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International Journal of Research in Management, Science & Technology (E-ISSN: 2321-3264)
Vol. 3, No. 1, April 2015
Available at www.ijrmst.org
Analysis of Price Using Black Scholes and
Greek Letters in Derivative European Option
Market
C. Rajanikanth#1, Dr. E. Lokanadha Reddy*2
#
Associate Professor, Department of MBA, Sri Venkateswara College of Engg. & Tech. (Autonomous), Chittoor,
Andhra Pradesh. India
*
Professor of Economics, Sri Venkateswara College of Engg. & Tech. (Autonomous), Chittoor,
Andhra Pradesh, India
#
[email protected]
*
[email protected]
Abstract – The measurement of risk in the market is so
crucial work and it cannot be used to denote for future
expectations. The market is completely moving with
random, wandering and it is efficient in changing of
volatility. The Chicago board of trade first introduced
derivatives and later it was developed all around the world
and most popularly used to hedge the risk. The American
option can be sold at any time before maturity, but
European can sell at only maturity date, these made the
researchers to concentrate to find the price of an option
in the future and its sensitivity through Greek letters. This
study is considered on Black-Scholes option pricing and
risk measure by using Delta, Vega, Theta, Rho, Gamma.
Keywords – Black-Scholes, Delta, Vega, Theta, Rho
and Gamma.
I.
INTRODUCTION
Black-Scholes model for calculating of an option was
introduced in 1973 in a paper entitled, "The Pricing of
Options and Corporate Liabilities" published in
the Journal of Political Economy. This method
developed by famous economists – Fischer Black,
Myron Scholes and Robert Merton – is perhaps the most
well-known options pricing model. Scholes and Merton
were awarded the Nobel Prize in Economics for their
work in finding a new method to determine the value of
derivatives in the year 1997.B-S model is used to
calculate the price of European put and call options.
Greek letters in option pricing are commonly used as the
sensitivities of an option price relative to changes in the
value of either a state variable or a parameter (Hull,
2009). Greek letter measures a different dimension to
the risk in an option position and, by analyzing Greek
letters, financial institutions can successfully manage
their risk. The option hedge ratio defined as the rate of
change of option price to the underlying price, Black
and Scholes, (1973) model offers an graceful and
effective way for option pricing and option hedging
2321-3264/Copyright©2015, IJRMST, April 2015
since it can give an analytic solution for option price, as
well as Greek letters, even though this model could
make certain pricing bias in realistic market. The blackScholes formula thus has been regarded as a benchmark
for option valuation and option hedging, and accepted
by many financial professionals including practitioners
who seek to manage their risk exposure.
Characteristically, option trader would use the Greek
letters under Black-Scholes framework as a benchmark
for properly adjusting option position so that all risks are
manageable.
II.
REVIEW OF LITERATURE
Xisheng Yu and Xiaoke Xie(2013): Greek letter
measures the sensitivity of an option price with respect
to the change in the value of a given underlying
parameter such as underlying asset’s price, value, time
etc. Analysis of Greek letters for European call and put
options within the Black-scholes. Hong-Yi Chen,
Cheng-Few Lee and Weikang Shih (2008): Explained
about the derivations of Greek letters for call and put
options on both dividends-paying stock and nondividends stock and the relationship between Greek
letters. Beata Stehlikova: Explained sensitivities by
using binomial option pricing model and relation with
different Greeks. Dr. S. Saravanan and G. Pradeep
Kumar (2012): Accuracy of the Black-Scholes option
pricing model with relation market prices and estimated
the stock option contracts prices helps in finding
whether the stock options are properly priced. Sanjana
Juneja(2013):Trading options and option strategies are
based on risk factors and it can be predicted by
calculating Black-Scholes and its Greeks.
III.
34
STATEMENT OF THE PROBLEM
International Journal of Research in Management, Science & Technology (E-ISSN: 2321-3264)
Vol. 3, No. 1, April 2015
Available at www.ijrmst.org
The volatility is considered an important function, this
influence the buyer and sellers in option market. The
indifference curves of investor are the dominating factor
in purchasing and selling of Call and Put option. These
calculations provides the solution to this problem by
calculating B-S Model and its Greek letters.
Delta
  e  q N(d1 )
  eq  N(d1 ) 1
Theta
IV. OBJECTIVES OF THE STUDY
1 .Analyzing price by using black-scholes model and its
sensitivities.
2. Analyzing volatility and guide investors to hedge risk
by estimating future price.
IV.

St  s
 N(d1 )  rX  e  r N(d 2 )
2 

St  s
 N(d1 )  rX  e  r N(d 2 )
2 
METHODOLOGY
The research Study is executed by calculating the option
prices using the Black-Scholes option pricing model for
the Call option and Put options and its sensitivities. The
secondary data collected from BSE and used Single
Strike Price Movement formulas to calculate option
prices and its sensitivities. Samples are taken from top
most 5 companies and assuming that 10 percent is risk
free and volatility is 10 percent respectively on nondividend paying stock.
Gamma
1
N  d1 
St  s 
1

N  d1 
St  s 

Vega
Formulas/Models Used:
Black-Scholes Model
  St   N  d1 
Ct  S t N d1   Xe  r N d 2 
Rho
Pt  Xe N  d2   St N  d1 
rho  X  e  r N(d 2 )
 r
s 
S   
ln  t    r 
2
X 
d2 
s 
,
rho   X  e  r N(d 2 )
2 
S  
ln  t    r  s 
2 
X 
d1 
2
s
  St   N  d1 


  d  
1
s
VI. RESULT ANALYSIS
Collected data is analyzed using the above mentioned
models and results drawn are tabulated as follows.
TABLE-1
CALCULATION OF PREMIUM AND ITS SENSITIVITIES
Companies
Particulars
Price of the underlying
Risk-free interest rate (%)
R : risk free rate of interest
Strike price
Annual volatility (%)
2321-3264/Copyright©2015, IJRMST, April 2015
Apollo
Tyers
Call Option
Bank
Baroda
Put Option
Bosch Ltd
Call
Option
Sun TV
Put
Option
Ambuja
Cement Ltd
Call Option
173.75
10.0
0.1
180
10.0
26222.20
10.0
0.1
26200
10.0
177.20
10.0
0.1
180
10.0
434.25
10.0
0.1
450
10.0
260.90
10.0
0.1
280
10.0
35
International Journal of Research in Management, Science & Technology (E-ISSN: 2321-3264)
Vol. 3, No. 1, April 2015
Available at www.ijrmst.org
o : volatility
Time to expiration (days left)
T-t : time to expiration
Price of Call Option
Price of Put Option
Delta for Call Option
Delta for Put Option
Theta for Call Option
Theta for Put Option
Gamma for Call Option
Gamma for Put Option
Vega for Call Option
Vega for Put Option
Rho for Call Option
Rho for Put Option
0.1
15
0.0411
0.10
5.61
0.063
-0.937
-0.017
0.032
0.035
0.035
0.044
0.044
0.004
-0.069
From the above table, some of the inferences
drawn are mentioned below.
1. It was observed that Apollo Tyers call option value
0.10 and put value 5.61.Delta 0.063 for call showing
market is increasing and for put value -0.937 and
showing underlying is increasing. Theta value -0.017
and 0.032 because time to expiry is near. Gama for both
call and put is 0.035 based on time to expiry. Vega is
positive 0.044 when volatility increases. Both call and
put value 0.004 and -0.069 changes very small friction
based on 1 unit change in interest rate.
2. It was observed that Bank Baroda call option value
197.66 and put option 118.10.Delta 0.584 for call
showing market is increasing and for put value-0.416
and showing underlying is increasing. Theta value 13.606 and -6.444 because time to expiry is near. Gama
for both call and put is 0.001 based on time to expiry.
Vega is positive 15.141 when volatility increases. Both
call and put value 3.314 and -2.416 changes very small
friction based on 1 unit change in interest rate.
3. It was observed that Bosch Ltd call option value 0.64
and put option 2.70.Delta 0.288 for call showing market
is increasing and for put value-0.712 and showing
underlying is increasing. Theta value -0.055 and -0.005
because time to expiry is near. Gama for both call and
put is 0.095 based on time to expiry. Vega is positive
0.122 when volatility increases. Both call and put value
0.021 and -0.053 changes very small friction based on 1
unit change in interest rate.
4. It was observed that Sun TV call option value 1.07
and put option 13.26.Delta 0.167 for call showing
market is increasing and for put value-0.833 and
showing underlying is increasing. Theta value -0.072
and 0.050 because time to expiry is near. Gama for both
call and put is 0.020 based on time to expiry. Vega is
2321-3264/Copyright©2015, IJRMST, April 2015
0.1
0.1
0.1
0.1
8
15
29
1
0.0219
0.0411
0.0795
0.0027
197.66
0.64
1.07
0.00
118.10
2.70
13.26
19.02
0.584
0.288
0.167
0.000
-0.416
-0.712
-0.833
-1.000
-13.606
-0.055
-0.072
-0.000
-6.444
-0.005
0.050
-0.077
0.001
0.095
0.020
0.000
0.001
0.095
0.020
0.000
15.141
0.122
0.306
0.000
15.141
0.122
0.306
0.000
3.314
0.021
0.057
0.000
-2.416
-0.053
-0.298
-0.008
positive 0.306 when volatility increases. Both call and
put value 0.057 and -0.298 changes very small friction
based on 1 unit change in interest rate.
5. It was observed that Sun TV call option value 0.00
and put option 19.02.Delta 0.000 for call showing
market is increasing and for put value-1.000 and
showing underlying is increasing. Theta value -0.000
and -0.077 because time to expiry is near. Gama for
both call and put is 0.000 because one day maturity.
Vega is 0.000 when volatility low .Both call and put
value 0.000 and -0.008 changes very small friction
based on 1 unit change in interest rate.
VII. SUGGESTIONS





The investor should wait for a time to increase in
underlying value to make profits in Apollo Tyres.
Stike price should increase in Bank Baroda.
The investor is advised to purchase a call option in
Bosch Ltd and Amudha Cement Ltd.
The investor can generate in-the-money in Sun TV
option .
By calculating sensitivities one can evaluate the
price of option exactly.
VIII. CONCLUSIONS
Each company values are different it operates with
demand factors for a particular industry. The options,
either call or put in European type moves with nonlinear payoff for both parties. This makes the investors
to understand how to price an option strategically and
make in-the-money in the option market. The BlackScholes model gives you a price of option for forward,
based on some parameters (spot price, strike price,
interest, volatility etc).The Greek letters are used to
understand to identify the market price fluctuation or
simply it is used to calculate risk sensitivities towards
36
International Journal of Research in Management, Science & Technology (E-ISSN: 2321-3264)
Vol. 3, No. 1, April 2015
Available at www.ijrmst.org
price change. This paper educated the investors how to
behave in the option market.
REFERENCES
[1]
[2]
[3]
[4]
John C. Hull “Options, Futures and other derivatives” Prentice
Hall, Seventh edition.
Black and Scholes, “The pricing of options and corporate
liabilities” Journal of political economy, May 1973.
Xisheng Yu and Xiaoke Xie” On Derivations of Black-Scholes
Greek Letters” ISSN 2222-1697 (Paper) ISSN 2222-2847
(Online) Vol.4, No.6, 2013.
Hong-Yi Chen, Rutgers University, Cheng-Few Lee, Rutgers
University, and Weikang Shih, Rutgers University, USA
2321-3264/Copyright©2015, IJRMST, April 2015
[5]
[6]
[7]
[8]
[9]
37
“Derivations and Applications of Greek Letters – Review and
Integration” 2008.
Beata Stehlikova ”Black-Scholes model: Greeks - sensitivity
analysis” 2015.
Dr. S. Saravanan and G. Pradeep Kumar “Estimation of Stock
Option Prices Using Black-Scholes Model” Volume No. 2
(2012), Issue No. 11 (Nov) ISSN 2231-5756.
Sanjana Juneja “Understanding the greeks and their use to
measure risk” Volume No. 3 (2013), Issue No. 10 (Oct) ISSN
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www.bscindia.com