| A call option gives the holder the right, but not the obligation, to buy some | stock at a given exercise-price, on or before some given maturity date . | | The exercise-price depends on the interest-rate, the standard-deviation of the value of the stock, | and the number of months till the maturity-date. | | If the option is European, it can only be used ( exercised ) at the maturity date. | | If the option is American, it can be used at any date up to and including the maturity date. some-stock with price $ some-price has an option maturing in some-number months at exercise price $ some-ex-price the standard deviation ( volatility ) of the value of that-stock is estimated to be some-sigma the current interest rate is some-rate per month that-price / that-ex-price = some-ratio price ratio that-ratio and volatility that-sigma at that-rate for that-number months has d1 some-d1-value for input that-d1-value the cumulative univariate normal distribution value is some-d1-cumulative that-price * that-d1-cumulative = some-worth-part1 that-rate * that-number = some-rate-times-time 2.718 raised to the power that-rate-times-time = some-result that-ex-price * that-result = some-worth-part2a price ratio that-ratio and d1 that-d1-value with volatility that-sigma for that-number months has d2 some-d2 for input that-d2 the cumulative univariate normal distribution value is some-d2-cumulative that-worth-part2a * that-d2-cumulative = some-worth-part2 that-worth-part1 - that-worth-part2 = some-calculated-value that-calculated-value rounded to 3 place(s) after the decimal point is some-value ---------------------------------------------------------------------------------------------------------------------- call that-stock current that-price volatility that-sigma that-number months ex price that-ex-price is worth that-value the log to base 2.718 of some-ratio = some-log-ratio some-rate * some-number = some-rate-times-number that-ratio + that-rate-times-number = some-numerator the square root of that-number = some-sqrt-number some-sigma * that-sqrt-number = some-denominator that-numerator / that-denominator = some-d1-part1 0.5 * that-denominator = some-d1-part2 that-d1-part1 + that-d1-part2 = some-d1-value --------------------------------------------------------------------------------------------------------- price ratio that-ratio and volatility that-sigma at that-rate for that-number months has d1 that-d1-value price ratio some-ratio and volatility some-sigma at some-rate for some-number months has d1 some-d1-value the square root of that-number = some-sqrt-number that-sigma * that-sqrt-number = some-sigma-times-sqrt-number that-d1-value - that-sigma-times-sqrt-number = some-d2 ------------------------------------------------------------------------------------------------------------------ price ratio that-ratio and d1 that-d1-value with volatility that-sigma for that-number months has d2 that-d2 some-input is within range to apply the cumulative univariate normal distribution formula for that-input the value of pre-n is some-pre-n-value that-input is less than 0 1 - that-pre-n-value = some-n-value ---------------------------------------------------------------------------------------- for input that-input the cumulative univariate normal distribution value is that-n-value some-input is within range to apply the cumulative univariate normal distribution formula for that-input the value of pre-n is some-pre-n-value that-input is greater than or equal 0 ------------------------------------------------------------------------------------------- for input that-input the cumulative univariate normal distribution value is that-pre-n-value the cumulative univariate normal distribution with input below some-limit has value some-value some-input is less than that-limit ----------------------------------------------------------------------------------------- for input that-input the cumulative univariate normal distribution value is that-value the cumulative univariate normal distribution with input above some-limit has value some-value some-input is greater than that-limit ----------------------------------------------------------------------------------------- for input that-input the cumulative univariate normal distribution value is that-value the cumulative univariate normal distribution with input below some-lower-limit has value some-low-value the cumulative univariate normal distribution with input above some-upper-limit has value some-hi-value some-input is greater than or equal that-lower-limit that-input is less than or equal that-upper-limit ----------------------------------------------------------------------------------------------------------------- that-input is within range to apply the cumulative univariate normal distribution formula the cumulative univariate normal distribution with input this-type this-limit has value this-value ============================================================================================== below -6 0 above 6 1 b1 equals some-b1-value for numerical approximation to the cumulative univariate normal distribution b2 equals some-b2-value for numerical approximation to the cumulative univariate normal distribution b3 equals some-b3-value for numerical approximation to the cumulative univariate normal distribution b4 equals some-b4-value for numerical approximation to the cumulative univariate normal distribution b5 equals some-b5-value for numerical approximation to the cumulative univariate normal distribution for some-input the value of t is some-tvalue that-b5-value * that-tvalue = some-b5-value-times-t that-b5-value-times-t + that-b4-value = some-b5b4 that-b5b4 * that-tvalue = some-b5b4t that-b5b4t + that-b3-value = some-b5b4tb3 that-b5b4tb3 * that-tvalue = some-b5b4tb3t that-b5b4tb3t + that-b2-value = some-b5b4tb3tb2 that-b5b4tb3tb2 * that-tvalue = some-b5b4tb3tb2t that-b5b4tb3tb2t + that-b1-value = some-b5b4tb3tb2tb1 that-b5b4tb3tb2tb1 * that-tvalue = some-btvalue for that-input the value of b is some-bvalue that-bvalue * that-btvalue = some-bbt 1.0 - that-bbt = some-pre-n-value ----------------------------------------------------- for that-input the value of pre-n is that-pre-n-value the absolute value of some-input is some-abs-input p equals some-pvalue for numerical approximation to the cumulative univariate normal distribution that-abs-input * that-pvalue = some-abs-input-times-pvalue 1 + that-abs-input-times-pvalue = some-1-plus-abs-input-times-pvalue 1 / that-1-plus-abs-input-times-pvalue = some-tvalue ----------------------------------------------------------------------------- for that-input the value of t is that-tvalue c2 equals some-c2-value for numerical approximation to the cumulative univariate normal distribution some-input / 2.0 = some-input-div-2 0 - that-input = some-negated-input that-negated-input * that-input-div-2 = some-exponent 2.718 raised to the power that-exponent = some-result that-c2-value * that-result = some-bvalue ------------------------------------------------------------------------------------------------------ for that-input the value of b is that-bvalue some-input is less than 0 0 - that-input = an-abs-value -------------------------------------------------- the absolute value of that-input is that-abs-value some-input is greater than or equal 0 ---------------------------------------------- the absolute value of that-input is that-input this-constant equals this-value for numerical approximation to the cumulative univariate normal distribution =============================================================================================================== b1 0.31938153 b2 -0.356563782 b3 1.781477937 b4 -1.821255978 b5 1.330274429 p 0.2316419 c2 0.3989423 this-stock with price $ this-price has an option maturing in this-number months at exercise price $ this-exercise-price ===================================================================================================================== XYZ 5.25 10 2.48 the standard deviation ( volatility ) of the value of this-stock is estimated to be this-sigma ============================================================================================ XYZ 0.01 XYZ 0.46 XYZ 2.55 the current interest rate is this-rate per month ================================================ 0.002