Valuation of Options

8/6/2019 Valuation of Options

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PONDICHERRY UNIVERSITYSchool of management

DEPARTMENT OF COMMERCE

VALUATION OF OPTIONS ASSIGNMENT

NAME : P. PRAVEEN KUMARCLASS : II M.COM (B.F)SUBJECT : DERIVATIVES AND RISK MANAGEMENT

SUBMITTED TO Dr. Daniel Lazar


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VALUATION OF OPTIONS

In order to understand how options work in practice it is necessary to go back to the most fundamental financial dynamic: the balance between

expected return and risk. The problem for all investors is that it is only possible to receive a higher expected return if one also is prepared to take

on more risk. But what is acceptable riskand how should it be managed?

For a financial player the risk is that one a position never guarantees a return. This uncertainty about the future value of an asset is a central issue

within all investment decisions. A measure of an assets risk in this context is price movements or volatility, which refers to the averagedeviation from the assets historical average value change. In other words, the risk of a stock is dependent on how much and how fast the pricemoves on the exchange. The problem is that risk is not entirely uniform. A stock portfolio is usually connected to three different types of risks;

company risk, industry risk and market risk. The company and industry specific risks basically all those factors that can affect the uniquecompany or its whole industry negatively can be eliminated through diversification, achieved mainly by including stocks from severalcompanies from different industries in the portfolio. The third risk, the market risk, is common for all assets on the market and cannot be

diversified away.

For options the situation is different, since options do not imply the purchasing of assets. Instead, one invests in the opportunity to share the

future price change of a stock. Thereby completely new rules are introduced in comparison with trading stocks only rules that for an outsiderjust may seem risky and complicated, but which the initiated find as logical as any other mathematical dynamics.

The value of an option is determined by its chance to be exercised with profit on the expiry day. This consists of two parts: the real value and thetime value. The real value is the value that is possible to touch. A call option has a real value if the underlying stocks price exceeds theoptions strike price. For put options it is the other way a round, in that they have real value if the strike price instead exceeds the value of thestock. Conversely the time value is the value of the possibility that good news will occur during the time to maturity in order for an option to

have a real value on the expiry day. Time value changes during the maturity period and will always be zero on the expiry day. Options that havea real value are said to be in the money and are called plus options by professionals, while options that completely miss real value are out-of-

the-money and are referred to as minus options. Options where the strike price and stock price corresponds are at -the-money and are calledpari options.

The value of the option can be estimated with a mathematical formula named Black & Scholes after its inventors. In the formula the price is

calculated as a function of the underlying stock value, the strike price, the time to maturity and the level of the risk free interest rate, among

others. All terms in the equation can be determined relatively easy except one: the stocks volatility. The risk measure that is interesting when


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one deal with options is the so called implied volatility, which contains the premium that the market has set on the option . Contrary tohistorical volatility, implied volatility measures the markets expectations on the future changes in the stock price. This is crucial, as it is whenan investor has a different opinion to the general market about the future risk of a stock that it becomes possible to enter and make money from

an option, since its premium then will be different than what it should be.

There are a large number of strategies that one can use in order to profit from the possibilities of options. Learning the differences and

advantages between different options is critical to success, as options can appear superficially similar. It is seldom enough just to look at theoptions premium and the strike price. Real professionals also look at how sensitive the options are for the market climate. By using Black &Scholes formula one can derive several important sensitivity measures commonly mentioned as the Greeks since they have been providedwith Greek lettersthat can clearly tell how an option will react from different market conditions.

An options valuation method developed by Cox, et al, in 1979. The binomial option pricing model uses an iterative procedure, allowing for the

specification of nodes, or points in time, during the time span between the valuation date and the options expiration date.

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.

FUNDAMENTAL VALUE:

FUNDAMENTAL VALUE OF CALL = (Market price of underlying equityStrike price in the call) * 100

FUNDAMENTAL VALUE OF PUT = (Strike price on the putMarket price of underlying equity) * 100

VALUATION OF OPTIONS MODELS

BLACK-SCHOLES METHODBINOMIAL METHOD

1. OPTION EQUIVALENT2. RISK NEUTRAL

ONE STEP & n STEP


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BINOMIAL MODEL

In finance, the binomial options pricing model (BOPM) provides a generalizable numerical method for the valuation ofoptions. The binomial

model was first proposed by Cox, Rossand Rubinstein (1979). Essentially, the model uses a "discrete-time" (lattice based) model of the varying

price over time of the underlying financial instrument. In general, Georgiadis showed that binomial options pricing models do not have closed-

form solutions.

The binomial model is a continuous time model in the limit. It is called binomial because it assumes that during the most period of timeshare prices will go to only one of two values. Although this assumption might seem to be a strange one on which to develop a practical

valuation model, it really is not if the investor thinks of a period of time as being very short and of the eventual expira tion date as being manyperiods from now.

This method can be shown under

1. Option Equivalent:

STEP 1: Estimate highest value of call assuming that spot price of u / l asset on expiration date will increase certain level in future.

Cu = SuE

Where,

Cu = Max call price

Su = Max expected spot price of u/l share as pursued by investor

E = Exercise price

STEP 2: Estimate lowest value of call assuming that spot price of u / l asset on expiration date will decrease certain level in future.

Cd = Max [ (SdE) or 0 ]
http://en.wikipedia.org/wiki/Financehttp://en.wikipedia.org/wiki/Numerical_analysishttp://en.wikipedia.org/wiki/Option_(finance)http://en.wikipedia.org/wiki/John_C._Coxhttp://en.wikipedia.org/wiki/Stephen_Ross_(economist)http://en.wikipedia.org/wiki/Mark_Rubinsteinhttp://en.wikipedia.org/wiki/Lattice_model_(finance)http://en.wikipedia.org/wiki/Underlyinghttp://en.wikipedia.org/w/index.php?title=Evangelos_Georgiadis_(mathematician)&action=edit&redlink=1http://en.wikipedia.org/w/index.php?title=Evangelos_Georgiadis_(mathematician)&action=edit&redlink=1http://en.wikipedia.org/wiki/Underlyinghttp://en.wikipedia.org/wiki/Lattice_model_(finance)http://en.wikipedia.org/wiki/Mark_Rubinsteinhttp://en.wikipedia.org/wiki/Stephen_Ross_(economist)http://en.wikipedia.org/wiki/John_C._Coxhttp://en.wikipedia.org/wiki/Option_(finance)http://en.wikipedia.org/wiki/Numerical_analysishttp://en.wikipedia.org/wiki/Finance


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Where,

Cd = Min call price

Sd = Min expected spot price of u/l share as pursued by investor

E = Exercise price

STEP 3: Calculation of hedge ratio

H = CuCd / SuSd

STEP 4: Estimating funds to be borrowed to create hedge Portfolio

B = dcuucd / (u-d) (1+r)t

Where,

d = Lowest multiple of current spot upto which spot price on exercise may decline

u = Highest multiple of current spot price to which spot price on exercise price may increase

r = lowest RF

t = time duration

STEP 5:

C = h*s-b


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EXAMPLE:

Current market price of the shares of X ltd is Rs.100 & option is being traded with exercise price of Rs.115 for a call option with 12 months to

expiration. This is expected that spot price of these shares at the end of 12m from now might increase by 60% of current spot price or it might

decline by 20% of the current spot price. If risk free rate of interest is 10% p.a. find out price of call option.

Solution:

Su = 160 (highest)

Sd = 80

E = 115

Cu = SuE

= 160115= 45.

Cd = [(SdE) or 0]

= (80115) or 0

= 0.

u = 160 / 100 = 1.6

d = 80 / 100 = 0.8

Hedge ratio = CuCd / SuSd

= 450 / 16080


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= 45 / 85

= 0.5625

= d Cu u Cd / (ud) * (1+r)t

= 0.8 * 451.6 * 0 / (1.60.8) (1+0.10)12/12.1

= 360 / 0.8 * 1.1

= 40.90

Call price = (h * S)B

= (0.5625 * 100)40.90

= 56.2540.90

= 15.35

2. Risk Neutral:

STEP 1: Estimate highest value of call assuming that spot price of u / l asset on expiration date will increase certain level in future.

Cu = SuE

Where,

Cu = Max call price

Su = Max expected spot price of u/l share as pursued by investor


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E = Exercise price

STEP 2: Estimate lowest value of call assuming that spot price of u / l asset on expiration date will decrease certain level in future.

Cd = Max [ (SdE) or 0 ]

Where,

Cd = Min call price

Sd = Min expected spot price of u/l share as pursued by investor

E = Exercise price

STEP 3: Expected Return for time

E(r) = p * (% increase in spot price) + (1 - p) * (% decrease in spot price)

STEP 4: Calculate future value of call option

F = Cu * p + ( Cd * (1 - p) )

STEP 5: Calculate present value of call option

C = STEP 4 / (1+r)t


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Valuation of Put Option

P = CS + ( E / ert )

EXAMPLE:

Current market price of the shares of X ltd is Rs.100. in an option with an exercise price of Rs.115 for a call option with 12 m to expiration. It is

expected that spot price of the shares at the end of 12 m will increase by 60% of current spot free or it might decline by 20% of current spot free.

If Rf is 10% p.a. find out the prive of call option.

Solution:

Su = 160

Sd = 45

E = 115

Cu = 45

Cd = 0

E (r) = p * (% Inc in spot price)

(1p) * (% dec in spot price)

0.10 = p (0.60) + (1p) * (- 0.20)

0.10 = p (0.6) + (1p) * (.20)

0.10 = 0.60 p + (1p) * - (.20)


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= 0.60 p + (- 0.20 + 0.20 p)

= 0.60 p + 0.20 p + (-0.20)

0.10 = 0.80 p0.20

0.10 + 0.20 = 0.80 p

0.30 = 0.80 p

P = 0.30 / 0.80

= 0.375

1p = 10.375

= 0.625

4) Fv of call option

= Cu * p + [Cd * (1p)]

= (45 * 0.375) + (0 * 0.625)

= 16.875

5) PV of call option

= Exp. Future value / (1+r)t

= 16.875 / (1+.10)1

= 15.34


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N step:

Spot price at present is 20 which has a volatility of 20% from present spot price by the expiry of option contract period. If strike price is 22

provided then the outcome in 2 step binomial model.

D

B

A E outcome

C

F

The probability is assumed to be 0.575 for Cu & 0.425 for Cd.

Determine the outcome

So = 20

Su = 24

Cu = 2

Sd = 16

Cd = 0

Su = 28.8

Cu = 6.8

Sd = 19.2

Cd = 0

Sd =12.8

Cd= 0


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E (r)Rf is not given, so find it.

E (r) = p * (% inc in spot price) + (1p) * (% dec in spot price)

= 0.575 * (.20) + (10.575) * (- .20)

= 0.115 + (0.425)0.20

= 0.1150.085

= 0.03 * 100

= 3 %

FV of call option = Cu * p + [Cd * (1p)]

= (6.8 * 0.575) + (0 * 0.425)

= 3.91

PV of call option = FV of call option / (1 + r) t

= 3.91 / (1 + 0.03) 3/12

= 3.91 / (1.03)0.25

The probability is assumed to be 0.575 for Cu & 0.425 for Cd. [assume Rf = 12%]

Determine the outcome

E (r) = p * Cu + (1p) Cd

= (0.575 * 6.8) + (0.425 * 0)


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= 3.91

PV of call option = 3.91 / (1.12).25

= 3.8

The value of call option is 3.8 & it is for point B.

BLACK-SCHOLES MODEL

Fischer Black and Myron Scholes developed a precise model for determining the equilibrium value of an option. The model is widely used by

those who deal with options to search for situations where the market price of an option differs substantially from its fair value. In particular, the

model provides rich insight into the valuation of debt relation to equity.

Assumptions:

The Black Scholes option model is based on following assumptions:

There are no transactions costs and no taxes. The risk from interest rate is constant. The market operates constantly. The share prices are continuous, i.e. there are no jumps in the share prices; if one plots a graph of the share price against time, the graph

must be smooth. To be more specific, the share is log normally distributed for any finite time interval.

The share pays no dividends. The option is of European type, that is, options that can be exercised only at maturity. Shares can be sold short without penalty and short sellers receive the full proceeds from the transaction.

Given these assumptions, the equilibrium value of an option can be determined. Should the actual price of the option differ from that given by

model, the investor could establish a riskless hedged position and a return in excess of the short-term interest rate. As enterprise entered the

scene, the excess return would eventually be driven out and the price of the option would equal the value given by the model.


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STEP 1:

d1 = n * (so / E) + (r +(0.5 *S.D2)) * t / S.D t

STEP 2:

d2 = d1S.D t

STEP 3:

Nd1

Nd2

STEP 4:

Co = So * (Nd1)E * e rt * (Nd2)

STEP 5:

Po = CoSo + E * E rt

EXAMPLE:

Spot price of shares of X company is Rs.60 with an exercise price of Rs.60 with time to expiration 6 months. Risk free rate of return 12% & S.D

of return of the share is 30%. Calculate the price of call option.

Solution:


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d1 = n * (so / E) + (r +(0.5 *S.D2)) * t / S.D t

= n (60 / 60) + (0.12 + 0.5 * 0.3) * 0.5 / 0.3 0.5

= n (1) + (.12 + 0.045) * 0.5 / 0.3 * 0.7071

= 0 + (.12 + 0.045) * 0.5 / 0.3 * 0.707

= 0 + (0.165) * 0.5 / 0.2121

= 0.389

d2 = d1S.D t

= 0.3890.30 0.5

= 0.3890.212

= 0.177

(Nd1) (Nd2)Normal distribution

Nd1 = 0.648

Nd2 = 0.5675

Value of call option

Co = So * (Nd1)E * e rt * (Nd2)

= 60 * (0.648)60 * (2.7182) * (0.5675)


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= 38.8860 * (2.71828) * (0.5675)

= 38.8860 * 0.94 * 0.5675

= 38.8832.007

= Rs. 6.87

Value of Put option

Po = CoSo + E * E rt

= 6.8760 + 60 * (2.71828) -0.12 * 0.3

= - 53.13 + 60 * 0.94

= - 53.13 + 56.4

= 3.27

BUTTERFLY SPREAD OPTION

You have give 3 call options on a stock at an exercise price of Rs.40, 45 & 50 with the expiration date in 3 months & the premium of RS.4, Rs.2

& Rs.1 respectively. Show how the option can be used to create a butterfly spread. Construct a table with different market prices & show how

profits changes with the stock prices ranging from Rs.30 to Rs.60 for the butterfly spread.

Market

prices

Pay off when

K=40

Pay off when

K=45

Pay off when

K=50

premium Total pay off

30 _ _ _ -1 -1


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32 _ _ _ -1 -1

34 _ _ _ -1 -1

36 _ _ _ -1 -1

38 _ _ _ -1 -1

40 _ _ _ -1 -1

42 +2 _ _ -1 +1

44 +4 _ _ -1 +3

46 +6 -2 _ -1 +3

48 +8 -6 _ -1 +1

50 +10 -10 _ -1 -1

52 +12 -14 +2 -1 -1

54 +14 -18 +4 -1 -1

56 +16 -22 +6 -1 -1

58 +18 -26 +8 -1 -1

60 +20 -30 +10 -1 -1


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PUT-CALL PARITY RELATIONSHIP

There is a relationship between the price of a call option and the price of o put option on the same underlying instrument with the same strike

prices and the same expiration date. To see this relationship, commonly referred to as the put-call parity relationship, let us assume a put and call

option on the same underlying stock (stock XYZ), with month to expiration and with a strike price of Rs.100. The call price and put price are

assumed to be Rs.5 and Rs.3, respectively.

Consider this strategy:

Buy stock XYZ at price of Rs.100

Sell a call option at a price of Rs.5

Buy a put option at a price of Rs.3

The strategy involves:

Long stock XYZ

Short the call option.

Long the put option.

Table shows the profit and loss profile at the expiration date for this strategy for selected stock prices. For the long stock position, there is no

profit. That is because at a price above Rs.100, stock XYZ will be called from the investor at a price of Rs.100, and at a price below 100, stock

XYZ will be put by the investor at a price of Rs.100. No matter what stock XY Zs price is at the expiration date, this strategy will produce aprofit of Rs.2 without anybody making any net investment. Ignoring (1) the cost of financing the long position in stock XYZ and the long

position and (2) the return from investing the proceeds from the sale of the call, this situation cannot exist in an efficient market. By

implementing the strategy to capture the Rs.2 profit, the actions of market participants will have one or more of the following consequences

which tend to eliminate the Rs.2 profit: (1) the price of the stock XYZ will increase, (2) the call option price will drop, and / or (3) the put option

price will drop.


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TABLE:

Profit/loss profile for a strategy involving a long position in stock XYZ, short call option position, and long put option position

Assumptions:

Price of stock XYZ = Rs.100

Call option price = Rs.5

Put option price = Rs.3

Strike price = Rs.100

Time to expiration = 1 month

Price of stockXYZ at expire

date

Profit fromstock XYZ*

price receivedfor call

price paid forput

overall profit

Rs.150 0 5 -3 2

0 5 -3 2

120 0 5 -3 2

110 0 5 -3 2

100 0 5 -3 2

90 0 5 -3 2


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80 0 5 -3 2

70 0 5 -3 2

*There is no profit, because at price above Rs.100, stock XYZ will be called from the investor at price of Rs.100, and at a price below 100, stock

XYZ will be put by the investor at a price of Rs.100.

Assuming stock XYZs price does not change; the call price and the put price will tend toward equality. However, this is true only when weignore the time value of money (financing cost, opportunity cost, cash payments, and reinvestment income). Also, the illustration does not

consider the possibility of early exercise of the option. Thus, we have been considering a put-call parity relationship applicable for only

European options.

It can be shown that the put-call parity relationship for an option where the underlying stock makes cash dividend is:

Put option price-call option price = present value of the strike price + present value of dividends

price of underlying stock.

This relationship is actually the put-call parity relationship for European options; it is approximately true for American options. If this

relationship does not hold, arbitrage opportunities exist. That, is portfolios consisting of long and short positions in the stock and related options

that provide an extra return (practical) certainty will exist.


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PRACTICAL VALUATION OF OPTIONS (ABB & RELIANCE COMPANY)

STEP 1: Calculation of Theoretical price of Options.

STEP 2: Comparison of Theoretical price with Actual price.

ABB: TABLE CALCULATIONS

BINOMIAL -

Symbol Date ExpiryStrike

Price (E)

Spot

Value (S )CU CD H U D B

Call price

(Theoretical

Price)

Settlement Price

(Actual Price)Difference

Cu = Su

E;

Su=811.2

Cd = MAX [

(Sd-E) OR 0];

Sd=616.95

H = CuCd /SuSd

U = Su / S D = Sd / S

B = dcu

ucd / (u-

d) (1+r)t

C = (h*s) - b

ABB 28-Feb-11 26-May-11 420 666.65 391.20 196.95 1.00 1.22 0.93 415.84 250.81 256.80 5.99

ABB 28-Feb-11 26-May-11 1060 666.65 -248.80 0.00 -1.28 1.22 0.93 -782.38 -71.48 1.60 73.08

ABB 28-Feb-11 26-May-11 460 666.65 351.20 156.95 1.00 1.22 0.93 455.45 211.20 218.95 7.75

ABB 28-Feb-11 26-May-11 480 666.65 331.20 136.95 1.00 1.22 0.93 475.25 191.40 200.55 9.15

ABB 28-Feb-11 26-May-11 500 666.65 311.20 116.95 1.00 1.22 0.93 495.05 171.60 182.65 11.05

ABB 28-Feb-11 26-May-11 520 666.65 291.20 96.95 1.00 1.22 0.93 514.85 151.80 165.35 13.55

ABB 28-Feb-11 26-May-11 540 666.65 271.20 76.95 1.00 1.22 0.93 534.65 132.00 148.80 16.80

ABB 28-Feb-11 26-May-11 560 666.65 251.20 56.95 1.00 1.22 0.93 554.46 112.19 133.05 20.86

ABB 28-Feb-11 26-May-11 580 666.65 231.20 36.95 1.00 1.22 0.93 574.26 92.39 118.25 25.86

ABB 28-Feb-11 26-May-11 600 666.65 211.20 16.95 1.00 1.22 0.93 594.06 72.59 104.40 31.81

ABB 28-Feb-11 26-May-11 620 666.65 191.20 0.00 0.98 1.22 0.93 601.25 54.93 91.60 36.67

ABB 28-Feb-11 26-May-11 640 666.65 171.20 0.00 0.88 1.22 0.93 538.36 49.19 79.95 30.76

ABB 28-Feb-11 26-May-11 660 666.65 151.20 0.00 0.78 1.22 0.93 475.47 43.44 69.30 25.86

ABB 28-Feb-11 26-May-11 680 666.65 131.20 0.00 0.68 1.22 0.93 412.57 37.69 59.75 22.06

ABB 28-Feb-11 26-May-11 700 666.65 111.20 0.00 0.57 1.22 0.93 349.68 31.95 51.25 19.30


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ABB 28-Feb-11 26-May-11 720 666.65 91.20 0.00 0.47 1.22 0.93 286.79 26.20 43.70 17.50

ABB 28-Feb-11 26-May-11 740 666.65 71.20 0.00 0.37 1.22 0.93 223.90 20.46 37.10 16.64

ABB 28-Feb-11 26-May-11 760 666.65 51.20 0.00 0.26 1.22 0.93 161.00 14.71 31.35 16.64

ABB 28-Feb-11 26-May-11 780 666.65 31.20 0.00 0.16 1.22 0.93 98.11 8.96 26.40 17.44

ABB 28-Feb-11 26-May-11 800 666.65 11.20 0.00 0.06 1.22 0.93 35.22 3.22 22.10 18.88

ABB 28-Feb-11 26-May-11 820 666.65 -8.80 0.00 -0.05 1.22 0.93 -27.67 -2.53 18.40 20.93

ABB 28-Feb-11 26-May-11 840 666.65 -28.80 0.00 -0.15 1.22 0.93 -90.56 -8.27 15.30 23.57

ABB 28-Feb-11 26-May-11 860 666.65 -48.80 0.00 -0.25 1.22 0.93 -153.46 -14.02 12.65 26.67

ABB 28-Feb-11 26-May-11 880 666.65 -68.80 0.00 -0.35 1.22 0.93 -216.35 -19.77 10.45 30.22

ABB 28-Feb-11 26-May-11 900 666.65 -88.80 0.00 -0.46 1.22 0.93 -279.24 -25.51 8.55 34.06

ABB 28-Feb-11 26-May-11 920 666.65 -108.80 0.00 -0.56 1.22 0.93 -342.13 -31.26 7.00 38.26

ABB 28-Feb-11 26-May-11 940 666.65 -128.80 0.00 -0.66 1.22 0.93 -405.03 -37.00 5.75 42.75

ABB 28-Feb-11 26-May-11 960 666.65 -148.80 0.00 -0.77 1.22 0.93 -467.92 -42.75 4.65 47.40

ABB 28-Feb-11 26-May-11 980 666.65 -168.80 0.00 -0.87 1.22 0.93 -530.81 -48.50 3.80 52.30

ABB 28-Feb-11 26-May-11 1000 666.65 -188.80 0.00 -0.97 1.22 0.93 -593.70 -54.24 3.10 57.34

ABB 28-Feb-11 26-May-11 1020 666.65 -208.80 0.00 -1.07 1.22 0.93 -656.60 -59.99 2.50 62.49

ABB 28-Feb-11 26-May-11 1040 666.65 -228.80 0.00 -1.18 1.22 0.93 -719.49 -65.73 2.00 67.73

ABB 28-Feb-11 26-May-11 440 666.65 371.20 176.95 1.00 1.22 0.93 435.64 231.01 237.75 6.74


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BLACKSCHOLES


Price

Spot

ValueLN SD SD2 RF 0.5 TIME

SQT

TIMED1 D2 ND1 ND2 e^(-rt)

C

(Theoretical

Price)

Settlement

Price

(Actual

Price)

Difference

n *

(so / E)

3

Month

d1 = n *

(so / E) + (r+(0.5

*S.D2)) * t /

S.D t

d2 =d1S.D

t

2.71828^ (-

.06*.25)

Co = So *(Nd1)E *e rt * (Nd2)

ABB 28-Feb-11 26-May-11 420 666.65 0.46 0.17 0.03 0.06 0.50 0.25 0.50 5.75 5.67 1.00 1.00 0.99 252.95 256.8 3.85

ABB 28-Feb-11 26-May-11 1060 666.65 -0.46 0.17 0.03 0.06 0.50 0.25 0.50 -5.33 -5.42 0.00 0.00 0.99 0.00 1.6 1.60

ABB 28-Feb-11 26-May-11 460 666.65 0.37 0.17 0.03 0.06 0.50 0.25 0.50 4.66 4.58 1.00 1.00 0.99 213.55 218.95 5.40

ABB 28-Feb-11 26-May-11 480 666.65 0.33 0.17 0.03 0.06 0.50 0.25 0.50 4.16 4.07 1.00 1.00 0.99 193.85 200.55 6.70

ABB 28-Feb-11 26-May-11 500 666.65 0.29 0.17 0.03 0.06 0.50 0.25 0.50 3.67 3.58 1.00 1.00 0.99 174.15 182.65 8.50

ABB 28-Feb-11 26-May-11 520 666.65 0.25 0.17 0.03 0.06 0.50 0.25 0.50 3.20 3.11 1.00 1.00 0.99 154.46 165.35 10.89

ABB 28-Feb-11 26-May-11 540 666.65 0.21 0.17 0.03 0.06 0.50 0.25 0.50 2.74 2.66 1.00 1.00 0.99 134.80 148.8 14.00

ABB 28-Feb-11 26-May-11 560 666.65 0.17 0.17 0.03 0.06 0.50 0.25 0.50 2.31 2.23 0.99 0.99 0.99 115.25 133.05 17.80

ABB 28-Feb-11 26-May-11 580 666.65 0.14 0.17 0.03 0.06 0.50 0.25 0.50 1.89 1.81 0.97 0.96 0.99 96.00 118.25 22.25

ABB 28-Feb-11 26-May-11 600 666.65 0.11 0.17 0.03 0.06 0.50 0.25 0.50 1.48 1.40 0.93 0.92 0.99 77.40 104.4 27.00

ABB 28-Feb-11 26-May-11 620 666.65 0.07 0.17 0.03 0.06 0.50 0.25 0.50 1.09 1.01 0.86 0.84 0.99 59.98 91.6 31.62

ABB 28-Feb-11 26-May-11 640 666.65 0.04 0.17 0.03 0.06 0.50 0.25 0.50 0.71 0.63 0.76 0.73 0.99 44.39 79.95 35.56

ABB 28-Feb-11 26-May-11 660 666.65 0.01 0.17 0.03 0.06 0.50 0.25 0.50 0.34 0.26 0.63 0.60 0.99 31.19 69.3 38.11

ABB 28-Feb-11 26-May-11 680 666.65 -0.02 0.17 0.03 0.06 0.50 0.25 0.50 -0.02 -0.10 0.49 0.46 0.99 20.71 59.75 39.04

ABB 28-Feb-11 26-May-11 700 666.65 -0.05 0.17 0.03 0.06 0.50 0.25 0.50 -0.36 -0.45 0.36 0.33 0.99 12.97 51.25 38.28

ABB 28-Feb-11 26-May-11 720 666.65 -0.08 0.17 0.03 0.06 0.50 0.25 0.50 -0.70 -0.78 0.24 0.22 0.99 7.64 43.7 36.06

ABB 28-Feb-11 26-May-11 740 666.65 -0.10 0.17 0.03 0.06 0.50 0.25 0.50 -1.03 -1.11 0.15 0.13 0.99 4.24 37.1 32.86

ABB 28-Feb-11 26-May-11 760 666.65 -0.13 0.17 0.03 0.06 0.50 0.25 0.50 -1.35 -1.43 0.09 0.08 0.99 2.22 31.35 29.13

ABB 28-Feb-11 26-May-11 780 666.65 -0.16 0.17 0.03 0.06 0.50 0.25 0.50 -1.66 -1.74 0.05 0.04 0.99 1.09 26.4 25.31

ABB 28-Feb-11 26-May-11 800 666.65 -0.18 0.17 0.03 0.06 0.50 0.25 0.50 -1.96 -2.05 0.02 0.02 0.99 0.51 22.1 21.59

ABB 28-Feb-11 26-May-11 820 666.65 -0.21 0.17 0.03 0.06 0.50 0.25 0.50 -2.26 -2.34 0.01 0.01 0.99 0.23 18.4 18.17


24/30

ABB 28-Feb-11 26-May-11 840 666.65 -0.23 0.17 0.03 0.06 0.50 0.25 0.50 -2.55 -2.63 0.01 0.00 0.99 0.09 15.3 15.21

ABB 28-Feb-11 26-May-11 860 666.65 -0.25 0.17 0.03 0.06 0.50 0.25 0.50 -2.83 -2.91 0.00 0.00 0.99 0.04 12.65 12.61

ABB 28-Feb-11 26-May-11 880 666.65 -0.28 0.17 0.03 0.06 0.50 0.25 0.50 -3.10 -3.19 0.00 0.00 0.99 0.01 10.45 10.44

ABB 28-Feb-11 26-May-11 900 666.65 -0.30 0.17 0.03 0.06 0.50 0.25 0.50 -3.37 -3.46 0.00 0.00 0.99 0.01 8.55 8.54

ABB 28-Feb-11 26-May-11 920 666.65 -0.32 0.17 0.03 0.06 0.50 0.25 0.50 -3.64 -3.72 0.00 0.00 0.99 0.00 7 7.00

ABB 28-Feb-11 26-May-11 940 666.65 -0.34 0.17 0.03 0.06 0.50 0.25 0.50 -3.89 -3.98 0.00 0.00 0.99 0.00 5.75 5.75

ABB 28-Feb-11 26-May-11 960 666.65 -0.36 0.17 0.03 0.06 0.50 0.25 0.50 -4.15 -4.23 0.00 0.00 0.99 0.00 4.65 4.65

ABB 28-Feb-11 26-May-11 980 666.65 -0.39 0.17 0.03 0.06 0.50 0.25 0.50 -4.39 -4.48 0.00 0.00 0.99 0.00 3.8 3.80

ABB 28-Feb-11 26-May-11 1000 666.65 -0.41 0.17 0.03 0.06 0.50 0.25 0.50 -4.63 -4.72 0.00 0.00 0.99 0.00 3.1 3.10

ABB 28-Feb-11 26-May-11 1020 666.65 -0.43 0.17 0.03 0.06 0.50 0.25 0.50 -4.87 -4.96 0.00 0.00 0.99 0.00 2.5 2.50

ABB 28-Feb-11 26-May-11 1040 666.65 -0.44 0.17 0.03 0.06 0.50 0.25 0.50 -5.10 -5.19 0.00 0.00 0.99 0.00 2 2.00

ABB 28-Feb-11 26-May-11 440 666.65 0.42 0.17 0.03 0.06 0.50 0.25 0.50 5.20 5.11 1.00 1.00 0.99 233.25 237.75 4.50


25/30

RELIANCE: TABLE CALCULATIONS

BINOMIAL -


Price (E)

Spot

Value (S )CU CD H U D B

Call price

(Theoretical

Price)

Settlement Price

(Actual Price)Difference

Cu = Su

E;

Su=1085.6

Cd = MAX [

(Sd-E) OR 0];

Sd=895.5

H = CuCd / Su

Sd

U =

Su /

S

D = Sd

/ S

B = dcu

ucd / (u-

d) (1+r)t

C = (h*s) - b

RELIANCE 28-Feb-11 26-May-11 560 964.25 525.6 335.5 1.00 1.13 0.93 554.46 409.79 417.05 7.26

RELIANCE 28-Feb-11 26-May-11 1340 964.25 -254.4 0 -1.34 1.13 0.93 -1186.53 -103.87 2.50 106.37

RELIANCE 28-Feb-11 26-May-11 600 964.25 485.6 295.5 1.00 1.13 0.93 594.06 370.19 378.00 7.81

RELIANCE 28-Feb-11 26-May-11 620 964.25 465.6 275.5 1.00 1.13 0.93 613.86 350.39 358.50 8.11

RELIANCE 28-Feb-11 26-May-11 640 964.25 445.6 255.5 1.00 1.13 0.93 633.66 330.59 339.05 8.46

RELIANCE 28-Feb-11 26-May-11 660 964.25 425.6 235.5 1.00 1.13 0.93 653.47 310.78 319.65 8.87

RELIANCE 28-Feb-11 26-May-11 680 964.25 405.6 215.5 1.00 1.13 0.93 673.27 290.98 300.35 9.37

RELIANCE 28-Feb-11 26-May-11 700 964.25 385.6 195.5 1.00 1.13 0.93 693.07 271.18 281.20 10.02

RELIANCE 28-Feb-11 26-May-11 720 964.25 365.6 175.5 1.00 1.13 0.93 712.87 251.38 262.25 10.87

RELIANCE 28-Feb-11 26-May-11 740 964.25 345.6 155.5 1.00 1.13 0.93 732.67 231.58 243.55 11.97

RELIANCE 28-Feb-11 26-May-11 760 964.25 325.6 135.5 1.00 1.13 0.93 752.48 211.77 225.20 13.43

RELIANCE 28-Feb-11 26-May-11 780 964.25 305.6 115.5 1.00 1.13 0.93 772.28 191.97 207.25 15.28

RELIANCE 28-Feb-11 26-May-11 800 964.25 285.6 95.5 1.00 1.13 0.93 792.08 172.17 189.80 17.63

RELIANCE 28-Feb-11 26-May-11 820 964.25 265.6 75.5 1.00 1.13 0.93 811.88 152.37 172.90 20.53

RELIANCE 28-Feb-11 26-May-11 840 964.25 245.6 55.5 1.00 1.13 0.93 831.68 132.57 156.70 24.13

RELIANCE 28-Feb-11 26-May-11 860 964.25 225.6 35.5 1.00 1.13 0.93 851.49 112.76 141.25 28.49

RELIANCE 28-Feb-11 26-May-11 880 964.25 205.6 15.5 1.00 1.13 0.93 871.29 92.96 126.70 33.74

RELIANCE 28-Feb-11 26-May-11 900 964.25 185.6 0 0.98 1.13 0.93 865.65 75.78 112.90 37.12

RELIANCE 28-Feb-11 26-May-11 920 964.25 165.6 0 0.87 1.13 0.93 772.36 67.61 100.10 32.49

RELIANCE 28-Feb-11 26-May-11 940 964.25 145.6 0 0.77 1.13 0.93 679.08 59.45 88.20 28.75

RELIANCE 28-Feb-11 26-May-11 960 964.25 125.6 0 0.66 1.13 0.93 585.80 51.28 77.35 26.07


26/30

RELIANCE 28-Feb-11 26-May-11 980 964.25 105.6 0 0.56 1.13 0.93 492.52 43.12 67.40 24.28

RELIANCE 28-Feb-11 26-May-11 1000 964.25 85.6 0 0.45 1.13 0.93 399.24 34.95 58.45 23.50

RELIANCE 28-Feb-11 26-May-11 1020 964.25 65.6 0 0.35 1.13 0.93 305.96 26.78 50.40 23.62

RELIANCE 28-Feb-11 26-May-11 1040 964.25 45.6 0 0.24 1.13 0.93 212.68 18.62 43.25 24.63

RELIANCE 28-Feb-11 26-May-11 1060 964.25 25.6 0 0.13 1.13 0.93 119.40 10.45 36.90 26.45

RELIANCE 28-Feb-11 26-May-11 1080 964.25 5.6 0 0.03 1.13 0.93 26.12 2.29 31.30 29.01

RELIANCE 28-Feb-11 26-May-11 1100 964.25 -14.4 0 -0.08 1.13 0.93 -67.16 -5.88 26.50 32.38

RELIANCE 28-Feb-11 26-May-11 1120 964.25 -34.4 0 -0.18 1.13 0.93 -160.44 -14.05 22.25 36.30

RELIANCE 28-Feb-11 26-May-11 1140 964.25 -54.4 0 -0.29 1.13 0.93 -253.72 -22.21 18.65 40.86

RELIANCE 28-Feb-11 26-May-11 1160 964.25 -74.4 0 -0.39 1.13 0.93 -347.00 -30.38 15.55 45.93

RELIANCE 28-Feb-11 26-May-11 1180 964.25 -94.4 0 -0.50 1.13 0.93 -440.29 -38.54 12.90 51.44

RELIANCE 28-Feb-11 26-May-11 1200 964.25 -114.4 0 -0.60 1.13 0.93 -533.57 -46.71 10.65 57.36

RELIANCE 28-Feb-11 26-May-11 1220 964.25 -134.4 0 -0.71 1.13 0.93 -626.85 -54.87 8.75 63.62

RELIANCE 28-Feb-11 26-May-11 1240 964.25 -154.4 0 -0.81 1.13 0.93 -720.13 -63.04 7.20 70.24

RELIANCE 28-Feb-11 26-May-11 1260 964.25 -174.4 0 -0.92 1.13 0.93 -813.41 -71.21 5.85 77.06

RELIANCE 28-Feb-11 26-May-11 1280 964.25 -194.4 0 -1.02 1.13 0.93 -906.69 -79.37 4.75 84.12

RELIANCE 28-Feb-11 26-May-11 1300 964.25 -214.4 0 -1.13 1.13 0.93 -999.97 -87.54 3.85 91.39

RELIANCE 28-Feb-11 26-May-11 1320 964.25 -234.4 0 -1.23 1.13 0.93 -1093.25 -95.70 3.10 98.80

RELIANCE 28-Feb-11 26-May-11 580 964.25 505.6 315.5 1.00 1.13 0.93 574.26 389.99 397.50 7.51


27/30

BLACKSCHOLES


Price

Spot

ValueLN SD SD2 RF 0.5 TIME

SQT

TIMED1 D2 ND1 ND2 e^(-rt)

C

(Theoretical

Price)

Settlement

Price

(Actual

Price)

Difference

n *(so /

E)

3

Month

d1 = n *

(so / E) + (r+(0.5

*S.D2)) * t /

S.D t

d2 =d1

S.D t

2.71828^ (-

.06*.25)

Co = So *(Nd1)E *e rt * (Nd2)

RELIANCE 28-Feb-11 26-May-11 560 964.25 0.54 0.16 0.03 0.06 0.50 0.25 0.50 7.02 6.94 1.00 1.00 0.99 412.65 417.05 4.40

RELIANCE 28-Feb-11 26-May-11 1340 964.25 -0.33 0.16 0.03 0.06 0.50 0.25 0.50 -3.89 -3.97 0.00 0.00 0.99 0.00 2.50 2.50

RELIANCE 28-Feb-11 26-May-11 600 964.25 0.47 0.16 0.03 0.06 0.50 0.25 0.50 6.16 6.08 1.00 1.00 0.99 373.25 378.00 4.75

RELIANCE 28-Feb-11 26-May-11 620 964.25 0.44 0.16 0.03 0.06 0.50 0.25 0.50 5.75 5.67 1.00 1.00 0.99 353.55 358.50 4.95

RELIANCE 28-Feb-11 26-May-11 640 964.25 0.41 0.16 0.03 0.06 0.50 0.25 0.50 5.35 5.27 1.00 1.00 0.99 333.85 339.05 5.20

RELIANCE 28-Feb-11 26-May-11 660 964.25 0.38 0.16 0.03 0.06 0.50 0.25 0.50 4.97 4.89 1.00 1.00 0.99 314.15 319.65 5.50

RELIANCE 28-Feb-11 26-May-11 680 964.25 0.35 0.16 0.03 0.06 0.50 0.25 0.50 4.59 4.51 1.00 1.00 0.99 294.45 300.35 5.90

RELIANCE 28-Feb-11 26-May-11 700 964.25 0.32 0.16 0.03 0.06 0.50 0.25 0.50 4.23 4.15 1.00 1.00 0.99 274.75 281.20 6.45

RELIANCE 28-Feb-11 26-May-11 720 964.25 0.29 0.16 0.03 0.06 0.50 0.25 0.50 3.88 3.80 1.00 1.00 0.99 255.05 262.25 7.20

RELIANCE 28-Feb-11 26-May-11 740 964.25 0.26 0.16 0.03 0.06 0.50 0.25 0.50 3.54 3.46 1.00 1.00 0.99 235.35 243.55 8.20

RELIANCE 28-Feb-11 26-May-11 760 964.25 0.24 0.16 0.03 0.06 0.50 0.25 0.50 3.20 3.12 1.00 1.00 0.99 215.66 225.20 9.54

RELIANCE 28-Feb-11 26-May-11 780 964.25 0.21 0.16 0.03 0.06 0.50 0.25 0.50 2.88 2.80 1.00 1.00 0.99 196.00 207.25 11.25

RELIANCE 28-Feb-11 26-May-11 800 964.25 0.19 0.16 0.03 0.06 0.50 0.25 0.50 2.56 2.48 0.99 0.99 0.99 176.38 189.80 13.42

RELIANCE 28-Feb-11 26-May-11 820 964.25 0.16 0.16 0.03 0.06 0.50 0.25 0.50 2.25 2.17 0.99 0.99 0.99 156.88 172.90 16.02

RELIANCE 28-Feb-11 26-May-11 840 964.25 0.14 0.16 0.03 0.06 0.50 0.25 0.50 1.95 1.87 0.97 0.97 0.99 137.61 156.70 19.09

RELIANCE 28-Feb-11 26-May-11 860 964.25 0.11 0.16 0.03 0.06 0.50 0.25 0.50 1.66 1.58 0.95 0.94 0.99 118.76 141.25 22.49

RELIANCE 28-Feb-11 26-May-11 880 964.25 0.09 0.16 0.03 0.06 0.50 0.25 0.50 1.37 1.29 0.91 0.90 0.99 100.56 126.70 26.14

RELIANCE 28-Feb-11 26-May-11 900 964.25 0.07 0.16 0.03 0.06 0.50 0.25 0.50 1.09 1.01 0.86 0.84 0.99 83.34 112.90 29.56

RELIANCE 28-Feb-11 26-May-11 920 964.25 0.05 0.16 0.03 0.06 0.50 0.25 0.50 0.81 0.73 0.79 0.77 0.99 67.42 100.10 32.68

RELIANCE 28-Feb-11 26-May-11 940 964.25 0.03 0.16 0.03 0.06 0.50 0.25 0.50 0.55 0.47 0.71 0.68 0.99 53.13 88.20 35.07

RELIANCE 28-Feb-11 26-May-11 960 964.25 0.00 0.16 0.03 0.06 0.50 0.25 0.50 0.28 0.20 0.61 0.58 0.99 40.70 77.35 36.65


28/30

RELIANCE 28-Feb-11 26-May-11 980 964.25 -0.02 0.16 0.03 0.06 0.50 0.25 0.50 0.02 -0.06 0.51 0.48 0.99 30.26 67.40 37.14

RELIANCE 28-Feb-11 26-May-11 1000 964.25 -0.04 0.16 0.03 0.06 0.50 0.25 0.50 -0.23 -0.31 0.41 0.38 0.99 21.82 58.45 36.63

RELIANCE 28-Feb-11 26-May-11 1020 964.25 -0.06 0.16 0.03 0.06 0.50 0.25 0.50 -0.48 -0.56 0.32 0.29 0.99 15.24 50.40 35.16

RELIANCE 28-Feb-11 26-May-11 1040 964.25 -0.08 0.16 0.03 0.06 0.50 0.25 0.50 -0.72 -0.80 0.24 0.21 0.99 10.31 43.25 32.94

RELIANCE 28-Feb-11 26-May-11 1060 964.25 -0.09 0.16 0.03 0.06 0.50 0.25 0.50 -0.96 -1.04 0.17 0.15 0.99 6.75 36.90 30.15

RELIANCE 28-Feb-11 26-May-11 1080 964.25 -0.11 0.16 0.03 0.06 0.50 0.25 0.50 -1.19 -1.27 0.12 0.10 0.99 4.29 31.30 27.01

RELIANCE 28-Feb-11 26-May-11 1100 964.25 -0.13 0.16 0.03 0.06 0.50 0.25 0.50 -1.42 -1.50 0.08 0.07 0.99 2.64 26.50 23.86

RELIANCE 28-Feb-11 26-May-11 1120 964.25 -0.15 0.16 0.03 0.06 0.50 0.25 0.50 -1.64 -1.72 0.05 0.04 0.99 1.57 22.25 20.68

RELIANCE 28-Feb-11 26-May-11 1140 964.25 -0.17 0.16 0.03 0.06 0.50 0.25 0.50 -1.87 -1.95 0.03 0.03 0.99 0.91 18.65 17.74

RELIANCE 28-Feb-11 26-May-11 1160 964.25 -0.18 0.16 0.03 0.06 0.50 0.25 0.50 -2.08 -2.16 0.02 0.02 0.99 0.51 15.55 15.04

RELIANCE 28-Feb-11 26-May-11 1180 964.25 -0.20 0.16 0.03 0.06 0.50 0.25 0.50 -2.30 -2.38 0.01 0.01 0.99 0.28 12.90 12.62

RELIANCE 28-Feb-11 26-May-11 1200 964.25 -0.22 0.16 0.03 0.06 0.50 0.25 0.50 -2.51 -2.59 0.01 0.00 0.99 0.15 10.65 10.50

RELIANCE 28-Feb-11 26-May-11 1220 964.25 -0.24 0.16 0.03 0.06 0.50 0.25 0.50 -2.71 -2.79 0.00 0.00 0.99 0.08 8.75 8.67

RELIANCE 28-Feb-11 26-May-11 1240 964.25 -0.25 0.16 0.03 0.06 0.50 0.25 0.50 -2.92 -3.00 0.00 0.00 0.99 0.04 7.20 7.16

RELIANCE 28-Feb-11 26-May-11 1260 964.25 -0.27 0.16 0.03 0.06 0.50 0.25 0.50 -3.12 -3.20 0.00 0.00 0.99 0.02 5.85 5.83RELIANCE 28-Feb-11 26-May-11 1280 964.25 -0.28 0.16 0.03 0.06 0.50 0.25 0.50 -3.31 -3.39 0.00 0.00 0.99 0.01 4.75 4.74

RELIANCE 28-Feb-11 26-May-11 1300 964.25 -0.30 0.16 0.03 0.06 0.50 0.25 0.50 -3.51 -3.59 0.00 0.00 0.99 0.00 3.85 3.85

RELIANCE 28-Feb-11 26-May-11 1320 964.25 -0.31 0.16 0.03 0.06 0.50 0.25 0.50 -3.70 -3.78 0.00 0.00 0.99 0.00 3.10 3.10

RELIANCE 28-Feb-11 26-May-11 580 964.25 0.51 0.16 0.03 0.06 0.50 0.25 0.50 6.58 6.50 1.00 1.00 0.99 392.95 397.50 4.55


29/30

-200

-100

0

100

200

300

400

500

Theoretical Price

Actual Price

0

50

100

150

200

250

300

350

400

450

Theoretical Price

Actual Price

ABB: CHARTS

RELIANCE: CHARTS

-100

-50

0

50

100

150

200

250

300

Theoretical Price

Actual Price

BINOMIAL0

50

100

150

200

250

300

Theoretical Price

Actual Price

BLACK - SCHOLES

BINOMIAL BLACK - SCHOLES


30/30

INTERPRETATION:

As far as ABB is considered all the option settlement prices are greater than the calculated theoretical price.

And in case ofRELIANCE also all the option settlement prices are greater than the calculated theoretical price.

KEY TERMS:

S.D: Standard Deviation is calculated from the daily returns of historical prices of equity.

STRIKE PRICE: The price at which a specific derivative contract can be exercised. Strike prices are mostly used to describe stock and index

options, in which strike prices are fixed in the contract. For call options, the strike price is where the security can be bought (up to the expiration

date), while for put options the strike price is the price at which shares can be sold.

SPOT PRICE: The current price at which particular securities can be bought or sold at a specified time and place.

SETTLEMENT PRICE: The average price at which a contract trades, calculated at both the open and close of each trading day.

RF: Bank rate is considered as risk free rate.

Su & Sd: Past 3 months equity highest and lowest closing price respectively.

REFERENCE:

WEBSITE:http://www.nseindia.com/

BOOK:INVESTMENT MANAGEMENT by V.K. Bhalla
http://www.nseindia.com/http://www.nseindia.com/http://www.nseindia.com/http://www.nseindia.com/

Documents

Valuation of Options