The Black-Scholes model for options pricing is the starting point understanding how to value derivatives, but the mathematics involved can be intimidating and at this point sophisticated options traders are working with more sophisticated models (and Monte Carlo simulations) anyways. But for anyone interested in the thought process behind the heavy math, Nobel Laureate Myron Scholes explains how to interpret the equation that bears his name in an interview with the CFA Institute’s Larry Cao.
Valuing options by creating an alternate portfolio
Scholes explains that the Black-Scholes model values an option by imagining that there is an alternate portfolio made up of stocks and bonds that perfectly mimics the behavior of the option over a short period of time. “With continuous hedging or replication there is perfect correlation with the between the option and the replicating portfolio,” says Scholes, explaining what is also called the no-arbitrage condition.
Scholes and his research partner Fischer Black showed that it was possible to create these alternate portfolios knowing only the volatility of the stock underlying the option and the interest rate, the work that Scholes was awarded his Nobel Prize for (Black had already passed away). You still have to integrate all those small time periods to get the valuation, which Scholes and Black did for the special case where interest rates and volatility are constant.
Extending Black-Scholes to be more realistic
The earliest form of the Black-Scholes model left out a lot of important real world details (which may be why Warren Buffett has said he doesn’t care for it): it doesn’t account for changing volatility or interest rates, dividend payments, and makes some assumptions about the distribution of returns that seem to work well but can’t be justified from first principles. In the decades since Scholes original paper came out researchers have worked on each of these different fronts, but extending it beyond the European style options that the pair originally studied is tough. He gives the example of the prepayment option on most mortgages, a non-traded option that poses a unique challenge.
“You cannot use the dynamic programming structure. You have to use combinations of forward and backward induction because we don’t know the optimal time to exercise the prepayment options,” says Scholes. He says that extensions to the model like this allow for many more non-traded options (eg executive stock options) to also be valued effectively.