Pieces of paper known as derivatives have been blamed for the last decade’s global financial crisis. A derivative derives its value from an asset such as a mortgage. ‘A positive derivative means that the graph is rising/And if it is less than zero, the result is not surprising,’ explains a calculus rhyme. When the house prices collapsed in the United States the value of mortgage derivatives approached zero. Not surprisingly, they became financial weapons of mass destruction. As some wit has said alcohol and calculus don’t mix, were these dodgy derivatives derived over gallons of wine?
Calculus is at the heart of financial derivatives. Calculus can be summed up by two basic ideas, the derivative and the integral. Simply put, the derivative is a way of measuring the change in one quantity in response to change in another quantity (the derivative of the position of a moving car with respect to time is its speed at that particular moment). The integral is the accumulation of an infinite number of tiny bits that make up a whole (the distance a car has travelled when only its speed is known).
Mathematician and songwriter Tom Lehrer explains it harmoniously in ‘The Derivative Song’ (The American Mathematical Monthly, May 1974):
You take a function of x and you call it y,
Take any x-nought that you care to try,
You make a little change and call it delta x,
The corresponding change in y is what you find nex’,
And then you take the quotient and now carefully
Send delta x to zero, and I think you’ll see
That what the limit gives us, if our work all checks,
Is what we call dy/dx,
It’s just dy/dx.
© Surendra Verma 2016