the right side of equation (2) such that the right side represents the number of z-scores in the CDF.

    With that concept in mind, let us throw the rest of the ingredients in. Black-Scholes uses the no-arbitrage argument that a rational investor can use the money he saves on stocks (by buying options) to buy bonds at the risk-free rate, r. So options prices must deteriorate at that rate, in order to mimic this behavior in our model, we must imagine that the stock increases by at that rate.

    Current Black-Scholes uses the assumption that stocks deteriorate at a rate equal to their percent dividend yield, q. This q will act opposite to r.

    The final assumption is that stocks have lognormal, not just normal, returns. This means that, while a normal dist. with Std Dev of 10% will have 1st, 2nd, and 3rd z-scores at 10%, 20%, and 30%, a lognormal dist with the same Std Dev will have 1st, 2nd, 3rd and 4th and so on z-scores at 10%, 20%, 40%, and 80%. It also prevents the stock price from becoming negative, because logarithms of negative numbers don't exist. As a consequence, the stock will drift downward at a rate of half of the variance. For example, because 90% * 110% = 99%.

    I will multiply equation (2) by -1 and add these ingredients.

    (click to enlarge)

    (3)

    Mind that Log(S(0)) - Log(NYSE:K) equals Log(S(0)/K).

    This gives us the term commonly referred to as d-two, the right side of the equation. The probability that equation (3) is true at the expiry of the option is then equal to the Cumulative Dist Function for z-score d-two.

    N(x) is the CDF for x amount of z-scores.

    A call option pays the difference between the stock price and strike at expiry if S(T) - K > 0, otherwise it pays expires worthless.

    (4)

    Remember before that we assumed the Stock would decrease at the rate of the dividend yield, q, and the an option would increase value over longer times because investors can save more money on stocks by buying options and T-Bills at rate, r. Tooling the equation such that this holds:

    (5)

    Where P is the number of periods until expiry.

    Equation (5) was for sake of building the equation since of course the call is at expiry, so P is zero. To remove the time constraint under Black-Scholes measures, we insert our probabilities.

    (6)

    We multiply the stock price by the Cumulative Dist. Function of d-one instead of d-two because weighing a variable, as opposed to a constant, K, in our probability measure adds another variance to every period.

    That is all! I will write an article in the near future about how the measure of total randomness affects the outputs of the Black-Scholes and how we can measure its error to give us actionable information when trading.

    Words of Caution: The model is intended for European Options but sees just as much use in American Options because it is still applicable. If requested, I will write on why that is.

    In other sources, you will find that capital T and lower case t are used to denote expiry time and current time, respectively. I use # of periods in this demonstration, but there is no difference.

    In application, make sure all of your time units are the same length. If you use annual T-Bill returns, make sure that your periods are in fractions of a year e.g. 100 days = 100/365.

    Disclosure: I have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours. I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it (other than from Seeking Alpha). I have no business relationship with any company whose stock is mentioned in this article.

    Themes: options
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