Except for the special cases of formula_30 and formula_31, no closed form expression for this probability distribution is known. The general case can be solved approximately by means of an asymptotic expansion in the parameter formula_32. Under typical market conditions, this parameter is small and the approximate solution is actually quite accurate. Also significantly, this solution has a rather simple functional form, is very easy to implement in computer code, and lends itself well to risk management of large portfolios of options in real time.