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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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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
RATE OF CHANGE IN CALL OPTION PRICE FOR EVERY TWO
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
RATE OF CHANGE IN CALL OPTION PRICE FOR EVERY TWO
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.