Black-Scholes Model: The Formula That Changed the Options Market
Have you ever thought about how much we depend on mathematics? Could you imagine that one elementary formula could cause a global financial crisis?
Options were very popular, but for a long time, traders couldn’t clearly understand their ultimate value, which deprived them of the opportunities to make big profits. The breakthrough that took options trading to a new level was the invention of American economists Fisher Black and Myron Scholes: Black-Scholes formula.
The mathematical model of Black-Scholes, introduced in the 1970s, has brought to life a new financial system based on options trading, futures, and derivatives. There was nothing left of the old classic stock markets in that new system.
The main idea of the Black-Scholes option pricing model is to determine the theoretical price of European options. The model implies that if there is a trading of the underlying asset in the market, then the market itself already sets the price of the option.
The fundamental definition of the value of an option was the possible volatility of the stock, in other words, the volatility of its price. Depending on the fluctuation of the asset, its price rises or falls, in a direct proportionality to the value of the option. On the other hand, if the price of the option is known, it is possible to assume the level of expected market volatility.
By the 20th century, the American commodity exchanges were used not only for futures but also options. Options were bought as a “hedge” against the sharp price increase. Later traders had a desire to resell these options, which was rather difficult because no one could answer the question: how much are these securities worth?
The CALL (sell) option price:
The PUT (buy) option price:
Notations:
C (S,t) — current CALL option price at the time before expiration;
S — current price of the underlying stock;
N (x) — probability of deviation reduction under the conditions of the normal standard distribution;
K — option striking price;
r — short-term risk-free interest rate (in percent);
T-t — option period;
σ — volatility of the underlying security (square root of the variance).
On April 28, 2012, the BBC published an article about the formula that changed the stock market and led to the financial crisis. It was about this model, the Black-Scholes model, which is used to evaluate derivatives and equity capital of financial companies.
