Black Scholes Model: A Formula That Changed The Stock Market

    It is amazing how much the existence of mankind depends on mathematics, if one simple formula can give rise to a global financial crisis.



    The BBC published an article on a formula that has changed the stock market and is to blame for the current financial crisis. We are talking about the Black-Scholes model , which is used to evaluate derivatives and equity of financial companies.

    The Black-Scholes mathematical model, introduced in the 70s, gave birth to a new financial system based on the trading of options, futures and derivatives. There was nothing from the old classic stock markets in this new system. The phenomenal success and widespread use of the formula led to the fact that Myron Scholes received the Nobel Prize in Economics in 1997 "for a new method for determining the value of derivative securities."

    Generally speaking, the first futures began to be applied in trade in the 17th century, on the Japanese rice exchange. Traders then began to conclude futures transactions, that is, to set the price of goods, which will be delivered in the future.

    By the 20th century, not only futures, but also options were in use on American commodity exchanges - the same agreement on the price for the future, but without an obligation to make a purchase. Options bought for "insurance" against a sharp increase in prices. Over time, traders had a desire to resell these options, which was difficult, because no one could answer the question: how much are these securities?

    For example, how much does a $ 100 annual rice option cost at the beginning of the year when the price of rice is $ 90, or a month before the option expires when the price of rice is the same $ 90? Unknown It brought to light the revolutionary Black-Scholes model, which takes into account market volatility.

    Call option price:
    where





    put option price:


    - the current value of the call option at time t before the expiration of the option;
    - current price of the underlying share;
    - the probability that the deviation will be less in the conditions of the standard normal distribution;
    - exercise price of the option;
    - risk-free interest rate;
    - time until the expiration of the option term (option period);
    - volatility (the square root of the variance) of the underlying stock.

    Stanford University Finance Professor Myron Scholes has been passionate about finance since childhood. While still young, he persuaded his mother to open an account so that he could trade on the stock market. At 27, he got a job at MIT and, together with his colleague Fischer Black, took up the puzzle of pricing options seriously. As already mentioned, the key to the solution was accounting directly in the formula for the limit of market volatility.

    Myron Scholism says that after a year and a half of work on the formula, they saw elements of options in all objects of the surrounding world.

    To the surprise of the authors themselves, the Black-Scholes model began to be used everywhere: in 2007, the volume of trade in derivatives in the world exceeded $ 1 quadrillion, which is ten times the cost of goods produced throughout the history of human civilization. How it all ended is well known.

    An unexpected change in market volatility has led to unpleasant consequences for financial markets. Now, some experts call this mathematical model a “dangerous invention,” which has oversimplified such a complex thing as asset valuation. The 1998 crisis showed that a strong change in volatility occurs more often than expected, and therefore all assets will have to be reevaluated with new ratios. Roughly speaking, the swollen bubble of the global economy needs to be shrunk back, even if it even means a long recession for developed countries. And all because of an oversimplified mathematical model.

    By the way, Knowles’s hedge fund Long-Term Capital Management collapsed in September 1998, less than a year after it received the Nobel Prize. The fund lost four billion dollars in Russia by joining the GKO financial pyramid built by the Russian government.

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