Applicability of Black Scholes model on Nifty 50 call options

 

Amit Garg*

Indian Statistical Institute, New Delhi – 110016

 

ABSTRACT:

Options trading volume is increasing by the day and Black Scholes model is most widely used for options pricing. As a step towards checking efficiency of Black Scholes model in Indian market, this paper attempted to apply this model to calculate theoretical price of Nifty 50 call options and found that there is statistically significant difference between theoretical prices calculated and market prices of Nifty 50 call options.

 

 

 

INTRODUCTION:

Nifty 50 is India’s one of the most actively traded index, owned and managed by NSE Indices Limited¹. A call option is a contract between buyer and seller such that buyer has right but no obligation to buy the underlying asset on a later date, at the price agreed upon today, underlying asset could be stock, bond, currency etc. The Black Scholes model is widely used for calculating theoretical prices of options.

 

The objective of this paper is to check applicability of Black Scholes model on Nifty 50 call options by looking for statistical difference between BS call prices calculated by the model and actual market prices.

 

DATA AND METHODOLOGY:

Data:

This study considers historical data of 1379 call options written on underlying security Nifty 50 during April 2018 to March 2019. Only the very highly traded options are considered. This data, along with daily Nifty 50 Index prices during the same period is taken from the website www.nse.com. Overnight MIBOR are taken from website www.fbil.org.in.

 

Hypotheses:

H₀: There is no difference between market call price and BS model call price.

H₁: There is difference between market call price and BS model call price.

 

Assumptions:

(a)     Overnight MIBOR are risk free interest rates.

(b)    Risk free interest rate during weekends and holidays are same as previous working days.

(c)     Historical closing prices of Nifty 50 Index and Nifty 50 call options are their market prices.

(d)    Data obtained from both the sources are reliable.

 

The Black Scholes model:

The most famous solution to Black Scholes differential equation is as follows²: c = S₀ N (d₁) -Ke ^-rTN (d₂)

 

where, c is BS model call price, S₀ is Index price at time 0, K is strike price of the option, r is continuously compounded risk free interest rate, T is time to maturity of the option in years,

 

, and is volatility per annum. This formula is used to calculate prices of 1379 call options.

 

 

Time to maturity:

Total number of days to maturity from historical data of calls options including weekends and holidays, is divided by 365 to obtain T.

 

Volatility:

Volatility is emperically calculated as³

where,  number of trading days in the year  248 in this study, and

 such that and is price of Index Nifty 50 on i^th trading day.

 

Continuously compounded risk free interest rate:

Considering calculation of price of some call option, r is calculated as where n is number of days to maturity, including weekends and holidays, and R_j is overnight MIBOR on (n-j) days before maturity.

 

RESULTS:

Paired sample t-test conducted on market prices and BS call prices. t-statistic obtained is 2.050953 and p-value obtained is 0.04046, since, p-value is less than 0.05, we can safely reject null hypothesis at 5% level of significance and conclude that there is statistically significant difference actual market prices of Nifty 50 call options and prices which are calculated by BS model.

 

DISCUSSIONS:

There are some limitations to this study as choice of risk free interest rates is very subjective, also, opening, high or low prices of Index and options could be used as market prices instead of closing prices. Still, results are in accordance with most of the studies which conclude that Black Scholes exhibits pricing errors in Indian market^4,5.

 

ABBREVIATIONS:

MIBOR: Mumbai Inter Bank Offered Rates BS: Black Scholes

 

REFERENCES:

1.        Available from URL: https://nseindia.com/products/ content/equities/indices/nifty_50.htm.

2.        John C. Hull. The Black – Scholes – Merton model. In John C. Hull’s. Options, Futures and Other Derivatives, Edited by Donna Battista. Pearson, England. 2012; 8^th ed: pp. 313-315.

3.        John C. Hull. The Black – Scholes – Merton model. In John C. Hull’s. Options, Futures and Other Derivatives, Edited by Donna Battista. Pearson, England. 2012; 8^th ed: pp. 304-307.

4.        Kumar Rajesh and Agrawal Rachna. A Close Look into Black-Scholes INDEX Nifty 50 Put Option Pricing Model: Evidence from Indian National Stock Exchange. Asian Journal of Management. 2018: 9(1); 407-412.

5.        Sharma Manish and Arora Kapil. Study of Relevance of Black-Scholes Model in Indian Stock Option Market. IJARIIE International Journal. 2015: 1(4); 324-334.=

 

 

 

Received on 17.07.2019            Modified on 21.08.2019

Accepted on 18.09.2019           ©A&V Publications All right reserved