Dirac Delta function and the Binary Option: A Spike all the way up to the Heavens
Rahul Bhattacharya
June 17, 2011
In the summer of 1992, a young associate,armed with an advanced degree in Math, had joined the FX options trading desk of a large investment bank on Wall Street as an analyst. In those days, it was rare for a trader or a quant (even the term “quant” wasn’t that prevalent) to have a math or a physics degree and in many banks MBAs generally ruled the trading floors. One day, there was some issue with a binary option position of a trader and the associate had to look at his position limits and risk limits to make a report to his boss. The boss was on leave so he had to go and see the head trader that afternoon.
The binary call, which was ATM, was on an FX pair and about to expire in a few hours’ time. The volatility was quite low and delta of the option was around 150%. The head trader asked the associate as to whether the delta was showing the correct value as according to his understanding delta is the hedge ratio and a 150% hedge ratio did not make sense to him. Obviously, the head trader was simply quizzing the associate and trying to test him. It was an extremely boring afternoon and the markets were rather quiet.
The associate, in a flush of excitement, blurted out that mathematically speaking, the delta of the binary is actually the derivative of the Heaviside function and therefore it’s a Dirac delta function. So the delta can be more or even significantly more than 100%. And even though it may seem that the “measure” is breaking down, it isn’t actually because the overall area, a kind of proxy measure, was still one.
The trader asked the associate if he was talking in English? “Heaviside function”, “measure”, “Dirac delta”? Was this bankingor option trading?
The trader then asked the associate to go an explain all this to the CEO of the bank, three floors above and warned him that the CEO was barely familiar with even high school math, let alone Measure Theory and Advanced Functions.
It is not known what happened afterwards. The associate, now the Global Head of FX in another bank will not tell us. But he could never forget what the trader told him that afternoon. The trader’s words, were as if etched in his mind; especially a phrase that he had used to symbolize the Dirac delta function.
This is what the trader had explained that day to the associate (not quoting verbatim):
Consider a rectangle with length “  ” and width “  ” such that the area is equal to one. And also consider that nothing exists in the vicinity of the rectangle, it's a ...void (all measures of area, volume, etc. outside this rectangle are zero). Now, no matter what the value of is, the area of the rectangle will always remain one and everything outside the rectangle will remain in a void, measuring zero. And all payoffs from the given binary option that can ever happen (whether you realize / get that payoff or not) happens within this rectangle. Now make the rectangle stand on its length, such that the length becomes the “base” and the width becomes the “height” on a  plane as shown below. And then keep decreasing the value of ,making the base smaller which will cause the height to increase, since height is inversely proportional to the base. However, no matter how small the base becomes (i.e. how small “ ” becomes) and how spikey or tall the height becomes (i.e. how large “ ” becomes), the area of this rectangle continues to remain one. And everything outside this rectangle continues to remain in a void.
If we now reduce the base of this rectangle to an infinitesimally small value, making it almost zero, then the height will become almost infinite - a tall spike reaching up to the heavens. The rectangle has now collapsed into a gigantically tall spike; simply a line standing on a point. The rectangle has become the Dirac delta function. Thus the Dirac delta function is contained within every rectangle in this universe and in every binary option. In fact, the Dirac delta function is contained within all financial derivatives.
Reference: For more on Dirac delta function and its application in trading binary options, see Nassim Taleb’s excellent book Dynamic Hedging.
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