Chapter 3_ Nagendran

8/2/2019 Chapter 3_ Nagendran

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CHAPTER III

SENSITIVITY ANALYSIS

3.1 INTRODUCTION

In any financial model, the understanding of the dependant variables and

the independent variables are necessary. The BS model is not a deterministic

model

3.2 SENSITIVITY TO VOLATILITY OF THE ASSET RETURNS

Out of the BS model uses

five except the dividends of the stocks during the life time of the options. Let us

start w v e of the stock. Theoretically, volatility

represents the risk of the invest nd h have mo impact on

the call option price. Vola ator

of the equation of d1 .3.35) e in Ch paragrap 2. Thus,

ascer e imp of it o all opt e may n easy by

obser quatio tself. A that is really traded at NSE was taken

and the volatility of the stock return varied5

from 0.17 to 0.97, keeping all other

variab stant ue. The nding ion prices lculated,

-------- --------- ----------- --------- ------------ -----------5

The volatilit s finaliz alyzing stock re f the 28compa

but, a stochastic process. Hence it is of more important to analyze the

variables and parameters of the model. In the BS model, except the volatility of

returns of the stock, all other factors are observable.

In the valuation of call option price there are two variables; the shareprice and the time to expiration and three parameters viz. volatility of the stock

returns, strike price and risk-free-interest rate. This chapter deals much on the

sensitivity of the model to its parameters and variables.

six determinants of the call option price, the

ith the olatility of th returns

ment a ence should re

tility appears both in the numerator and denomin

(1 xplained apter I h 1.3.1

taining th act n the c ion pric ot be

ving the e n i n option

les at con val correspo call opt are ca

------------ --- ----------- ---------- ----------

range of y i ed by an all the turns onies.


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CHART 3.1

SENSITIVITY OF THE CALL OPTION PRICE TO VOLATILITY OF STOCK

RNSRETU

SENSITIVITY OF VOLATILITY

0

50

100

150

200

2

300

0.

17

0.

19

0.

21

0.

23

0.

26

0.

28

0.

31

0.

35

0.

38

0.

42

0.

47

0.

51

0.

57

0.

62

0.

69

0.

76

0.

84

0.

92

VOLATILITY

C

ALL

OPTION

PRICE

50

1.01 0.88 1.07

and tabulated. For three different moneyness of 0.88, 1.01 and 1.08, the

sensitivity of call option price to volatility are calculated, studied, tabulated and

presented in the chart no.3.1.

be observed that the curve is flat

and the rate of change in the call option price is small related to the range of

From the Chart No.3.1 above, it may

volatility 0.17 to 0.27. After that, the slope of the curve increases steeper,

indicating the rate of change in the price is higher after this point. The curve is

not linear but upward sloping. Inference can be made as the volatility of stockreturn increases, the sensitivity is also increasing at a faster rate. The rate of

sensitivity is not uniform but in an increased rate as volatility increases.


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3.3 SENSITIVITY TO SHARE PRICE

magnitude of the

stock price is the main consideration for the trader to buy a call option or sell a

call option. Thus, theoretically, the sensitiv e to the stock

price is high. As per the derivation of the formula, the stock price appears in all

the e

BS model. For analysis in an opt

and parame ke n e st price d twelve

times in steps of rupees five and the call option prices are calculated.

As expected, the call o tion pr ses at a remarkable rate for each

increase in share price. The resultan ption s are n the chart

no. 3.2.

CHART 3.2

SENS Y OF T LL O PR SHA ICE

X = 1830, r = 0.0658 T = 0.0849

The share price is a very important variable, depends on which, the

payoff of the option is calculated. The expected direction and

ity of the call option pric

three equations (1.3.34, 1.3.35, and 1.3.36) calculating d1, d2 and Co in th

ion that is traded at the

pt constant a

NSE, all the variables

ockters are d only th is increase

p ice increa

t call o price given i

ITIVIT HE CA PTION ICE TO RE PR

SENSITIVITY O AR CE

100

110

120

130

CALL

OPTION

PRICE

F SH E PRI

80

90

1854

1859

1864

1869

1874

1879

1884

1889

1894

1899

1904

1909

1914

SHARE PRICE


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The movement of the call option price is a straight line and not a curve

as s i

option pric are directly proportional. A five rupee

ll option price by around Rs. 3.02. The

ange slightly increases from Rs. 3.02 to Rs. 3.60 from 1854 to 1914 for every

change in Rs. 5 in stock price. If considered in percentage terms rather than in

absolute terms, the 0.3% increase e increases the call option price

by aro

For different strike prices of Rs.1830 and Rs. 1890, the sensitivity of call

option

CHART 3.3

0.0658 T = 0.0849

een n the sensitivity of volatility. As the stock price increases, the call

e also increases. Both of them

increase in stock price increases the ca

ch

in stock pric

und 3%.

price to the stock price is checked. The curves are like straight line and

they are not parallel.

SENSITIVITY OF THE CALL OPTION PRICE TO SHARE PRICE

FOR DIFFERENT STRIKE PRICES. r =

SENSITIVITY OF SHARE PRICE

50

70

90

110

130

1854

1859

1864

1869

1874

1879

1884

1889

1894

1899

1904

1909

1914

SHARE PRICE

CALLOPTIONPRICE

X - 1830 X - 1890

The call option price sensit higher for the lesser strike price

than that of the higher strike price. In the case of strike price of 1830, the stock

price i 6.26,

which works out to 17.39%. But, for the st in

ivity is slightly

ncrease of 3.24% increases the call option price to the tune of 5

rike price of 1890, for the change


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3.24% reflected an increase of stock price by 46.41 which is 14.34% only.

Hence, the curves are of the same dire e

s

ther attempt i ade to keep the rate of change in stock price as

constant. Accordingly, the rate of change in share price is kept at 2% and all the

others are kept constant. The results are tabulated in table no. 3.1 below. Then,

the call option price increases at about 27.6% and slowly the rate of

increase in call option price decreases with further every 2% increas

price. The last increase only 9.46% compare ial increase of 27.65%.

This is evidenced from the table no. 3.1 and chart no. 3.4 shown below.

TABLE 3.1

E OF CHANG IN CALL OPTION PR R EVERY TW

PERCENTAGE CHANGE IN STOCK PRICE

ction but th sensitivity de creases with

increases in trike price.

Ano s m

initially

e in stock

is d to init

RAT E ICE FO O

SharePrice

SoRs.

Rate ofIncrease in

S0%

BSOptionPrice

Rs.

Rate ofincrease inoption price

%

increase inoption price

Rs.

1854.20 - 85.80 --1891.28 2 109.52 27.65 23.72

1929.11 2 136.63 24.75 27.11

1967.69 2 166.91 22.16 30.28

2007.05 2 200.06 19.86 33.15

2047.19 2 235.73 17.83 35.67

2088.13 2 273.58 16.05 37.85

2129.89 2 313.28 14.51 39.70

2172.49 2 354.55 13.18 41.28

2215.94 2 397.19 12.03 42.64

2260.26 2 441.03 11.04 43.84

2305.46 2 485.96 10.19 44.93

2351.57 2 531.92 9.46 45.96


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CHART 3.4

RATE OF CHANGE IN CALL OPTION PRICE FOR EVERY TWO

PERCENTAGE CHANGE IN STOCK PRICE

X = 1830, r = 0.0658 T = 0.0849

RATE OF CHANGE IN OPTION PRICE

0

5

10

15

20

25

30

2 2 2 2 2 2 2 2 2 2 2 2

% CHANGE IN SHARE PRICE

%CHANGEINOPTION

PRICE

3.4 SENSITIVITY TO STRIKE PRICE

From the equations (1.3.34), (1.3.35) and (1.3.36) derived in the chapter

1, t nd C0. In formulae of d1,

nd d2, strike priceX is in the denominator of log function and in C0 it is with the

le no. 3.2.

he strike price X, appears in the formula of d1, d2 a

a

negative sign. Thus, theoretically, the relationship between the call option price

and strike price is negative. Also, from the point of view of payoff of an investor,

the increase in strike price decreases the payoff and call options price should

be lower for increase in strike price.

The strike price is increased from 1770 to 1830 in steps of Rs.5, keeping

the other variables and parameters constant and the corresponding call option

prices are calculated and tabulated in the Tab


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TABLE 3.2

SE ENSITIVITY OF CALL OPTION PRICE TO CHANGE IN STRIKE PRIC

Strike

PriceX

Rs.

Rate of

Increase inX

%

BS

OptionPriceC0

Rs.

Increases

inC0

Rs.

Rate of

Increasein C0

%

1770 - 123.51 - -

1775 0.28 120.06 -3.45 -2.79

1780 0.28 116.66 -3.40 -2.83

1785 0.28 113.32 -3.34 -2.86

1790 0.28 110.04 -3.29 -2.90

1795 0.28 106.81 -3.23 -2.93

1800 0.28 103.63 -3.17 -2.97

1805 0.28 100.52 -3.12 -3.01

1810 0.28 97.46 -3.06 -3.04

1815 0.28 94.46 -3.00 -3.08

1820 0.28 91.51 -2.94 -3.12

1825 0.27 88.63 -2.89 -3.15

1830 0.27 85.80 -2.83 -3.19

From the above table, it is observed that the call option price is inversely

varying with strike price. When the strike price increases by Rs.5 the call option

price decreases by about Rs. 3. The rate of decrease is not constant but

call option price is directly proportional to the payoff of an option.

2

decreases from -2.79% to -3.19%. This is due to the base effect that the price

itself is decreasing. The reason for the above inverse relation may be due to the

fact explained below;

1. The

. The increase in strike price decreases the payoff and hence the call

option price is also decreasing

The same is shown in the chart no. 3.5.


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90

CHART 3.5

SENSITIVITY OF THE CALL OPTION PRICE TO STRIKE PRICE

SENSITIVITY TO STRIKE PRICE

80

90

100

110

120

130

1770

1775

1780

1785

1790

1795

1800

1805

1810

1815

1820

1825

1830

STRIKE PRICE

CALLOPTIONPRICE

To understand the sensitivity in a better manner, the strike price is

increased by a constant rate of 2% and the values of call option price are

calculated, and tabulated in the next page in Table No. 3.3.

Though the actual options y NSE with strike prices rounded

off to fifty paisa, for reased to 2% and

the actual values are

are offered b

analytical purpose the strike prices are inc

given without rounding off.


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TABLE


PERCENT CHANG RIK E

When the strike price increases by 2%, the call option price decreases

not at a constant rate but at a varying rate from 10.27 % to 42.51 %. Sensitivity

3.3

AGE E IN ST E PRIC

StrikePriceX

Rs

.

Rate of Incre Xase in

%

Rate of Increase in C0

%

Increase inC0

Rs.1567.94 - - -

1599.94 2 10.2 -- 7

1632.59 2 11.3 .12- 9 -1

1665.91 2 12.6 .26- 6 -1

1699.91 2 14.0 .41- 7 -1

1734.60 2 15.6 .56- 3 -1

1770.00 2 17.3 .70- 3 -1

1805.40 2 18.8 .48- 1 -1

1841.51 2 20.7 .89- 0 -1

1878.34 2 22.6 .99- 9 -1

1 .90 2 -24.77 -2.08915

1954.22 2 -26.92 -2.15

1993.31 2 -29.11 -2.20

2033.17 2 -31.34 -2.23

2073.84 2 -33.59 -2.25

2115.31 2 -35.84 -2.25

2157.62 2 -38.09 -2.24

2200.77 2 -40.31 -2.23

2244.79 2 -42.51 -2.20


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of option price is more for higher strike prices than the lower strike prices. The

same is exhibited in a detailed manner in chart no. 3.6.

CHART 3.6


PERCENTAGE CHANGE IN STRIKE PRICE

SENSITIVITY TO STRIKE PRICE

-45

-40

-35

-30EOTI

-25

-20

-15

-10

-5

0

2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

RATE OF CHANGE IN STRIKE PRICE %

RAT

FCHANG

EIN

OP

ONPRICE

%

It may be observed that the sensitivity curve in chart number 3.5 and 3.6

SE

e, the sensitivity should be positive.

Logically, as the time to expiration (life of the option) is more, the uncertainty is

are not similar, though the direction is same.

NSITIVITY TO TIME TO EXPIRATION

Theoretically, life of the option or time to expiration, T, appears in both

the equations of d1 and d2. It appears both in the numerator and the

denominator of the equations. In the numerator it appears as T and in the

denominator, as square root of T. Henc


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more and hence more risk. Thus, as life of the option increases the call option

price also increases and vice vers rical study also proves the same.

The sensitivity is shown in the chart no. 3.7.

SENSITIVITY OF CALL OPTION PRICE T P

a. The empi

CHART 3.7

O TIME TO EX IRATION

SENSITIVITY E OF OPT

20

80

10

15

20

25

30

35

40

45

50

55

60

65

70

75

80

85

90

CALL

OPTIO

N

PRICE

OF LIF ION

40

60

100

120

140

160

05

LIFE IN DAYS

From the above chart, it may be observed that the contour of sensitivity

is not a straight line. Initially, it curves upward and then moves like a straight

line. The sensitivity is directly proportional to the life of option. But the

magnitude of the sensitivity is very less when compared to share price, strike

price or volatility.


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3.6 SENSITIVITY TO RISK - FREE - INTEREST RATE

ble parameter is the opportunity cost for an investor; under

skless condition a risk-averse investor should get a return equal to risk-free-

interes

Risk-free-interest rate appears in all three equations namely d1 (1.3.34),

2 (1.3.35) and C0 (1.3.36). All are in the numerator side, therefore, should have

positive relation with the call option price.

As derived in the formula, the actual risk-free-interest rates are to be

ompounded interest rates before using in the BS

cordingly, the actual risk-free-interest rates from 1.50 to 12 are

pounded interest rates and used in an option

all others constant. The resultant call

es are tabulated in the table no.3.4 and shown in the chart no.3.8

This observa

ri

t rate. This rate is directly observed from the money market. In our Indian

economy, it is the MIBID / MIBOR rate.

d

a

converted into continuously c

model. Ac

converted into continuously com

that is really traded at NSE, keeping

option pric


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TABLE 3.4

SENSITIVITY OF CALL OPTION PRICE

TO RISK - FREE - INTEREST RATES

Real InterestRate

R%

ContinuouslyCompoundedInterest rate

r%

BSOptionPrice

C0 Rs.

Increasein C0

Rs.

Rate ofIncrease

in C0%

1.50 1.49 134.48 - -

2.00 1.98 134.90 0.414 0.31

2.50 2.47 135.31 0.413 0.31

3.00 2.96 135.72 0.411 0.30

3.50 3.44 136.13 0.410 0.30

4.00 3.92 136.54 0.408 0.30

4.50 4.40 136.94 0.407 0.30

5.00 4.88 137.35 0.405 0.30

5.50 5.35 137.75 0.404 0.29

6.00 5.83 138.16 0.403 0.29

6.50 6.30 138.56 0.401 0.29

7.00 6.77 138.96 0.400 0.297.50 7.23 139.36 0.399 0.29

8.00 7.70 139.75 0.397 0.29

8.50 8.16 140.15 0.396 0.28

9.00 8.62 140.54 0.395 0.28

9.50 9.08 140.94 0.393 0.28

10.00 9.53 141.33 0.392 0.28

10.50 9.98 141.72 0.391 0.2811.00 10.44 142.11 0.389 0.27

11.50 10.89 142.50 0.388 0.27

12.00 11.33 142.88 0.387 0.27


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From the table no. 3.4, it may be noted that the call option price

creased only by about 40 paisa for every 0.5 % change in actual risk-free-

meters. The 155.56% change

the

SENSITIVITY OF CALL OPTION PRICE TO ACTUAL RISK - FREE -

in

interest rate. Though the sensitivity is positive, the magnitude of the change is

small when compared to other variables and para

in risk-free-interest rate caused only 6.04% change in call option price. That

is, call option price is least sensitive to risk-free-interest rate.

CHART 3.8

INTEREST RATE

SENSITIVITY OF CALL OPTION PRICE TO RFR

E

130

132

1.

5

2.

0

2.

5

3.

0

3.

5

4.

0

4.

5

5.

0

5.

5

6.

0

6.

5

7.

0

7.

5

8.

0

8.

5

9.

0

9.

5

10.

0

10.

5

11.

0

11.

5

12.

0

RFR

134

136

138

OPTION

P

140

RIC

142

144

3.7 CONCLUSION

The sensitivities of call option price to all five determinants of the option

price are explained in the above paragraphs. To compare the sensitivity of

them, percentage of change in the variables and the corresponding percentage

of change in the call option price are calculated. Then the sensitivity is

calculated using the following formula.


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Percentage of change in the corresponding call option price6

Sensitivity = ---------------------------------------------------------------------------------Percentage of change in the independent variable

To make it comparable almost all the variables are kept constant except

the variable under study. The moneyness of 1.01 is taken to calculate the

percentage change of the variables like Risk-free-interest rate, Life of the option

and volatility of the returns of the stocks. While analyzing the sensitivity of the

share price and strike price, averages of volatility, risk-free-interest rate and life

of the options are used to minimize the base-effect. The results are given in the

table no. 3.5.

TABL

COMPARISON OF SENSITIVITY OF THE CALL OPTION PRICE TO THE

VARIABLES

E 3.5

Variables% change

in Variables

% change in

Option PriceSensitivity

SHARE PRICE S0 1.75 20.37 11.64

STRIKE PRICE X 1.63 -19.48 -11.95

VOLATILITY 74.63 64.08 0.86

LIFE 27.43 0.61T 44.74

RISK - FREE - INTERESTRATE

55.56 6.04 0.041r

-------------------------------------- ---------------------------------------------------------

Chan ll option price6Percentage of change in option price = -------------------------------------- x 100

Average call option price

-------------------

geinca


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The call option price is most sensitive to the share price and strike price,

almost in the same magnitude but in different direction. Increase in stock price

increases the call option p ase in ike price decreases the call

option price. Thus the us ess in classifications of call options is

proved correct in our study. This is due to the reason that increase in stock

price increases the payoff of the option and increase in strike price decreases

e payoff of it and vice versa.

The least sensitive variable is the risk-free-interest rate. The change in

more than 150 percentages increases the call option price only by about 0.04

percentages. Thus, it may be concluded that the sensitivity is almost negligible.

rice but incre the str

e of moneyn

th

Next, the volatility of stock returns is more sensitive compared to the

other variables like risk-free-interest rate and life of the option. This is due to the

reason that the volatility is the risk involved in the investment and hence more

volatility more price.

Life of the option also implies that longer the life more uncertainty and

are priced higher.

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