May 29

Intuition behind Black-Scholes Merton - 6 min screencast

by David Harper, CFA, FRM, CIPM

In six minutes, I try to explain the intuition behind the Black-Scholes-Merton. The sequence as follows:

We write a European call and seek to hedge with a portfolio that pays the same

Start with replicating portfolio: long delta (fractional) share of stock + short bond (future cash)
How much bond to be short? N(d2) gives the probability of the option being exercised; i.e., probability that option expires in the money. And we want bond face value = strike price. So, short position = N(d2)K. But we want the present value, so continuously compounded at rate of (r) over period of (T) gives = N(d2)K*EXP[(-r)(T)]
How much stock to be long? N(d1) = delta of a European call. So, N(d1)(Stock).
Black-Scholes is the portfolio that combines both: N(d1)(S0) - N(d2)(K)EXP[(-r)(T)]

If it helps, that is not quite how I remember it. I remember it by starting with the minimum value which is S-(K)EXP[(-r)(T)]. This is the lower bound on the European call. Then blend in the N(d1) and N(d2) which, as a sort of mnemonic, is like increasing the value of the call to account for volatility. So, in words, that is: option = minimum value plus volatility.

Here is the screencast: