comparative study of black scholes option pricing model and binomial option pricing model

Comparative study of Binomial Option Pricing Model and Black Scholes Option Pricing Model

Comparative study of Binomial Option Pricing Model and Black Scholes Option Pricing Model

PROJECT COMPLETION CERTIFICATE

This is to certify that project titled Comparitive Study of Binomial Option Pricing Model and Black Scholes Option Pricing Model is successfully done by Ms. Supriya Pramod Gunthey in partial fulfillment of her two years full time course Post Graduation Diploma in Management recognized by AICTE through the Prin. L. N. Welingkar Institute of Management Development & Research, Matunga, Mumbai.

This project in general is done under my guidance and I have validated the project conceptually and theoretically but not on duplicity.

___________________________(Signature of Faculty Guide)

Name: ______________________

Date: ______________________

ACKNOWLEDGEMENTS

"Gratitude is not a thing of expression; it is more matter of feeling."

There is always a sense of gratitude which one express towards others for their help and supervision in achieving the goals. This formal piece of acknowledgement is an attempt to express the feeling of gratitude towards people who were helpful to me in successfully completing this project.

First and foremost, I would like to thank our Group Director Prof. Dr. Uday Salunkhe for giving the second year students the time and resources for completion of the final year specialization project.

I would like to thank Prof. Kanu Doshi, Dean of Finance Department for his support and guidance.

I would like to express my sincere gratitude to Prof. Dr. Suyash Bhatt, my mentor, for his help and support. He was always there with his competent guidance and valuable suggestions throughout the pursuance of this research project. I would also like to thank him for his cooperation and for providing me with the helpful inputs which enabled me to complete the project in a hassle free manner.

CONTENTS

Executive Summary11. Introduction22. Literature Review73. Research Methodology104. Data Analysis155. Conclusion496.References & Bibliography507. Appendix51

Executive SummaryOptionsarederivativecontracts that give the holder the right, but not the obligation, to buy or sell theunderlyinginstrument at a specified price on or before a specified future date. Although the holder (also called the buyer) of the option is not obligated to exercise the option, the optionwriter(known as the seller) has an obligation to buy or sell the underlying instrument if the option is exercised. The price, or cost, of an option is an amount of money known as the premium. The buyer pays thispremiumto the seller in exchange for the right granted by the option. An option premium is its cost how much the particular option is worth to the buyer and seller. The option premium is the price the option buyer pays to the seller in order to have the right granted by the option, and it is the money the seller receives in exchange for writing the option. The theoretical value of an option, on the other hand, is the estimated value of an option a price generated by means of a model. It is what an option should currently be worth using all the known inputs, such as the underlying price, strike and days until expiration.Option traders utilize various option price models to attempt to set a current theoretical value. Option pricing theory has made vast strides since 1972, when Black and Scholes published their path-breaking paper providing a model for valuing European options. Black and Scholes used a replicating portfolio a portfolio composed of the underlying asset and the risk-free asset that had the same cash flows as the option being valued to come up with their final formulation. While their derivation is mathematically complicated, there is a simpler binomial model for valuing options that draws on the same logic.This project report is based on an attempt to study the valuation of options by implementing Binomial Option Pricing Model and Black Scholes Option Pricing Model to 30 Stocks of NSE. Further, the theoretical value of an option is compared to the option premium for those 30 stocks in order to find the accuracy of these models.

1. IntroductionAn option provides the holder with the right to buy or sell a specified quantity of an underlying asset at a fixed price (called a strike price or an exercise price) at or before the expiration date of the option. Since it is a right and not an obligation, the holder can choose not to exercise the right and allow the option to expire. There are two types of options viz., call options and put options. A call option gives the buyer of the option the right to buy the underlying asset at a fixed price, called the strike or the exercise price, at any time prior to the expiration date of the option. The buyer pays a price for this right. If at expiration, the value of the asset is less than the strike price, the option is not exercised and expires worthless. If, on the other hand, the value of the asset is greater than the strike price, the option is exercised the buyer of the option buys the asset [stock] at the exercise price. And the difference between the asset value and the exercise price comprises the gross profit on the option investment. The net profit on the investment is the difference between the gross profit and the price paid for the call initially.

A payoff diagram illustrates the cash payoff on an option at expiration. For a call, the net payoff is negative (and equal to the price paid for the call) if the value of the underlying asset is less than the strike price. If the price of the underlying asset exceeds the strike price, the gross payoff is the difference between the value of the underlying asset and the strike price and the net payoff is the difference between the gross payoff and the price of the call. This is illustrated in Figure 1 below:

If asset valuestrike price, the amount paid for the put is lost

Strike Price

Price of Underlying Asset

Figure 2: Payoff on Put Option

An option pricing model is a mathematical formula or model into which the following parameters are inserted: underlying stock or index price exercise price of the option expiry date of the option expected dividends (in cents for a stock, or as a yield for an index) to be paid over the life of the option expected risk free interest rate over the life of the option expected volatility of the underlying stock or index over the life of the option When the formula is applied to these variables, the resulting figure is called the theoretical fair value of the option.Option traders utilize various option price models to attempt to set a current theoretical value. Variables will fluctuate over the life of the option, and the option position's theoretical value will adapt to reflect these changes. Most professional traders and investors who trade significant option positions rely on theoretical value updates to monitor the changing risk and value of option positions and to assist with trading decisions.The value of an option is determined by a number of variables relating to the underlying asset and financial markets. Current Value of the Underlying Asset: Options are assets that derive value from an underlying asset. Consequently, changes in the value of the underlying asset affect the value of the options on that asset. Since calls provide the right to buy the underlying asset at a fixed price, an increase in the value of the asset will increase the value of the calls. Puts, on the other hand, become less valuable as the value of the asset increase.

Variance in Value of the Underlying Asset: The buyer of an option acquires the right to buy or sell the underlying asset at a fixed price. The higher the variance in the value of the underlying asset, the greater will the value of the option be1. This is true for both calls and puts. While it may seem counter-intuitive that an increase in a risk measure (variance) should increase value, options are different from other securities since buyers of options can never lose more than the price they pay for them; in fact, they have the potential to earn significant returns from large price movements.

Dividends Paid on the Underlying Asset: The value of the underlying asset can be expected to decrease if dividend payments are made on the asset during the life of the option. Consequently, the value of a call on the asset is a decreasing function of the size of expected dividend payments, and the value of a put is an increasing function of expected dividend payments. There is a more intuitive way of thinking about dividend payments, for call options. It is a cost of delaying exercise on in-the-money options. To see why, consider an option on a traded stock. Once a call option is in the money, i.e., the holder of the option will make a gross payoff by exercising the option, exercising the call option will provide the holder with the stock and entitle him or her to the dividends on the stock in subsequent periods. Failing to exercise the option will mean that these dividends are foregone.

Strike Price of Option: A key characteristic used to describe an option is the strike price. In the case of calls, where the holder acquires the right to buy at a fixed price, the value of the call will decline as the strike price increases. In the case of puts, where the holder has the right to sell at a fixed price, the value will increase as the strike price increases.

Time to Expiration on Option: Both calls and puts become more valuable as the time to expiration increases. This is because the longer time to expiration provides more time for the value of the underlying asset to move, increasing the value of both types of options. Additionally, in the case of a call, where the buyer has to pay a fixed price at expiration, the present value of this fixed price decreases as the life of the option increases, increasing the value of the call.

Risk Free Interest Rate Corresponding To Life of Option: Since the buyer of an option pays the price of the option up front, an opportunity cost is involved. This cost will depend upon the level of interest rates and the time to expiration on the option. The riskless interest rate also enters into the valuation of options when the present value of the exercise price is calculated, since the exercise price does not have to be paid (received) until expiration on calls (puts). Increases in the interest rate will increase the value of calls and reduce the value of puts.Effect on

FactorCall ValuePut Value

Increase in underlying assets valueIncreasesDecreases

Increase in strike priceDecreasesIncreases

Increase in variance of underlying assetIncreasesIncreases

Increase in time to expirationIncreasesIncreases

Increase in interest ratesIncreasesDecreases

Increase in dividends paidDecreasesIncreases

Table 1: Summary of Variables Affecting Call and Put Prices

There are two main models used in the market for pricing options: the Binomial Model and the Black Scholes model. For most traders these two models will give accurate enough results from which to work.The Binomial Option Pricing ModelFirst proposed by Cox, Ross and Rubinstein in a paper published in 1979, this solution to pricing an option is probably the most common model used for equity calls and puts today.The model divides the time to an options expiry into a large number of intervals, or steps. At each interval it calculates that the stock price will move either up or down with a given probability and also by an amount calculated with reference to the stocks volatility, the time to expiry and the risk free interest rate. A binomial distribution of prices for the underlying stock or index is thus produced.The model reduces possibilities of price changes, removes the possibility for arbitrage, assumes a perfectly efficient market, and shortens the duration of the option. Under these simplifications, it is able to provide a mathematical valuation of the option at each point in time specified.The binomial model takes a risk-neutral approach to valuation. It assumes that underlying security prices can only either increase or decrease with time until the option expires worthless. Due to its simple and iterative structure, the model presents certain unique advantages. For example, since it provides a stream of valuations for a derivative for each node in a span of time, it is useful for valuing derivatives such as American options which allow the owner to exercise the option at any point in time until expiration (unlike European options which are exercisable only at expiration). The model is also somewhat simple mathematically when compared to counterparts such as the Black-Scholes model, and is therefore relatively easy to build and implement with a computer spreadsheet.The Black Scholes ModelFirst proposed by Black and Scholes in a paper published in 1973, this analytic solution to pricing a European option on a non dividend paying asset formed the foundation for much theory in derivatives finance. The Black Scholes formula is a continuous time analogue of the binomial model.

The Black Scholes formula uses the pricing inputs to analytically produce a theoretical fair value for an option. The model has many variations which attempt, with varying levels of accuracy, to incorporate dividends and American style exercise conditions. However with computing power these days the binomial solution is more widely used.The major limitation of Black-Scholes model is that it cannot be used to accurately price options with an American-style exercise as it only calculates the option price at one point in time at expiration. It does not consider the steps along the way where there could be the possibility of early exercise of an American option.As all exchange traded equity options have American-style exercise (i.e. they can beexercised at any timeas opposed to European options which can only be exercised at expiration) this is a significant limitation.

2. Literature Review2.1. Backus, David K. and Foresi, Silverio and Wu, Liuren, Accounting for Biases in Black-Scholes (August 31, 2004).It is known that the prices of options commonly differ from theBlack-Scholesformula with respect to two parameters i.e. implied volatilities vary by strike price and maturity. Both these parameters are accounted for using Gram-Charlier expansions to approximate the conditional distribution of the logarithm of the price of the underlying security.Here, volatility is approximately a quadratic function of moneyness, a result used to infer skewness and kurtosis from implied volatilities variations. Evidence suggests that both kurtosisincurrency prices andbiasesinBlack-Scholesoption prices decline with maturity.2.2. Buraschi, Andrea and Jackwerth, Jens Carsten, Is Volatility Risk Priced in the Option Market? (March 1999).Rubinstein (1994) shows evidence of a significant time patternintheshape ofthevolatilitysmile afterthecrash of 1987 and proposes an implied binomial tree approach to overcomethe empirical limitations oftheBlack and Scholes model. This approach, and more generally the class of generalized deterministicvolatilitymodels,isbased ontheassumption thatthelocal volatilityoftheunderlying assetisa known function of time and ofthepath and level of the underlying asset price.Inthese economies, options are redundant assets. This observation is used as a testable restriction and three questions are asked. First,istheobserved dynamics ofthesmile consistent with deterministicvolatilitymodels? Second, if volatility is stochastic, so that two assets cannot dynamically complete the market, is volatility also priced and if so how importantisto model explicitlytheprice ofvolatilityinthedesign of risk management strategies? This question is addressed by testing ifthereturns ontheunderlying and on at-the-money options spantheasset pricesintheeconomy or if additional information is needed from returns on other options ortherisk free rate. Third, are there any differences in the spanning properties oftheoptionmarket before and afterthe1987 market crash?

For these three questions tests are conducted based on daily S&P500 index options data from April 1986--December 1995. All the tests suggest that in and out of the money options are needed for spanning purposes. This findingiseven strongerinthepost crash period and suggests that returns on away from the money options are driven by at least one additional economic factor compared to returns on at the money options. This findingisinconsistent withtheimplications of deterministicvolatility models based on generalized deterministic volatility. 2.3. Chance, Don M., A Synthesis of Binomial Option Pricing Models for Lognormally Distributed Assets (November 20, 2007).The finance literature has revealed no fewer than 11 alternative versions of thebinomialoptionpricingmodel forpricingoptions on lognormally distributed assets. These models are derived under a variety of assumptionsandin some cases require unnecessary information. This paper provides a reviewandsynthesis of thesemodels, showing their commonalitiesand differencesanddemonstrating how 11 diversemodelsall produce the same result in the limit. Some of themodelsadmit arbitrage with a finite number of time steps and some fail to capture the correct volatility. This paper also examines the convergence properties of each modelandfinds that none exhibit consistently superior performance over the others. Finally, it demonstrates how a general model that accepts any arbitrage-free risk neutral probability will reproduce theBlack-Scholes-Merton model in the limit.2.4. Feng, Yi and Kwan, Clarence C. Y. (2012) "Connecting Binomial and Black-Scholes Option Pricing Models: A Spreadsheet-Based Illustration,"Spreadsheets in Education (eJSiE): Vol. 5: Iss. 3, Article 2.The Black-Scholes option pricing model is part of the modern financial curriculum, even at the introductory level. However, as the derivation of the model, which requires advanced mathematical tools, is well beyond the scope of standard finance textbooks, the model has remained a great, but mysterious, recipe for many students. This paper illustrates, from a pedagogic perspective, how a simple binomial model, which converges to the Black-Scholes formula, can capture the economic insight in the original derivation. Microsoft Excelplays an important pedagogic role in connecting the two models. The interactivity as provided by scroll bars, in conjunction with Excel's graphical features, will allow students to visualize the impacts of individual input parameters on option pricing.2.5. Subrahmanyam, Marti G. and Peterson, Sandra and Stapleton, Richard C., An Arbitrage-Free Two-Factor Model of the Term Structure of Interest Rates: A Multivariate Binomial Approach (May 1998). NYU Working Paper No. FIN-98-070.A no-arbitragemodelofthetermstructure is built usingtwostochastic factors on each date, the short term interestrate andtheforward premium.Themodelis essentiallyan extension to two factorsofthelognormalinterestratemodelofBlack Karazinski. It allows for mean reversion in the short rate and intheforward premium.Themethod is computationally efficient for several reasons. First,interestratesare defined on a bankers' discount basis, as linear functionsofzero coupon bond prices, enabling us to usetheno-arbitragecondition to compute bond prices without resorting to cumbersome iterative methods. Second,themultivariate binomial methodologyofHo Stapleton Subrahmanyam is extended so that a multi period tree of rateswiththeno arbitrage property can be constructed using analytical methods.Themethod uses a recombiningtwo-dimensional binomial latticeofinterestratesthat minimizes the number ofstates andtermstructures. Third,theproblemofcomputing a large numberoftermstructures is simplified by using a limited numberofbucketratesin each termstructurescenario. In addition to these computational advantages, a key featureofthemodelis that it is consistent withtheobservedtermstructureofvolatilities implied bythe pricesofinterestrate caps and floors. 2.6. Vorst, Ton and Menkveld, Albert J., A Pricing Model for American Options with Stochastic Interest Rates.In this paper a new methodology to price American put options under stochastic interest rates is introduced. The method is a combination of an analytic approach and a binomial tree approach. A binomial tree for the forward risk adjusted tree is constructed and calculates analytically the expected early exercise value in each point. For American puts with stochastic interest rates the correlation between the stock price process and the interest rate process has different influences on the European option values and the early exercise premiums. This result in a non monotonic relation between this correlation and the American put option value. Furthermore, there is evidence that the early exercise premium due to stochastic interest rates is much larger than established before by other researchers.

3. Research MethodologyAn attempt to study the financial models Binomial Option Pricing Model and Black Scholes Option Pricing Model with reference to option pricing is done. Secondary Data is used for the research purpose. Secondary data is collected from NSE website as well as from various research papers, reports, books, Journals, Magazines, and News Papers etc. The above mentioned option pricing models are applied to 30 stocks listed in NSE. The theoretical fair value for both call and put of these 30 stocks is calculated and compared to the actual values. In order to calculate the theoretical values for both the option pricing models, Microsoft Excel plays an important pedagogic role.

3.1. The Binomial Option Pricing ModelIn order to carry out the analysis, the two step binomial tree is considered. The calculation of option price is done as follows: Consider the stock price at t0 to be S0. In the Binomial Method, the price can go either up or down. At t1 (after one time interval), the price can either be an up price or a down price. These prices can each go either up or down over the course of the next time interval. As we see that the possible prices quickly branch out over time, hence the term Binomial Tree is used for this technique. By making the number of time intervals between t0 and time of expiry T very large, we will get many possible stock prices at T and we will have a better approximation of the Brownian Random Walk, which is a time continuous model.

Figure 3: Two Step Binomial Tree

In order to get from S0 to Su, we have to multiply S0 by whats called the up ratio, labeled u. Similarly, to get from S0 to Sd, we have to multiply S0 by the down ratio, labeled d. These factors are constant throughout the tree. Also, if the stock takes an up move followed by a down move, itll arrive at the same price had the stock taken a down move followed by an up move. Hence, the order does not matter. u and d depend on two things: volatility of the stock and the length of a time interval. Cox, Ross, and Rubinstein chose the up and down ratios to be these:

Because d is the reciprocal of u, u*d = 1. Therefore, if S0 takes an up move followed by a down move or vice versa, the price will return to S0. If the probability of S0 rising to Su is p, then the probability of S0 falling to Sd must be 1-p, since one of those two outcomes must happen in this model. We can say that the expected price at t1 is the probability of the up move happening p times the up price plus the probability of the down move happening (1-p) times the down price.

In order to make the Binomial Method to be risk neutral, a riskless asset should grow by a factor of after delta t, with r as the risk free interest rate. So the expected value of S0 is: S0 = Also p is given as follows:

This is the risk-neutral transition probability of an up move. The u and d only depend on the volatility and the length of the time interval, so this probability only depends on volatility, the length of a time interval, and the riskless interest rate. All of these will remain constant throughout our binomial tree, so this probability will remain constant throughout the tree as well. In order to calculate the call values, the Strike Price is subtracted from the Stock Price. If the value obtained from subtracting the Strike Price from the Stock Price is less than zero, then the value is considered to be zero. In order to calculate the pull values, the Stock Price is subtracted from the Strike Price. If 333the value obtained from subtracting the Stock Price from the Strike Price is less than zero, then the value is considered to be zero. Thus, the theoretical values for call and put are calculated by using Binomial Option Pricing Model.

3.2. The Black Scholes Option Pricing ModelThe Black Scholes Option Pricing Model is used for the European options which can only be exercised at the expiration. The formula for Black Scholes Option Pricing Model:

The variable definitions are as follows: S: current stock price K: option strike price E: base of natural logarithms R: risk free interest rate T: time until option expiration : Standard deviation (sigma) of returns on the underlying security ln: natural logarithm N(d1) and N(d2): Cumulative standard normal distribution functionsThe cumulative distribution is shown in Figure 4:

Figure 4: Cumulative DistributionIn approximate terms, these probabilities yield the likelihood that an option will generate positive cash flows for its owner at exercise, i.e., when S>K in the case of a call option and when K>S in the case of a put option. The portfolio that replicates the call option is created by buying N(d1) units of the underlying asset, and borrowing Ke-rt N(d2). The portfolio will have the same cash flows as the call option and thus the same value as the option.The assumptions of the Black-Scholes Model are as follows: The stock pays no dividends during the options life If Black Scholes Option Pricing Model is applied to two securities, one with no dividends and the other with a dividend yield, the model will predict the same call premium

European exercise style A European option can only be exercised on the expiration date

Markets are efficient The Black Scholes Option Pricing Model assumes informational efficiency. No one can predict the direction of the market or of an individual stock. Put/call parity implies that everyone agrees on the option premium, regardless of whether the market is bullish or bearish

No transaction costs There are no commissions and bid-ask spreads

Interest rates remain constant Often the 30-day T-bill rate is used

Prices are lognormally distributed The logarithms of the underlying security prices are normally distributed This is a reasonable assumption for most assets on which options are available

4. Data AnalysisThe option prices for the 30 stocks listed in NSE are calculated by the Binomial Option Pricing Model and the Black Scholes Option Pricing Model. These theoretical values for each stock are compared with their actual values to find out the accuracy of each model.1. Company: ACC Limited

Strike Price (K)Stock Price (S)Risk Free Rate of Interest (R)Annualized Volatility ()Starting DateExpiry DateTime in MonthsTime in years (T)

11001136.48.77%31.82%04-Mar-1424-Apr-1420.167

Value of CallValue of Put

Binomial Option Pricing Model117.2749.18

Black Scholes Option Pricing Model87.4235.05

Actual Premium according to NSE14108.35

Table 2: Information about ACC Limited

Figure 5: Comparison of theoretical values and actual premium for call and put option for ACC LimitedFor ACC Limited, it is observed that the value of call by Black Scholes Option Pricing Model is closer to the actual premium according to NSE and the value of put by Binomial Option Pricing Model is closer to the actual premium according to NSE.2. Company: Adani Enterprises Limited


250268.68.77%51.56%04-Mar-1424-Apr-1420.167





Table 3: Information about Adani Enterprises Limited

Figure 6: Comparison of theoretical values and actual premium for call and put option for Adani Enterprises LimitedFor Adani Enterprises Limited, it is observed that the value of call by Black Scholes Option Pricing Model is closer to the actual premium according to NSE and the value of put by Black Binomial Option Pricing Model is closer to the actual premium according to NSE.3. Company: Ambuja Cements Limited


175168.58.77%38.60%04-Mar-1424-Apr-1420.167




Actual Premium according to NSE5.219.2

Table 4: Information about Ambuja Cements Limited

Figure 7: Comparison of theoretical values and actual premium for call and put option for Ambuja Cements LimitedFor Ambuja Cements Limited, it is observed that the value of call by Black Scholes Option Pricing Model is closer to the actual premium according to NSE and the value of put by Binomial Option Pricing Model is closer to the actual premium according to NSE.4. Company: Ashok Leyland Limited


1515.658.77%31.41%04-Mar-1424-Apr-1420.167





Table 5: Information about Ashok Leyland Limited

Figure 8: Comparison of theoretical values and actual premium for call and put option for Ashok Leyland LimitedFor Ashok Leyland Limited, it is observed that the value of both call and put by Binomial Option Pricing Model is closer to the actual premium according to NSE. 5. Company: Bharti Airtel Limited


330287.48.77%29.55%04-Mar-1424-Apr-1420.167





Table 6: Information about Bharti Airtel Limited

Figure 9: Comparison of theoretical values and actual premium for call and put option for Bharti Airtel LimitedFor Bharti Airtel Limited, it is observed that the value of call by Binomial Option Pricing Model is closer to the actual premium according to NSE and the value of put by Black Scholes Option Pricing Model is closer to the actual premium according to NSE.6. Company: Cipla Limited


400378.658.77%35.78%04-Mar-1424-Apr-1420.167





Table 7: Information about Cipla Limited

Figure 10: Comparison of theoretical values and actual premium for call and put option for Cipla LimitedFor Cipla Limited, it is observed that the value of call by Binomial Option Pricing Model is closer to the actual premium according to NSE and the value of put by Black Scholes Option Pricing Model is closer to the actual premium according to NSE.7. Company: DLF Limited


150140.78.77%38.28%04-Mar-1424-Apr-1420.167





Table 8: Information about DLF Limited

Figure 11: Comparison of theoretical values and actual premium for call and put option for DLF LimitedFor DLF Limited, it is observed that the value of call by Black Scholes Option Pricing Model is closer to the actual premium according to NSE and the value of put by Binomial Option Pricing Model is closer to the actual premium according to NSE.8. Company: Dr. Reddys Laboratories Limited


29002802.68.77%29.43%04-Mar-1424-Apr-1420.167





Table 9: Information about Dr. Reddys Laboratories Limited

Figure 12: Comparison of theoretical values and actual premium for call and put option for Dr. Reddys Laboratories LimitedFor Dr. Reddys Laboratories Limited, it is observed that the value of call by Black Scholes Option Pricing Model is closer to the actual premium according to NSE and the value of put by Binomial Option Pricing Model is closer to the actual premium according to NSE.9. Company: Exide Industries Limited


1201138.77%25.68%04-Mar-1424-Apr-1420.167





Table 10: Information about Exide Industries Limited

Figure 13: Comparison of theoretical values and actual premium for call and put option for Exide Industries LimitedFor Exide Industries Limited, it is observed that the value of call by Black Scholes Option Pricing Model is closer to the actual premium according to NSE and the value of put by Binomial Option Pricing Model is closer to the actual premium according to NSE.10. Company: The Federal Bank Limited


85079.658.77%37.21%04-Mar-1424-Apr-1420.167





Table 11: Information about The Federal Bank Limited

Figure 14: Comparison of theoretical values and actual premium for call and put option for The Federal Bank LimitedFor The Federal Bank Limited, it is observed that the value of call by Black Scholes Option Pricing Model is closer to the actual premium according to NSE and the value of put by Binomial Option Pricing Model is closer to the actual premium according to NSE.11. Company: Housing Development Finance Corporation Limited


900821.18.77%26.11%04-Mar-1424-Apr-1420.167




Actual Premium according to NSE2072

Table 12: Information about Housing Development Finance Corporation Limited

Figure 15: Comparison of theoretical values and actual premium for call and put option for Housing Development Finance Corporation LimitedFor Housing Development Finance Corporation Limited, it is observed that the value of both call and put by Black Scholes Option Pricing Model is closer to the actual premium according to NSE.12. Company: ICICI Bank Limited


110010688.77%32.26%04-Mar-1424-Apr-1420.167





Table 13: Information about ICICI Bank Limited

Figure 16: Comparison of theoretical values and actual premium for call and put option for ICICI Bank LimitedFor ICICI Bank Limited, it is observed that the value of both call and put by Black Scholes Option Pricing Model is closer to the actual premium according to NSE.13. Company: IDFC Limited


10099.88.77%34.55%04-Mar-1424-Apr-1420.167





Table 14: Information about IDFC Limited

Figure 17: Comparison of theoretical values and actual premium for call and put option for IDFC LimitedFor IDFC Limited, it is observed that the value of call by Black Scholes Option Pricing Model is closer to the actual premium according to NSE and the value of put by Binomial Option Pricing Model is closer to the actual premium according to NSE.14. Company: Infosys Limited


380037998.77%19.05%04-Mar-1424-Apr-1420.167





Table 15: Information about Infosys Limited

Figure 18: Comparison of theoretical values and actual premium for call and put option for Infosys LimitedFor Infosys Limited, it is observed that the value of call by Black Scholes Option Pricing Model is closer to the actual premium according to NSE and the value of put by Binomial Option Pricing Model is closer to the actual premium according to NSE.15. Company: Jindal Steel & Power Limited


280248.78.77%38.36%04-Mar-1424-Apr-1420.167





Table 16: Information about Jindal Steel & Power Limited

Figure 19: Comparison of theoretical values and actual premium for call and put option for Jindal Steel & Power LimitedFor Jindal Steel & Power Limited, it is observed that the value of both call and put by Black Scholes Option Pricing Model is closer to the actual premium according to NSE.16. Company: Kotak Mahindra Bank Limited


740676.38.77%30.9104-Mar-1424-Apr-1420.167





Table 17: Information about Kotak Mahindra Bank Limited

Figure 20: Comparison of theoretical values and actual premium for call and put option for Kotak Mahindra Bank LimitedFor Kotak Mahindra Bank Limited, it is observed that the value of call by Black Scholes Option Pricing Model is closer to the actual premium according to NSE and the value of put by Binomial Option Pricing Model is closer to the actual premium according to NSE.17. Company: LIC Housing Finance Limited


210208.158.77%30.57%04-Mar-1424-Apr-1420.167





Table 18: Information about LIC Housing Finance Limited

Figure 21: Comparison of theoretical values and actual premium for call and put option for LIC Housing Finance LimitedFor LIC Housing Finance Limited, it is observed that the value of call by Black Scholes Option Pricing Model is closer to the actual premium according to NSE and the value of put by Binomial Option Pricing Model is closer to the actual premium according to NSE.18. Company: Mahindra & Mahindra


10009548.77%29.50%04-Mar-1424-Apr-1420.167





Table 19: Information about Mahindra & Mahindra

Figure 22: Comparison of theoretical values and actual premium for call and put option for Mahindra & MahindraFor Mahindra & Mahindra, it is observed that the value of call by Black Scholes Option Pricing Model is closer to the actual premium according to NSE and the value of put by Binomial Option Pricing Model is closer to the actual premium according to NSE.19. Company: Maruti Suzuki India Limited


17001600.058.77%39.51 %04-Mar-1424-Apr-1420.167





Table 20: Information about Maruti Suzuki India Limited

Figure 23: Comparison of theoretical values and actual premium for call and put option for Maruti Suzuki India Limited For Maruti Suzuki India Limited, it is observed that the value of both call and put by Binomial Option Pricing Model is closer to the actual premium according to NSE. 20. Company: Oil & Natural Gas Corporation Limited


300296.48.77%28.79%04-Mar-1424-Apr-1420.167





Table 21: Information about Oil & Natural Gas Corporation Limited

Figure 24: Comparison of theoretical values and actual premium for call and put option for Oil & Natural Gas Corporation LimitedFor Oil & Natural Gas Corporation Limited, it is observed that the value of call by Black Scholes Option Pricing Model is closer to the actual premium according to NSE and the value of put by Binomial Option Pricing Model is closer to the actual premium according to NSE.21. Company: Punjab National Bank


600564.88.77%39.04%04-Mar-1424-Apr-1420.167





Table 22: Information about Punjab National Bank

Figure 25: Comparison of theoretical values and actual premium for call and put option for Punjab National BankFor Punjab National Bank, it is observed that the value of call by Black Scholes Option Pricing Model is closer to the actual premium according to NSE and the value of put by Binomial Option Pricing Model is closer to the actual premium according to NSE.22. Company: Ranbaxy Laboratories Limited


400362.38.77%61.95%04-Mar-1424-Apr-1420.167





Table 23: Information about Ranbaxy Laboratories Limited

Figure 26: Comparison of theoretical values and actual premium for call and put option for Ranbaxy Laboratories LimitedFor Ranbaxy Laboratories Limited, it is observed that the value of both call and put by Binomial Option Pricing Model is closer to the actual premium according to NSE. 23. Company: Reliance Communications Limited


130111.78.77%47.55%04-Mar-1424-Apr-1420.167





Table 24: Information about Reliance Communications Limited

Figure 27: Comparison of theoretical values and actual premium for call and put option for Reliance Communications LimitedFor Reliance Communications Limited, it is observed that the value of call by Binomial Option Pricing Model is closer to the actual premium according to NSE and the value of put by Black Scholes Option Pricing Model is closer to the actual premium according to NSE.24. Company: Sun Pharmaceuticals Industries Limited


670620.158.77%29.06%04-Mar-1424-Apr-1420.167





Table 25: Information about Sun Pharmaceuticals Industries Limited

Figure 28: Comparison of theoretical values and actual premium for call and put option for Sun Pharmaceuticals Industries Limited For Sun Pharmaceuticals Industries Limited, it is observed that the value of call by Black Scholes Option Pricing Model is closer to the actual premium according to NSE and the value of put by Binomial Option Pricing Model is closer to the actual premium according to NSE.25. Company: Tata Motors Limited


4004138.77%37.75%04-Mar-1424-Apr-1420.167





Table 26: Information about Tata Motors Limited

Figure 29: Comparison of theoretical values and actual premium for call and put option for Tata Motors LimitedFor Tata Motors Limited, it is observed that the value of call by Black Scholes Option Pricing Model is closer to the actual premium according to NSE and the value of put by Binomial Option Pricing Model is closer to the actual premium according to NSE.26. Company: Tata Consultancy Services Limited


22002240.28.77%33.43%04-Mar-1424-Apr-1420.167





Table 27: Information about Tata Consultancy Services Limited

Figure 30: Comparison of theoretical values and actual premium for call and put option for Tata Consultancy Services LimitedFor Tata Consultancy Services Limited, it is observed that the value of both call and put by Black Scholes Option Pricing Model is closer to the actual premium according to NSE.27. Company: Unitech Limited


1011.458.77%45.86%04-Mar-1424-Apr-1420.167





Table 28: Information about Unitech Limited

Figure 31: Comparison of theoretical values and actual premium for call and put option for Unitech LimitedFor Unitech Limited, it is observed that the value of both call and put by Black Scholes Option Pricing Model is closer to the actual premium according to NSE.28. Company: Voltas Limited


130140.358.77%52.24%04-Mar-1424-Apr-1420.167





Table 29: Information about Voltas Limited

Figure 32: Comparison of theoretical values and actual premium for call and put option for Voltas LimitedFor Voltas Limited, it is observed that the value of call by Black Scholes Option Pricing Model is closer to the actual premium according to NSE and the value of put by Binomial Option Pricing Model is closer to the actual premium according to NSE.29. Company: Yes Bank Limited


4003218.77%35.13%04-Mar-1424-Apr-1420.167





Table 30: Information about Yes Bank Limited

Figure 33: Comparison of theoretical values and actual premium for call and put option for Yes Bank LimitedFor Yes Bank Limited, it is observed that the value of call by Binomial Option Pricing Model is closer to the actual premium according to NSE and the value of put by black Scholes Option Pricing Model is closer to the actual premium according to NSE.30. Company: Zee Entertainment Enterprises Limited


260270.18.77%33.47%04-Mar-1424-Apr-1420.167





Table 31: Information about Zee Entertainment Enterprises Limited

Figure 34: Comparison of theoretical values and actual premium for call and put option for Zee Entertainment Enterprises LimitedFor Zee Entertainment Enterprises Limited, it is observed that the value of both call and put by Black Scholes Option Pricing Model is closer to the actual premium according to NSE.Call ValuePut Value

Sr. No.Company NameModel closer to the actual premium according to NSEActual Premium according to NSEModel closer to the actual premium according to NSEActual Premium according to NSE

1.ACC LimitedBlack Scholes87.4214Binomial49.18108.35

2.Adani Enterprises LimitedBlack Scholes34.4820

Binomial19.1832.55

3.Ambuja Cements LimitedBlack Scholes8.845.2

Binomial15.4319.2

4.Ashok Leyland LimitedBinomial1.712.15

Binomial0.630.65

5.Bharti Airtel LimitedBinomial9.812.45

Black Scholes40.7326.9

6.Cipla LimitedBinomial27.4631.05Black Scholes31.119.2

7.DLF LimitedBlack Scholes5.864.65Binomial15.5822.8

8.Dr. Reddy's Laboratories LimitedBlack Scholes109.7

19.5

Binomial195.72

238.95

9.Exide Industries LimitedBlack Scholes2.65

2.1

Binomial9.1

18.95

10.The Federal Bank LimitedBlack Scholes3.15

3.9

Binomial8.7

9.95

11.Housing Development Finance Corporation LimitedBlack Scholes12.47

20

Black Scholes78.31

72

12.ICICI Bank LimitedBlack Scholes48.8339.55Black Scholes64.8756.15

13.IDFC LimitedBlack Scholes6.224.5Binomial5.711.2

14.Infosys LimitedBlack Scholes146.05172.25Binomial96.58111.9

15.Jindal Steel & Power LimitedBlack Scholes6.227.3

Black Scholes33.4531.55

16.Kotak Mahindra Bank LimitedBlack Scholes14.9916.1

Binomial77.5290.55

17.LIC Housing Finance LimitedBlack Scholes10.938.45

Binomial11.2521.35

18.Mahindra & MahindraBlack Scholes32.4912.8

Binomial75.72123.95

19.Maruti Suzuki India LimitedBinomial126.22161.15

Binomial177.19189.65

20.Oil & Natural Gas Corporation LimitedBlack Scholes14.25

7.7

Binomial15.64

29.2

21.Punjab National BankBlack Scholes24.9725.1

Binomial61.9294.9

22.Ranbaxy Laboratories LimitedBinomial42.33

41.35

Binomial68.51

109.85

23.Reliance Communications LimitedBinomial8.06

8.25

Black Scholes19.68

14

24.Sun Pharmaceuticals Industries LimitedBlack Scholes14.34

10.6

Binomial62.52

72.95

25.Tata Motors LimitedBlack Scholes35.3120Binomial22.5947.9

26.Tata Consultancy Services LimitedBlack Scholes159.4143

Black Scholes87.2755

27.Unitech LimitedBlack Scholes1.831.9Black Scholes0.230.35

28.Voltas LimitedBlack Scholes18.535.95Binomial10.0326.85

29Yes Bank LimitedBinomial7.127.9Black Scholes74.9393.2

30.Zee Entertainment Enterprises LimitedBlack Scholes22.32

18

Black Scholes8.45

6

Table 32: Comparison of values obtained from Binomial Option Pricing Model and Black Scholes Option Pricing Model with actual premium according to NSE from 04-Mar-2014 to 24-Apr-2014

Sr. No.Company NameStrike Price (K)Stock Price (S)Risk free rate of interest (R)Annualized Volatility ()Starting DateExpiry DateTime in Months (T)

1.ACC Limited11001136.48.77%31.82%04-Mar-1424-Apr-142

2.Adani Enterprises Limited

250

268.68.77%51.56%

04-Mar-1424-Apr-14

2

3.Ambuja Cements Limited

175

168.58.77%38.60%

04-Mar-1424-Apr-14

2

4.Ashok Leyland Limited1515.658.77%31.41%04-Mar-1424-Apr-142

5.Bharti Airtel Limited330287.48.77%29.55%04-Mar-1424-Apr-142

6.Cipla Limited400378.658.77%35.78%04-Mar-1424-Apr-142

7.DLF Limited150140.78.77%38.28%04-Mar-1424-Apr-142

8.Dr. Reddy's Laboratories Limited

2900

2802.68.77%29.43%

04-Mar-1424-Apr-14

2

9.Exide Industries Limited

120

1138.77%25.68%

04-Mar-1424-Apr-14

2

10.The Federal Bank Limited85079.658.77%37.21%04-Mar-1424-Apr-142

11.Housing Development Finance Corporation Limited

900

821.18.77%26.11%

04-Mar-1424-Apr-14

2

12.ICICI Bank Limited110010688.77%32.26%04-Mar-1424-Apr-142

13.IDFC Limited10099.88.77%34.55%04-Mar-1424-Apr-142

14.Infosys Limited380037998.77%19.05%04-Mar-1424-Apr-142

15.Jindal Steel & Power Limited280248.78.77%38.36%04-Mar-1424-Apr-142

16.Kotak Mahindra Bank Limited

740

676.38.77%30.91

04-Mar-1424-Apr-14

2

17.LIC Housing Finance Limited

210

208.158.77%30.57%

04-Mar-1424-Apr-14

2

18.Mahindra & Mahindra10009548.77%29.50%04-Mar-1424-Apr-142

19.Maruti Suzuki India Limited17001600.058.77%39.51 %04-Mar-1424-Apr-142

20.Oil & Natural Gas Corporation Limited

300

296.48.77%28.79%

04-Mar-1424-Apr-14

2

21.Punjab National Bank600564.88.77%39.04%04-Mar-1424-Apr-142

22.Ranbaxy Laboratories Limited400362.38.77%61.95%04-Mar-1424-Apr-142

23.Reliance Communications Limited

130

111.78.77%47.55%

04-Mar-1424-Apr-14

2

24.Sun Pharmaceuticals Industries Limited

670

620.158.77%29.06%

04-Mar-1424-Apr-14

2

25.Tata Motors Limited4004138.77%37.75%04-Mar-1424-Apr-142

26.Tata Consultancy Services Limited

2200

2240.28.77%33.43%

04-Mar-1424-Apr-14

2

27.Unitech Limited1011.458.77%45.86%04-Mar-1424-Apr-142

28.Voltas Limited130140.358.77%52.24%04-Mar-1424-Apr-142

29Yes Bank Limited4003218.77%35.13%04-Mar-1424-Apr-142

30.Zee Entertainment Enterprises Limited

260

270.18.77%33.47%

04-Mar-1424-Apr-14

2

Table 33: Details of all the 30 stocks

5. ConclusionFrom the data analysis, it can be seen that for most of the stocks the premium for call is closer to the theoretical value obtained from the Black Scholes Option Pricing Model and the premium for put is closer to the theoretical value obtained from Binomial Option Pricing Model. It should be noted that while quantifiable factors can explain much of the observable price behavior, supply and demand still play an important part and can override predictive option values created by using pricing models. A sustained imbalance of competing bids and offers can drive prices away from theoretically expected values. Imbalances can be caused by factors such as a sudden political event or unexpected news regarding a particular stock. These factors cannot be quantified and can have an effect on both option prices and the accuracy of price modeling. That being said, theoretical options pricing is a valuable tool that helps investors and traders anticipate price movements for option positions.

6. References and Bibliography1. Backus, David K. and Foresi, Silverio and Wu, Liuren, Accounting for Biases in Black-Scholes (August 31, 2004).2. Buraschi, Andrea and Jackwerth, Jens Carsten, Is Volatility Risk Priced in the Option Market? (March 1999).3. Chance, Don M., A Synthesis of Binomial Option Pricing Models for Lognormally Distributed Assets (November 20, 2007).4. Feng, Yi and Kwan, Clarence C. Y. (2012) "Connecting Binomial and Black-Scholes Option Pricing Models: A Spreadsheet-Based Illustration,"Spreadsheets in Education (eJSiE): Vol. 5: Iss. 3, Article 2.5. Subrahmanyam, Marti G. and Peterson, Sandra and Stapleton, Richard C., An Arbitrage-Free Two-Factor Model of the Term Structure of Interest Rates: A Multivariate Binomial Approach (May 1998). NYU Working Paper No. FIN-98-070.6. Vorst, Ton and Menkveld, Albert J., A Pricing Model for American Options with Stochastic Interest Rates.7. Options, Futures and Derivatives, 7th Edition by John. C. Hull, Sankarshan Basu8. http://www.investopedia.com9. http://en.wikipedia.org10. http://www.asx.com.au11. http://www.nseindia.com12. http://www.ssrn.com

7. Appendix7.1. Black Scholes Option Pricing Model Calculations:The calculation for finding the theoretical value for call and put option for a stock was done in Microsoft Excel. Given below is the screenshot of the excel sheet which shows the calculation for Adani Enterprises Limited.

Figure 35: Calculation for Black Scholes Option Pricing Model

The cell formulas used in the excel sheet are as follows:B8 =B7/12

E3 =EXP(B6*B8)

E4 =EXP(-B6*B8)

E7 =((LN(B4/B3))+((B6+((B10*B10)/2))*B8))/(B10*(SQRT(B8)))

E8 =E7-(B10*SQRT(B8))

E10 =NORMSDIST(E7)

E11 =NORMSDIST(E8)

For Call Value:

B16 =(B4*E10)-(B3*E4*E11)

For Put Value:

B21 =(B4*(E10-1))-(B3*E4*(E11-1))

Table 34: Cell Formulas for Black Scholes Option Pricing Model

7.2. Binomial Option Pricing Model Calculations:The calculation for finding the theoretical value for call and put option for a stock was done in Microsoft Excel. Given below is the screenshot of the excel sheet which shows the calculation for Adani Enterprises Limited.

Figure 36: Calculation for Binomial Option Pricing ModelThe cell formulas used in the excel sheet are as follows:B5 =B4+(B4*E12)E3 =EXP(B11*B13)

B6 =B4-(B4*E13)E4 =EXP(-B11*B13)

B7 =B5+(E12*B5)E6 =(E3-E10)/(E9-E10)

B8 =B5-(B5*E13)E7 =1-E6

B9 =B6-(B6*E13)E9 =EXP(B15*SQRT(B13))

B13 =B12/12E10 =1/E9

E12 =((E9*100)-100)/100

E13 =(100-(E10*100))/100

For Call Value:For Put Value:

B21 =B7-B3B34 =B3-B7

B22 =B8-B3B35 =B3-B8

B23 =B9-B3B36 =B3-B9

B25 =B5-B3B38=B3-B5

B26 =B6-B3B39=B3-B6

B28 =B4-B3B41=B3-B4

C21 =IF(B21

comparative study of black scholes option pricing model and binomial option pricing model

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