*The Business Professor*, updated March 25, 2019, last accessed August 14, 2020, https://thebusinessprofessor.com/lesson/black-scholes-formula-definition/.

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**Black – Scholes Formula – Definition **

The Black-Scholes is a formula also known as Black-Scholes-Merton formula. The economists used it the first time for option pricing. It basically estimates a theoretical value of options in European-style with the help of current stock prices, the option’s strike price, expected dividends, expected interest rates, expected volatility and expiry time.

**A Little More on What is the Black Scholes Method**

There were 3 economists who introduced this formula, namely Robert Merton, Fischer Black and Myron Scholes. It is widely used and most popular model of options pricing. In 1973, their research paper published with the title ‘The Pricing of Options & Corporate Liabilities’. They introduced it in this publication of the ‘Journal of Political Economy’.

Black left the world 2 years before Merton and Scholes got the Nobel Prize of 1997 in the subject of Economics for their outstanding work. They brought forth a new way to find the derivatives’ value. The Nobel Prize is generally not awarded after death. However, Black’s contribution was highly acknowledged by the Nobel Committee regarding the Black Scholes Formula.

There are specific assumptions for the Black Scholes Model, i.e.

- The option is only European. It can just be exercised at expiry.
- During the lifespan of the option, no dividend will be paid out.
- There is market efficiency, means the movement of the markets is unpredictable.
- To buy an option, transaction costs will not be applied at all.
- The rate is risk-free. This rate and the volatility are constant and known.
- There is a normal distribution of the returns of the model.

Note: The original model of Black Scholes did not take into consideration the impacts of dividends paid in the lifespan of the option. The company adopts the model frequently for dividends to account by finding the date value of the ex-dividend for the underlying stock.

**Black-Scholes Formula**

The Black Scholes Model considers multiple variables, i.e.

- the present underlying price
- Price of options strike
- Time till expiry, which is shown being a yearly percentage.
- Implied volatility
- Interest rates which are risk-free.

The model has 2 part. The 1st part is SN(d1) multiplies the price with call premium variation in response to the underlying price variation. So, this part explains the expected benefit of buying the underlying outright. The 2nd part N(d2)K*e**-rt* gives the present value of making payment of the exercise price in expiry. The point to be noted is that application of the BS model (Black Scholes) is only on the European options which we can exercise on expiry day only. To calculate the value of the option, subtract both parts as given in the equation.

The maths part of the formula is tricky and may be intimidating. Luckily, one does not have to be an expert in mathematics for using this model one’s own strategy.

Options investors can access many of the options calculators online. Several commercial platforms present analysis instruments for robust options, e.g. spreadsheets and indicators that make the calculations and give the price values of the options as output. Fig 5 shows a sample calculator for the Black Scholes Model. The user provides 5 variables as input, i.e. stock price, volatility, strike price, interest rate (risk-free) and time in days. To show output, there’s written ‘Get a quote’. Simply click on it and the results will be before you.

The formula of the Black Scholes Merton Model estimates only European call options, mainly for equity (exercising on expiry date only). It integrates factors, including price volatility of the underlying stock, the relation in present price and the exercise price of the option, expected interest rates, expiry time of the option and expected dividends.

The mathematicians of the United States namely Fischer Black & Myron Scholes developed the complicated algorithm of BS Formula in 1973. Afterwards, Robert Martin modified it. The researchers have continuously improved this algorithm, i.e. Garman Kohlhagen, Barone Adesi and Whaley and Rubinstein, Cox and Ross.

**References of The Black Scholes Formula**

- http://www.businessdictionary.com/definition/Black-Scholes-option-pricing-model.html
- https://www.investopedia.com/university/options-pricing/black-scholes-model.asp
- https://en.wikipedia.org/wiki/Black%E2%80%93Scholes_model
- https://economictimes.indiatimes.com/definition/black-scholes-model

**Academic Research on the Black Scholes Formula **

**Pricing **European currency **options**: A comparison of the modified **Black**–**Scholes model **and a random variance **model**, **Chesney, M., & Scott, L. (1989). ***Journal of Financial and Quantitative Analysis***, ***24***(3), 267-284.** The authors use the modified form of the BS model and option pricing method of the random variance in order to analyse the prices of ECO (European Currency Options) traded in Geneva. The options that we cannot exercise early are call & puts on the exchange rate of Dollar/Swiss franc. The statistical analysis shows that the model is biased in the perspective of the strike price, volatility and maturity time.

An empirical examination of the **Black**‐**Scholes **call **option pricing model**, **MacBeth, J. D., & Merville, L. J. (1979). ***The Journal of Finance***, ***34***(5), 1173-1186. **This paper was published after 6 years of the development of Black Scholes Formula. The authors make a statistical analysis of this pricing model of the call option.

Tests of the **Black**‐**Scholes **and Cox call **option valuation models**, **MacBeth, J. D., & Merville, L. J. (1980). ***The Journal of Finance***, ***35***(2), 285-301. **In this research, the authors perform several tests on the BS and Cox model of call option valuation.

The GARCH **option pricing model**, **Duan, J. C. (1995). ***Mathematical finance***, ***5***(1), 13-32. ** The author creates a model of option pricing and its related delta formula with respect to the process of GARCH asset return (Generalized Autoregressive Conditional Heteroskedastic). The research uses the LRNVR (Locally Risk-Neutral Valuation Relationship). It has been depicted under specific assumptions and preferences. It reflects the changes in the volatility of the asset parsimoniously. It explains well the systemic biases linked with the BS model.

**Pricing Black**–**Scholes options **with correlated credit risk, **Klein, P. (1996). ***Journal of Banking & Finance***, ***20***(7), 1211-1229.** This paper offers an improved version of pricing options for vulnerable BS model with the appropriate assumptions regarding business concepts. They derive an analytical pricing method that creates a correlation in the option’s asset and credit risk as well as liabilities. The ratio of nominal claims is endogenous to the formula. The author clears the idea with examples.

**Valuation **of American call **options **on dividend-paying stocks: Empirical tests, **Whaley, R. E. (1982). ***Journal of Financial Economics***, ***10***(1), 29-58. ** This study evaluates the pricing function of the equation of valuation for the US call options on securities with dividends that are known. Then, it makes its comparison with 2 approximation methods. The authors have developed a new evaluation method for the efficiency of the options market and they have tested it as well.

The calculation of implied variances from the **Black**‐**Scholes model**: A note, **Manaster, S., & Koehler, G. (1982). ***The Journal of Finance***, ***37***(1), 227-230. **This paper explains the Black Scholes formula to calculate the implied variances in an easy and quick way.

**Pricing **warrants: An empirical study of the **Black**‐**Scholes model **and its alternatives, **Lauterbach, B., & Schultz, P. (1990). ***The Journal of Finance***, ***45***(4), 1181-1209. ** This research takes a sample of more than twenty-five thousand warrant prices daily using warrant pricing method to statistically investigate potential issues. Black and Scholes presented this model being an extended call option formula. One issue is especially important. The assumption of constant variance is biased. The author shows more correct expected price with the help of constant elasticity of this model.

The information content of **option prices **and a test of market efficiency, **Chiras, D. P., & Manaster, S. (1978). ***Journal of Financial Economics***, ***6***(2-3), 213-234. **Merton generalized BS option pricing formula for payments of dividends. It estimates implied variances of expected stock returns. These are better for prediction instead of the ones got from historical information of stock price. A trading technique exploits the data of the implied variances. It produces fairly high returns.

On valuing American call **options **with the **Black**‐**Scholes **European **formula**, **Geske, R., & Roll, R. (1984). ***The Journal of Finance***, ***39***(2), 443-455. **Statistical papers on option pricing show systematic differences in values and market prices created by BS Model. These are relevant to the maturity time, variance and exercise price. The American variant of the BS model can elaborate these biases.

A contingent-claims **valuation **of convertible securities, **Ingersoll Jr, J. E. (1977). ***Journal of Financial Economics***, ***4***(3), 289-321. **This paper evaluates the convertible bonds pricing and the same for the preferred stocks. The criteria of dominance determine the optimal strategies for conversion and call of these stocks. The traders use methods of Black Scholes formula for pricing the convertible stocks as a collection of claims on the company overall.