Analysis of Asset Allocation

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Options: Black and Scholes Model
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To derive a fair value for an option's premium, the volatility must be estimated. But also two other statistical assumptions must also be incorporated:

  • The risk free interest rate over the life of the option.

  • If the option is on a yield bearing asset, the dividend or yield  must be known or estimated during the life of the option.

Based on the above, one of the best known model is the Black and Scholes one. Without entering into details, the Black and Scholes model is based on the assumption that that stock price changes follow a normal (Gaussian) curve.

The Black and Scholes model can be used to estimate European Style options. To compute the premium with this model (for non income bearing instruments), you have to follow these steps:


  • N(d) = cumulative normal integral (figure you can find in  a normal table).
  • r = risk free interest rate.
  • S = measure of volatility (see above)
  • X = underlying price.
  • K = Strike price.
  • t = annualized time to expiration.
  • e = exponent.
  • ln = natural logarithm.
  • C = call premium.
  • P = put premium.

First step: compute the volatility (S).

Volatility can be measured different ways. One of these is to compute standard deviation of log percentage change in annualized prices. This is compute on historical data (closing price) of the underlying security.

With Pt = closing price at the date t.

Compute all the R for your time series 

R = Pt+1 / Pt

afterward compute the average of R (sum of R divided by number of R) noted AR.

The standard deviation S is:

Standard deviation should be computed on a one year sample. If you have a shorter time interval, you should adjust it by a time factor. This factor is the square root  of the number of periods in a year (square root of 52 for the weekly time factor).

SDV will be used as S here under.

The volatility can be also measured based on option's premium. This is what is called implied volatility.

You don' know how to compute and use statistical indicators? Visit our Statistics Tutorial.

Second step: compute d1.

d1 = [ln(X / K) + (S2 / 2) t] / [S (t)0.5]

Third step: from d1, compute d2.

d2 = d1 - S (t)0.5

Fourth step: compute the premium for a call C or a put P.

C = [X N(d1) - K N(d2) e-rt

P = K N(-d2) e-rt - X N(-d1)
 with N() value taken from the normal table.

To convert the above formulas for use with income earning assets (bonds, dividend on shares,...), one would need to subtract an expected future yield from the underlying price.

If you have a normal table (you can use the EXCEL function NORMDIST as well), by following the above steps, you can compute the theoretical value of an European style option.

American style options will be studied in the next topic.

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