Statistics Black-Scholes Model

T. Dhasaratharaman*

Statistician, Kauvery Hospital Cantonment, Trichy

*Correspondence: dhasa.cst@kauveryhospital.com

Background

The Black Scholes model is a mathematical model to check price variation over time of financial instruments such as stocks, which can be used to compute the price of the European call option. This model assumes that the price of assets, which are heavily traded, follows a geometric Brownian motion having a constant drift and volatility. In the case of a stock option, the Black Scholes model incorporates the constant price variation of the underlying stock, the time value of money, the strike price of the option and it is time to expiry.

The Black Scholes Model was developed in 1973 by Fisher Black, Robert Merton and Myron Scholes and it is still widely used in European financial markets. It provides one of the best ways to determine fair prices of options.

Input

The Black Scholes model requires five inputs.

  1. Strike price of an option
  2. Current stock price
  3. Time to expiry
  4. Risk-free rate
  5. Volatility

Assumption

The Black Scholes model assumes the following points.

  1. Stock prices follow a lognormal distribution.
  2. Asset prices cannot be negative.
  3. No transaction cost or tax.
  4. Risk-free interest rate is constant for all maturities.
  5. Short selling of securities with the use of proceeds is permitted.
  6. No riskless arbitrage opportunity present.

Formula: C= SN (D1) Ke-rtN(D2)

Here,

  1. d1=lnStK+r+22tt,
  2. d2 = d1 t,
  3. C – call option price,
  4. S= current stock price or price of the underlying security,
  5. K= strike price,
  6. r = risk-free interest rate,
  7. t = time to maturity,
  8. N= normal distribution,

Limitation

The Black Scholes model has the following limitations.

  1. Only applicable to Indian options as American options could be exercised before their expiry.
  2. Constant dividends and constant risk-free rates may not be realistic.
  3. Volatility may fluctuate with the level of supply and demand of option thus being constant may not be true

Example

Step I: To determine the historical volatility, daily log returns have been calculated by using the moving average method.

Daily return = ln (today’s closing price/yesterday’s closing price)

Daily Standard deviation (SD) = (Variance of daily returns) 0.5

Historical volatility = Daily SD (250) 0.5

(250 trading days in a year is taken for the above calculation purpose)

Step II: To derive the fair value of call and put options of single strike prices, first we collect all required data in the Black formula from the NSE and then apply them in the BSOPM. The next action is to determine the variations between the model value and the actual market prices.

Step III: The last step is the comparison of the fair option premium with the actual price of the option premium.

Monthly log-returns of the corresponding scrips have been used to find out the historical volatility:

Monthly return = ln [(this month’s closing price)/(last month’s closing price)]

Volatility = Standard deviation of the monthly returns

Of note, 7.4 per cent is the risk-free rate of return, which has been used in this study. This Rf is the current yield on 10-year government bonds issued by the Indian Government. The time to maturity is calculated as the fractional value of the number of days remaining to the maturity date. NSC and BSE websites are referred for collecting the data, i.e. spot prices of the different stocks. BSOPM has been used then to determine the call and put option fair price using the single strike price of all the stocks. The following hypothesis has been framed and a paired sample test has been conducted to derive whether there is a significant difference between BSOPM price and real market price.

Null hypothesis (H0): There is no significant difference between BSOPM prices and actual market prices.

Alternate hypothesis (Ha): There is a significant difference between BSOPM prices and actual market prices at a 95 per cent level of confidence.

If P-value 0.05, then the null hypothesis is accepted.

 

Conclusion

A total of 60 hypotheses are framed (30 for call option contracts and 30 for put option contracts) and tested using the paired sample t-test. The paired t-test compares the means and standard deviations of the two series of numbers and determines if there is any significant difference between the two series of numbers. The following stocks are chosen for the analysis.

Dhasaratharaman

Mr. T. Dhasaratharaman

Assistant Manager – Statistician

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