Black Scholes option model

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					Notes for use of Black & Scholes option pricing model

The Black & Scholes worksheet is designed to produce four output values:
theoretical values of a call and a put, together with the hedge ratios (delta
values or slopes of the option price curve) corresponding to the call and put

Input numbers include the price of the asset underlying the put and call options,
dividend or interest income on the asset stated as an annual amount, the
option's strike price, the risk-free rate of interest (genarally the rate of interest
on short-term Treasury securities), time to expiration of the option stated as
a decimal proportion of one year, and the standard deviation of returns on the
underlying asset.

In the case of options on futures contracts the dividend and risk-free rate of
interest may be entered as zero, because no dividend or interest is to be
received on the asset before the option's expiration and there is no asset cost
to be discounted. The model may be used for options on common stocks or
bonds which will have dividend cash flows.

The standard deviation of asset returns in the March O.J. example was
found by trial and error. After the other input values were entered, standard
deviation figures were tested until the model generated put and call prices
that matched the actual market prices on Dec. 3, 2008. The result, a standard
deviation of 0.555, is the implied standard deviation (with implied variance
equal to 0.308) for the March 09 O.J. futures contract on Dec. 3, 2008.

For the worksheet model, data entry figures are shown in black, calculation
formula cells are blue, and output formula cells are red.

Many research papers and textbooks contain the Black & Scholes option
pricing formulas. A very clear review of these formulas is found online at

Source of data for futures and option prices: Chicago Mercantile Exchange.
                       March 09 O.J. Futures Theoretical Option Values, Dec 3, 2008

                 Days to expiration                                                                      80
                 Futures price                                                F=                   76.3500                 Black = input
                 Dividend (zero for futures)                                  D=                      0.000                Blue = calculation formulas
                 Adjusted asset price                                         F=                   76.3500                 Red = output formulas
                 Strike prioe                                                 S=                   75.0000
                 Risk-free rate of interest                                   R=                     1.00%                 D and R may be entered as
                 Time to expiration, proportion of year                       T=                       0.22                zero for futures options
                 Standard deviation of asset returns                          st dev =              0.5550
                 Variance of underlying asset returns                         (st dev)^2 =          0.3080                 st dev = value found by trial
                 Call market price               8.65                         Call =                8.6472                 and error so that put and call
                 Put market price                7.35                         Put =                 7.1308                 values approximately match
                 Hedge ratio, call (delta)                                    Hc =                 0.58218                 market prices.
                 Hedge ratio, put (delta)                                     Hp =                -0.52158

 ln(Ps/E) =        0.0178399               d1 = 0.2074962                              -d1 =   -0.2075
  (S^2)/2 =        0.1540125               d2 = -0.0541333                             -d2 = 0.054133
S(T^0.05) =        0.2616295            N(d1) = 0.5821798                           N(-d1) = 0.41782
  e^(-RT) =            0.9978           N(d2) = 0.4784243                           N(-d2) = 0.521576

 Calculation of N(d1) and N(d2) for call option

                           d1 =     0.2074962 standard deviations from the mean

                             y = 0.9541395
                             x = 0.3904459
                            w = 0.5821798
                       N(d1) =                        0.5821798 cumulative probability to d1

                           d2 = -0.0541333 standard deviations from the mean

                           y=       0.9876158
                           x=       0.3983582
                          w=        0.5215757
                       N(d2) =                       0.4784243 cumulative probability to d2


Calculation of N(-d1) and N(-d2) for put option

                         - d1 = -0.2074962 standard deviations from the mean

                            y=      0.9541395
                            x=      0.3904459
                            w=      0.5821798

                      N(- d1) -                       0.4178202 cumulative probability to - d1
                         - d2 = 0.0541333 standard deviations from the mean

                            y=      0.9876158
                            x=      0.3983582
                            w=      0.5215757

                     N(- d2) =                        0.5215757 cumulative probability to - d2