It is still famous after all these years. And still as beautiful. I’m referring to the Black-Scholes equation for the value or price of a call and put option on a stock.
A Brief History
The foundations of Mathematical Finance is the theory of stochastic process linked to such luminaries as Louis Bachelier (French), Norbert Weiner (American) and Kiyoshi Ito (Japanese). Its moment of glory came in 1973 when Fischer Black and Myron Scholes published a formula for the ‘fair price’ of call and put options on stocks. At that time, Black was an MIT applied mathematician with a strong interest in economics and finance, and Scholes was a professor of finance at the University of Chicago.
A stock option is a derivative product that “rides” on the value of a stock. A call option gives the buyer the right (but not the obligation) to buy the stock at a pre-determined price on a pre-determined date in the future, while a put option gives the buyer the right but not the obligation to sell the stock at a pre-determined price on a pre-determined date. Calls (puts) are useful for hedging against anticipated increaes (decreases). Of course, they can also be used to speculate on stock price movements.
The pre-determined price of a call or put option is known as the strike or exercise price. By a quirk of terminology, options which can only be exercised on one date (the exercise or expiration date, denoted by T) are called European options. The original Black-Sholes formula was meant this type of options.
Black and Scholes discovered their pricing formula using a partial differential equation, the form of which rests on two key assumptions. The first assumption is that the stock price jumps around at very small (infinitesimal) time intervals in the manner of a geometric Brownian motion. Roughly speaking, a Brownian motion particle behaves like the motion of pollen gains suspended in water, a phenomenon first documented by the English botanist, Robert Brown in 1827). A Brownian motion is also known as a Wiener processes, after Norbert Weiner (1894 – 1994) who proved it mathematical properties.
The second assumption is that a riskless portfolio can be formed by combining a short position in one option and a long position in certain number of shares of the stock at every point in time. This portfolio is called a delta-hedged portfolio. Importantly, using Ito’s lemma, an important result in stochastic theory due to Kiyoshi Ito (1915 – 2008), they showed the value of a delta-hedged portfolio has zero risk. Hence, its rate of return is exactly equal to that of a risk-free asset such as Treausry bills.
With the two key assumptions, Black and Scholes derived the following partial differential equation:
where V is the price of the option (as a function of two variables: the stock price S and time t), r is the risk-free interest rate and σ is the volatility of the log returns of the underlying stock.
It is helpful to rewrite the above equation as follows:
Then the left side consists of a “time-decay” term representing the change in the price of the option V due to time t increasing and a convexity term capturing how the option’s value changes relative to the price of the stock (the term with the second derivative). The right hand side representsthe risk-free return from a long position in the optionand a short position consisting of ∂V/∂S shares of the stock. To simplify notations, express this equation in terms of the ‘greeks’.
This equation says that the riskless return over any infinitesimal time interval (right-hand side terms) can be expressed as the sum of theta and a term incorporating gamma. For a European option, theta is typically negative, reflecting the loss in value due to having less time for exercising the option, while gamma is positive as the gamma term reflects the gains from holding the option. Over any infinitesimal time interval, the loss from theta and the gain from the gamma term cancels out, so the result is a riskfree rate of return.
The celebrated Black-Scholes formula for an option’s price is obtained by solving the above partial differential equation (which, interestingly is related to the heat equation in physics!). The solution requires that certain boundary conditions be satisfied. Conveniently, those boundary conditions are exactly the option value at the expiration date, T. They can be written as:
Where C is the call option price and P is the put option price, and T is the expiration date. E is just a reminder that we are dealing with European options.
Black and Scholes showed that the functional form of the analytic solution to their derived partial differential equation with the mentioned boundary conditions of a European call option is:
This formula gives the price of European call options for a non-dividend-paying stock. The inputs going into the formula are S = price of security, T = date of expiration, t = current date, X = exercise price, r = risk-free interest rate and σ = volatility (standard deviation of the underlying asset). The function N(・) represents the cumulative distribution function for a normal or Gaussian distribution and may be thought of as ‘the probability that a random variable is less or equal to its input (i.e. d₁ and d₂) for a normal distribution’. Being a probability, the of value N(・) in other words will always be between 0 ≤ N(・) ≤ 1. The inputs d₁ and d₂ are given by:
Intuitively, the two terms in the sum given by the Black-Scholes formula may be thought of as ‘the current price of the stock weighted by the probability that you will exercise your option to buy the stock’ minus ‘the discounted price of exercising the option weighted by the probability that you will exercise the option’. In other words, the two terms in the formula captures ‘what you are going to get’ minus ‘what you are going to pay’.
For a European put option, the equivalent functional form is:
The Black-Scholes formula is a tour-de-force in intellectual thinking, for behind the complicated is the simple thought experiment: can we form a portfolio consisting of long and short positions in the option and the underlying stock such that this portfolio is in effect has the same return over time and thus, risk-free? It is this ingenious thought experiment which led to the exact form of the partial differential equation that allowed Black and Scholes to derive their formula.
As luck would have it, Black and Scholes published their famous paper in the same year (1973) the Chicago Board of Trade (CBOT) introduced the trading of stock options. Fischer Black would surely have received the Nobel Prize in Economic Science were if not for the fact that he died in 1995, two years before the Prize was awarded to Myron Scholes and the mathematical economist, Robert C. Merton, another pioneer in continuous-time finance and modern option pricing.
Further study
For more details on the derivation of the Black-Scholes partial differential equation, see the Wikipedia entry at https://en.wikipedia.org/wiki/Black%E2%80%93Scholes_model