Is the mathematical Constant "pi" hidden in the Black-Scholes Equation?
May 21, 2011
In an interview (allegedly, a Goldman Sachs interview for the position of a trainee trader) the interviewer asked the interviewee if the mathematical constant (pi) is hidden in the famous Black-Scholes equation for pricing call options.
The answer is "yes". This was shown in a seminal result by Brenner and Subrahmanyam in 1988 which has now come to be known as the Brenner-Subrahmanyam approximation.
There exists a nice (and very simple) formula for at the money (ATM) or at the money forward (ATMF) call optionswhich traders (mostly in the FX options market) use from time to time to get a handle on the on the price of a call option on an underlying. This approximation works on a non-dividend paying stock(or a currency pair where the strike is equal to the forward) and when the interest rate is very low (almost zero).
Alternatively, this approximation for ATM option exists when for a non-dividend paying equity we define at the money level as when the spot is equal to the discounted strike, i.e. And for an FX pair, we define ATMF as the strike equal to the forward, i.e.
For an ATM option on a non-dividend paying stock, with zero or near zero interest rate, we have and the Black-Scholes formula becomes:
In the above formula, and are given by:
For, and we get, .
If we make use of the above expressions for and , and the Taylor Series expansion of around 0 as shown below
We would get the expression for ATM Call option as:
Since, , we get the approximation for the ATM (or ATMF) call option price as:
This is the famous Brenner-Subrahmanyam (1988) approximation to the Black-Scholes formula and is often used by traders to get ATM (or ATMF) call prices or implied volatilities under the assumption of zero interest rates.
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