This article will explain the Black-Scholes formula in simple words. The Black-Scholes model is a mathematical model of the dynamics of a financial market containing derivative investment instruments.
From the partial differential equation in the model (known as the Black-Scholes equation), the Black-Scholes formula can be derived. It gives a theoretical assessment of the price of European-style options and shows that the option has a unique price regardless of the risk of the security and its expected return (instead of replacing the expected yield of the security with a risk-neutral rate).
The formula led to a boom in options trading and ensured the mathematical legitimacy of the Chicago Options Exchange and other options markets around the world. It is widely used, although often with adjustments and corrections, by the options market participants. In the pictures in this article, you can see examples of the Black-Scholes formula.
History and Essence
Based on work previously developed by market researchers and practitioners such as Louis Bachelier, Sheen Kassuf and Ed Thorpe, Fisher Black and Miron Scholes in the late 1960s, demonstrated that a dynamic portfolio review eliminates the expected return of security.
In 1970, after they tried to apply the formula to the markets and suffered financial losses due to the lack of risk management in their professions, they decided to concentrate in their field, the academic environment. After three years of effort, the formula, named after their release, was finally published in 1973 in an article entitled “Pricing Options and Corporate Commitments” in the Journal of Political Economy. Robert S. Merton was the first to publish an article expanding the mathematical understanding of the option pricing model and coined the term “Black-Scholes pricing model”.
Merton and Scholes received the 1997 Nobel Memorial Prize in Economic Sciences for their work, a committee citing the fact that they opened a dynamic revision that was not risk-sensitive, as a breakthrough that separates the option from basic security risk. Despite the fact that he did not receive the award due to his death in 1995, the Swedish academician mentioned Black as a participant. In the picture below you can see a typical Black Scholes calculation formula.
Options
The main idea of this model is to hedge an option by correctly buying and selling the underlying asset and, as a result, eliminating the risk. This type of insurance is called “continuously updated delta hedging.” It is the foundation of more complex strategies, such as those used by investment banks and hedge funds.
Management of risks
The assumptions of the model were softened and generalized in many directions, which led to the many models that are currently used in pricing derivative instruments and risk management. It is the understanding of the model, as shown in the Black-Scholes formula, that is often used by market participants, in contrast to actual prices. This information includes the absence of arbitration boundaries and risk-free pricing (due to ongoing review). In addition, the Black-Scholes equation, a partial differential equation that determines the price of an option, allows pricing using numerical methods when an explicit formula is not possible.
Volatility
The Black-Scholes formula has only one parameter that cannot be directly observed in the market: the average future volatility of the underlying asset, although it can be found at the price of other options. Since the value of the parameter (whether it is “put” or “challenge”) increases in this parameter, it can be inverted to obtain a “volatility surface”, which is then used to calibrate other models, for example, over-the-counter derivatives.
Based on these assumptions, suppose that derivatives are also traded on this market. We indicate that this security will have a certain payment on a certain date in the future, depending on the value accepted by the stock before that date. Surprisingly, the price of a derivative is currently fully determined, although we do not know which way the price of stocks will go in the future.
For a special case of the European call or put option, Black and Scholes showed that it is possible to create a hedged position consisting of a long position in the stock and a short position in the option, the value of which will not depend on the price of the stock. Their dynamic hedging strategy led to a partial differential equation that determined the price of an option. Its solution is given by the Black-Scholes formula.
Difference of terms
The Black Scholes formula for excel can be interpreted by first breaking the call option into the difference between the two binary options. The call option exchanges cash for an asset upon expiration, while a call asset with or without an asset simply gives an asset (no cash in exchange), and a call with a bank transfer simply returns money (without exchanging an asset ) The Black-Scholes formula for an option is the difference of two terms, and these two terms are equal to the value of the binary call options. These binary options are sold much less frequently than vanilla options, but they are easier to parse.
In practice, some sensitivity values are usually given in abbreviated terms to fit the scale of probable parameter changes. For example, rho is often reported divided by 10,000 (change by 1 basis point), vega by 100 (change by 1 volume point) and theta by 365 or 252 (1-day decline based on either calendar days or trading days in a year )
The model described above can be expanded for variable (but deterministic) rates and volatility. The model can also be used to evaluate European options on dividend payment instruments. In this case, closed solutions are available if the dividend is a known proportion of the share price. American stock options and stock options paying a known cash dividend (in the short term, more realistic than a proportional dividend) are more difficult to evaluate, and a choice of decision methods (for example, grids and grids) is available.
Approximation
Useful approximation: although volatility is not constant, model results often help establish hedges in the correct proportions to minimize risk. Even if the results are not entirely accurate, they serve as a first approximation to which adjustments can be made.
The foundation for more advanced models: The Black-Scholes model is reliable in the sense that it can be adjusted to cope with some of its failures. Instead of considering some parameters (such as volatility or interest rates) as constant, we consider them as variables and, thus, add sources of risk.
This is reflected in the Greeks (changing the option value to change these parameters or is equivalent to the partial derivatives of these variables), and hedging these Greeks reduces the risk caused by the inconsistent nature of these parameters. However, other defects cannot be eliminated by changing the model, in particular, tail risk and liquidity risk, and instead they are managed outside the model, mainly by minimizing these risks and stress testing.
Explicit modeling
Explicit modeling: this function means that instead of assuming a priori volatility and calculating prices from it, you can use the model to determine volatility, which gives the implied volatility of the option at given prices, terms and strike prices. By solving volatility over a specific set of duration and strike prices, you can build a surface of implied volatility.
In this application of the Black-Scholes model, we obtained the transformation of coordinates from the price region to the region of volatility. Instead of specifying option prices in dollars per unit (which are difficult to compare by strikes, durations, and coupon frequency), option prices can be quoted in terms of implied volatility, which leads to trading volatility in option markets.