Risk Latte - Using Series Expansion to Calculate Option Probabilities

Using Series Expansion to Calculate Option Probabilities

Rahul Bhattacharya
August 30, 2006

In a recent interview for the position of a trainee trader in a prominent Wall Street firm the following question was asked by the Head of Trading (who likes to be called "HOT") to the interviewee:

Assuming interest rates to be negligible what is the probability that a one year at the money (ATM) call option with volatility of 10% will finish in the money?

No Excel spreadsheet was to be used for this and not even a calculator was allowed. The interviewee was asked to calculate the value of the above with just a paper and a pencil. How could such a calculation be done? The trainee a math major, who ultimately got the job, couldn't do it and gave up. His reasoning was that the information was not sufficient to calculate the probability and with the proper values of the stock and strike plus the interest rates he needed to use Excel spreadsheet or an HP calculator to estimate the value of the probability of the option finishing in the money. Basically, he needed to calculate the cumulative normal probability distribution.

Here is a simplified - paper and pencil - algorithm for the same.

Firs of all if the interest rates are negligible (assume zero value) then for an at the money option (call or put does not matter) the value of d(2) is given by:

This is simple to remember. Now, for a call option that the probability that the option will finish in the money at maturity is given by and to estimate this cumulative standard normal density function we need a scientific calculator or an Excel spreadsheet. But this density function can be easily approximated by the Taylor series expansion:

Ignoring higher order terms we get:

The above is also easy to remember. Therefore all we need to calculate the probability of the option finishing in the money, i.e. is the volatility estimate and the term to maturity.

The above can easily be done using a pencil and a paper, provided we remember the short hand formulas. Therefore, the probability that the option will finish in the money is 48%.

Option traders have to be extremely quick on their feet and many a times they need to do calculations in their head or at most on the back of a napkin. Perhaps not any more! But that was the world when some of us were option traders more than a decade ago. There was Lotus 1-2-3 and some models but many a times all we had was a pencil and a used napkin and we had to make quick guesses for approximate values of options. We used to remember many of the short hand formulas in our head.

Of course, the world has now changed and it is all Excel and VBA and models and sophis and Murex and Bloomberg and Kondor+ and what not. All math is done by computers and one can hardly find a pencil on a traders desk.

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