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Risk Management Assignment

Module 5

On

Black and Scholes model with real time data of TATA steel ltd

Under the Guidance of Prepared by

PROF. (Dr.) RAJKUMARI SONI Jugal R patel

Heta mehta

PARUL INSTITUTE OF ENGINEERING & TECHNOLOGY (MBA Dept.)

LIMDA, DIST. VADODARA.

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TABLE OF CONTENT

Sr. No. Particulars Page No.

1. Basic Introduction and assumption 3

2. Equations of Black-Scholes Model 4

3. Calculation of call and put prices based on real time data 6

4 Conclusion 9

5 Bibliography 10

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Basic introductionThe Black-Scholes Option Pricing Formula

In 1973 Fisher Black and Myron Scholes brilliantly developed the Black-

Scholes option pricing formula. Assuming that the price of the underlying stock

follows a lognormal random variable, they found a closed form solution for the price

of a European call and European put. Essentially their method was to extend the

arbitrage pricing approach developed in Section III by letting the length of a period in

the binomial model go to zero. Using the binomial approximation found in Section V,

they were able to find the following formulas for pricing European puts and calls.

Black-Scholes Model AssumptionsThe BlackScholes model of the market for a particular stock makes

the following explicit assumptions:

There is no arbitrage opportunity (i.e., there is no way to make a risklessprofit).

It is possible to borrow and lend cash at a known constant risk-freeinterest rate.

It is possible to buy and sell any amount, even fractional; of stock (thisincludes short selling).

The above transactions do not incur any fees or costs (i.e., frictionlessmarket).

The stock price follows a geometric Brownian motion with constant driftand volatility.

The underlying security does not pay a dividend.

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Equations of Black-Scholes Model

Let

S0= Todays stock price.

t = Duration of the option.

Sx = Exercise or strike price.

r = risk-free rate. This rate is assumed to be continuously compounded.

= Annual volatility of stock

y = percentage of stock value paid annually in dividends.

For a European call option the price is computed as follows:.

C0 = S0 N (d1) SX N(d2) e-rt

Where,

T

TrXSd

)2/2()/ln( 01

TdT

TrXSd

10

2

)2/2()/ln(

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For put option the price is computed as follows:

)()( 102 dNSdNeXprT

Where,

N(d1) is the probability P(xd1). Since a

normal distribution is systematic, we can write:

)( 1dN = P (xd1) = 1-N(d1)

)( 2dN = P (xd2) = 1-N(d2)

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Calculation of call and put prices based on real time data ofTATA steel limited.

S0 = 299.8

SX = 301.45

r = 8%

t = 43/365

= 32.246

Call price:Step:1_calculation of d1 and d2.

TTrXSd

)2/2

()/ln( 01

= ln(299.8 / 301.45) + {0.08 + (0.32*0.32/2)}*43/365

0.32 * 118.0

= - 0.0055 + 0.0155

0.1098

= 0.0911

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TdT

TrXSd

10

2

)2/2()/ln(

= (-0.0187)

Step:2 value of N(d1) and N(d2).

As per the normal distribution value of both the variable are as follows:

N(d1)= N(0.0911)=0.5359

N(d2)=N(-0.0187)= 0.4920

Step:3 calculation of call price.

C0 = S0 N (d1) SX N(d2) e-rt

= 13.83 INR

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Terminal value of the call option:

Terminal value of call = Max(STSX, 0)

= Max(299.8 - 301.45,0)

= Max(-1.65,0)

= 0

Time value of call = Call price + Terminal value

= 13.83 + 0

= 13.83

Gain or loss from purchasing call option:

GC

= Max [(ST

- SXC

0), - C

0]

= Max [(299.8 - 301.4513.83), - 13.83]

= Max [(-15.48), - 13.83]

= - 13.83

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Put price:We have value of N(d1) and N(d2) which are 0.6443 and 0.3632 respectively

)( 1dN =

P (xd1) = 1-N (d1)

= 1- 0.5359

=0.4641

)( 2dN = P (xd2) = 1-N(d2)

= 1- 0.4920

=0.5080

Put price:

)()( 102 dNSdNeXprT

=12.46 INR

Terminal value of the put option:

Terminal value of put = Max (SXST, 0)

= Max (301.45 - 299.8, 0)

= Max (1.65, 0)

= 1.65

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Time value of put = Put price + Terminal value

= 12.46 + 1.65

= 14.11

Gain or loss from purchasing put option:

GP = Max [P0, SXST - P0]

= Max (-12.46, 301.45 - 299.812.46)

= Max (-12.46,-10.81)= -10.81

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Conclusion:

As per the assumptions Black and Scholes showed that it is possible to create a hedgedposition, consisting of a long position in the stock and a short position in the option, whose

value will not depend on the price of the stock.

To buy a one share of TATA steel ltd investors have to pay call price of Rs 84.7 with an

expiry of 30th may 2013.

To buy one put share of TATA steel ltd the investors have to pay a Put price of Rs. 83.52with an expiry of 30th May 2013.

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Bibliography

Books:

Derivatives and risk management : By Sundaram Janakiramanan Derivatives and risk management : By Rajiv Shrivastava

Websites:

www.nseindia.com www.moneycontrol.com

http://www.nseindia.com/http://www.nseindia.com/http://www.moneycontrol.com/http://www.moneycontrol.com/http://www.moneycontrol.com/http://www.nseindia.com/