REVISION CLASS
DERIVATIVES
01 – DERIVATIVES
DERIVATIVES
:-
Derivatives is a financial instrument which derives its value from an underlying
asset.
:-
Underlying asset means share, stock, bonds, currency, commodity, stock index etc.
:-
Derivative is an instrument for betting.
:-
We will discuss this chapter in two parts.
Part I - Option Contract.
Part II – Forward & Future Contract
Part I : Option Contract
We will discuss option contract in three points
i.
Basics.
ii.
Valuation of Option
iii.
Option Strategy
(1) Basics
(i)
Option contract is a contract in which option holder has right but not obligation to
buy or sell an underlying asset at predetermine price (Exercise price or strike price)
on maturity. An option premium is to be paid in advance & such premium is
transferred to option writer by stock exchange.
(ii)
There are two parties in option contract
Option Holder or Option Buyer
Right buy not obligation
An option premium to be paid in
advance.
(iii)
unlimited profit & maximum
loss premium amount.
(iv)
Loves volatility.
(i)
(ii)
Option Writer or Option Seller
Obligation but not right.
Margin money is required to be
deposited at stock exchange.
(iii)
Unlimited
loss & maximum
profit is premium amount.
(iv)
Hates volatility.
(i)
(ii)
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REVISION CLASS
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(iii)
There are two types of options
(a) Call Option
- Right to buy
- Expected to price rise
(b) Put Option
- Right to sell
- Expected to price fall
(iv)
Types of options on the basis of cash flows
(a)
European Option:- European option can be exercised only on maturity.
(b)
American Option:- American option can be exercised on or before maturity.
Premium amount of American option is more than European.
Example – 01
Mr. E is interested in buying a share of I.T.C. he is however afraid that the price of the
share may move down. Hence, he does not purchase a share but buys a call option o 1
share of I.T.C. at a strike price of ₹300 by paying an option premium of ₹35.
Required:(i)
Determine the breakeven point price of Mr. E.
(ii)
Determine the Profit/Loss if the price on maturity is:- 250, 270, 290, 300, 320, 340,
350.
Solution:
(i)
Calculation of breakeven point.
BEP
= EP + Premium
= ₹300 + ₹35
= ₹335
(ii)
Calculation of Profit/loss
E = ₹300,
Market
Price
250
270
290
Premium = ₹35
Exercise or not
Lapsed
Lapsed
Lapsed
Gross Pay
off
0
0
0
Premium
(35)
(35)
(35)
Net
Pay off
(35)
(35)
(35)
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REVISION CLASS
DERIVATIVES
300
320
340
350
Lapsed
Exercised
Exercised
Exercised
0
20
40
50
(35)
(35)
(35)
(35)
(35)
15
5
15
Example – 02
Mr. G is hoping that the price of a share of ACC is going to fall. He purchases a put option
at an exercise price of ₹480. He pays a premium of ₹40.
Required:(i)
Determine the breakeven point to Mr. G.
(ii)
Compute Profit/Loss for Mr. G if the price on maturity is:- ₹400, 420, 440, 480,
490, 500, 530.
Solution:
(i)
Sales Calculation of breakeven point.
BEP
= EP – Premium
= ₹480 − ₹40
= ₹440
(ii)
Calculation of Profit/Loss
Market
Price
400
420
440
480
490
500
530
Exercise
or not
Yes
Yes
Yes
No
No
No
No
Gross
Pay off
80
60
40
0
0
0
0
Premium
(40)
(40)
(40)
(40)
(40)
(40)
(40)
Net Pay
off
40
20
0
(40)
(40)
(40)
(40)
Question – 01
The equity share of SSC Ltd. is quoted at ₹310. A three month call option is available at a
premium of ₹8 per share and a three month put option is available at a premium of ₹ 7 per
share.
Ascertain the net payoffs to the option holder of a call option and a put option, considering
that:
(i)
The strike price in both cases is ₹320; and
(ii)
The share price on the exercise day is ₹300, 310, 320, 330 and 340.
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REVISION CLASS
DERIVATIVES
Also, indicate the price range at which the call and the put options may be gainfully
exercised.
(Exam Nov – 2018)
Solution:
Call option Holder
Calculation of Net Pay off
E = 320,
Premium = 8
Market Price
300
310
320
330
340
Exercise or
not
No
No
No
Yes
Yes
Gross Pay off
Premium
Net Pay off
0
0
0
10
20
(8)
(8)
(8)
(8)
(8)
(8)
(8)
(8)
2
12
Gross Pay off
Premium
Net Pay off
10
20
0
0
0
(7)
(7)
(7)
(7)
(7)
13
3
(7)
(7)
(7)
Put option Holder
Calculation of Net Pay off
E = 320,
Premium = 7
Market Price
300
310
320
330
340
Exercise or
not
Yes
Yes
No
No
No
Call option is gainfully exercised when price of share it on maturity is use than 328 (320 +
8).
Put option is gainfully exercised when price of share it on maturity is use than 313 (320 −
7).
(v)
In the money , At the money, Out of the Money, Intrinsic value & Time value
In the money (ITM), At the money ( ATM ), Out of the money (OTM)
Call
Put
EP < CMP
ITM
OTM
EP = CMP
ATM
ATM
EP > CMP
OTM
ITM
There are two parts of option premium:- Intrinsic value & Time value (Volatility
premium)
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REVISION CLASS
DERIVATIVES
Intrinsic value: If option is in the money, then difference between CMP & EP is
called intrinsic value. If option is out of the money & At the money than Intrinsic
value will be zero.
Time value or Volatility premium : If option is in the money then Time value =
premium amount – Intrinsic value If option is Out of the money & At the money
then whole of the premium amount is time value.
(vi)
Participants in Derivative Market
There are three participants or players in Derivative Market.
(a) Hedgers
:- Existing Exposure
:- To avoid risk
:- Take Long or short position
(b) Speculators
:- No existing exposure
:- For making profit on the basis of price expectation.
:- Take long or short position
:- They may loose
(c) Arbitrageurs
:- No existing exposure
:- For making profit on the basis of mispricing
:- They are sophisticated investors & use skill to make profit
:- Take long & short position simultaneously
:- Loss is not possible
(vii)
Short selling (Stock lending & Borrowing Scheme)
(a)
Definition:- Short selling is a speculative activity is designed to make profit
on the basis of bearish price expectation.
(b)
Explanation:- In short selling, short seller borrow stock from stock lender &
sell it at current market price with a view to buy later on at lower price &
return to stock lender.
(c)
Sources of Return:Page 5
REVISION CLASS
DERIVATIVES
− Price depreciation
− Interest on selling amount
(d)
Sources of Risk:− Price Appreciation
− Dividend (Short seller compensates dividend amount to stock lender)
− Stock lending charges
(e)
Legal Status:- Short selling is prohibited in some Countries. In some
Countries like US & India allow short selling with some restriction.
In India stock Lending & Borrowing scheme (SLBS) of SEBI regulates short
selling activities.
Question – 02
Mr. A is holding 1,000 shares of face value of ₹100 each of M/s. ABC Ltd. He wants to hold
these shares for long term and have no intention to sell.
On 1st January 2020, M/s. XYZ Ltd. has made short sales of M/s. ABC Ltd.’s shares and
approached Mr. A to lend his shares under Stock Lending Scheme with following terms:
(i)
Shares to be borrowed for 3 months from 1st January 2020 to 31st March 2020.
(ii)
Lending Charges/Fees of 1% to be paid every month on the closing price of the
stock quoted in Stock Exchange and
(iii)
Bank Guarantee will be provided as collateral for the
2020.
value as on 1 st January
Other Information :
(a)
Cost of Bank Guarantee is 8% per annum.
(b)
On 29th February 2020 M/s. ABC Ltd. declared dividend of 25%.
(c)
Closing price of M/s. ABC Ltd.’s shares quoted in Stock Exchange on various dates
are as follows :
Date
1st January 2020
31st January 2020
29th February 2020
31st March 2020
Share Price in
Scenario – 1 Bullish
1,000
1,020
1,040
1,050
Share Price in Scenario
– 2 Bullish
1,000
980
960
940
You are required to find out:
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REVISION CLASS
DERIVATIVES
(i)
Earnings of Mr. A through Stock Lending Scheme in both the scenarios,
(ii)
Total earnings of Mr. A during 1st January 2020 to 31st March 2020 in both the
scenarios,
(iii)
What is the profit or loss to M/s. XYZ by shorting the shares using through Stock
Lending Scheme in both the scenarios?
Solution:
(i) Earning of Mr. A lending Scheme
Bullish
31 Jan
(1080 × 1%) = 10.20
Bearish
(980 × 1%) = 9.80
29 Feb
(1040 × 1%) = 10.40
(960 × 1%)
= 9.60
31 March
(1050 × 1%) = 10.50
(940 × 1%)
= 9.40
Earning per share
₹31.10
₹28.80
(X) No. of share
1000
1000
Earnings
₹31100
₹28800
Bullish
31.10
Bearish
28.30
(+) Dividend Income
(100 × 25)
25
25
Total Earning per share
₹56.70
₹53.80
(X) No. of share
1000
1000
Total Earnings
₹56,100
₹53,800
(ii) Total Earning of Mr. A
Lending charger
(iii) Calculation of Profit/Loss to M/s. XYZ Ltd.
Bullish
Profit/Loss on share
(1000-1050)
Bearish
(1000-940)
(50)
60
Lending charger
(31.10)
(28.80)
Bank Guarantee
(20)
(20)
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REVISION CLASS
DERIVATIVES
(1000× 8% ×
3
12
)
(X) No. of share
Loss
(vii)
101.10
11.20
1000
1000
Profit 11200
₹101100
Expected value of Option
− Expected price of share
= ∑ Price × probability
− Expected value of option
= ∑ Gross payoff + probability
Or
∑ Intrinsic value x probability
Question – 05
Equity share of PQR Ltd. is presently quoted at ₹ 320. The Market Price of the share after
6 months has the following probability distribution:
Market Price
₹ 180
260
280
320
400
Probability
0.1
0.2
0.5
0.1
0.1
A put option with a strike price of ₹ 300 can be written.
You are required to find out expected value of option at maturity (i.e. 6 months)
(SM New Syllabus & PM)
Solution:
Expected Value of option.
Price
180
260
280
320
400
Exercise or
Gross pay off
not
Yes
120
Yes
40
Yes
20
No
0
No
0
Expected value of option
Probability
0.1
0.2
0.5
0.1
0.1
Gross pay off ×
Probability
12
8
10
0
0
₹ 30
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REVISION CLASS
DERIVATIVES
Question – 06
You as an investor had purchased a 4 month call option on the equity shares of X Ltd. of ₹
10, of which the current market price is ₹ 132 and the exercise price ₹ 150. You expect the
price to range between ₹ 120 to ₹ 190. The expected share price of X Ltd. and related
probability is given below:
Expected Price (₹)
Probability
120
0.05
140
0.20
160
0.50
180
0.10
190
0.15
COMPUTE:
(i)
Expected Share price at the end of 4 months.
(ii)
Value of Call Option at the end of 4 months, if the exercise price prevails.
(iii)
In case the option is held to its maturity, what will be the expected value of the call
option?
(MTP March – 2022, SM & PM)
Solution:
(i)
Expected share price at the end of 4 months
(120 × 0.05) + (140 × 0.2) + (160 × 0.5) + (180 × 0.1) + (190 × 0.15) = ₹160.50
ii)
Value of option = 150 − 150 = 0
iii)
Expected Value of option
Price
120
140
160
180
190
Exercise or
not
No
No
Yes
Yes
Yes
Gross pay
off
0
0
10
30
40
Probability
0.05
0.20
0.50
0.10
0.15
Expected value of option
Gross pay off ×
Probability
0
0
5
3
6
₹ 14
(2) Valuation of Option or Option Pricing
In this topic, we calculate value of option & compare with market price of option i.e.
premium & decide whether option should be purchased or not?
- Premium Amt. > Value of option
- Premium Amt. < Value of option
Overpriced
Not buy
Underpriced
Buy
There are three methods to calculate value of option.
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REVISION CLASS
DERIVATIVES
(i)
Binomial Model
- Risk neutral probability approach
- Delta hedging or Risk free portfolio approach
- Replicating portfolio approach
(ii)
Put call parity theorem (PCPT)
(iii)
Black – Scholes Model (BSM)
(i) Binomial Model
(a)
Risk Neutral Probability Approach :- As per Binomial Model (Name Suggested),
Only two possible price of stock on maturity i.e.
- Maximum price or upper price of stock (us)
- Minimum price or lower price of stock (ds)
(b)
Following step are applied to calculate value of option
Step – 1 : Standard notation or given
Step – 2 : Calculate risk neutral probability
P=
R−d
u−d
Step – 3 : Binomial Tree
Step – 4 : Calculate value of option
- Value of call
Co =
Cup +Cd (1−P)
R
- Value of put
Po =
PuP +Pd (1−P)
R
Question – 08
The current market price of an equity share of Penchant Ltd is ₹420. Within a period of 3
months, the maximum and minimum price of it is expected to be ₹500 and ₹400
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REVISION CLASS
DERIVATIVES
respectively. If the risk free rate of interest be 8% p.a., what should be the value of a 3
months Call option under the “Risk Neutral” method at the strike rate of ₹450?
Given e0.02 = 1.0202
(SM & PM)
Solution:
Step 1: Given
S
= ₹420
us
=
ds
=
E
= 450
R
=8×
500
= 1.1905
420
400
420
= 0.9524
3
= 2%
12
E0.02 = 1.0202
Step 2: Risk Neutral Probability
P=
e rt −d 1.0202 −0.9524
=
= 0.2844
u−d
1.1905−0.9524
Step 3: Binomial Tree
Step 4: Value of Call Option
Co
=
=
Cup + Cd (1−p)
e rt
50×0.2844 +(0×0.7156)
1.0202
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REVISION CLASS
DERIVATIVES
= ₹13.94
Question – 09
Sumana wanted to buy shares of ElL which has a range of ₹ 411 to ₹ 592 a month later.
The present price per share is ₹ 421. Her broker informs her that the price of this share
can sore up to ₹ 522 within a month or so, so that she should buy a one-month CALL of
ElL. In order to be prudent in buying the call, the share price should be more than or at
least ₹ 522 the assurance of which could not be given by her broker.
Though she understands the uncertainty of the market, she wants to know the probability
of attaining the share price ₹ 592 so that buying of a one-month CALL of EIL at the
execution price of ₹ 522 is justified. Advice her. Take the risk-free interest to be 3.60% and
e0.036 = 1.037.
(SM New Syllabus& PM)
Solution:
Step 1: Given
S
= ₹ 421
u
=
d
=
R
=
592
421
= 1.406
411
= 0.976
421
e0.02
= 1.037
Step 2: Risk Neutral Probability
P
ert -d
=
u-d
=
1.037-0.976
1.406-0.976
= 0.1419
Probability of use in price is 0.1419
Question – 11
A two year tree for a share of stock in ABC Ltd., is as follows:
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REVISION CLASS
DERIVATIVES
Consider a two years American call option on the stock of ABC Ltd., with a strike price of
₹98. The current price of the stock is ₹100. Risk free return is 5 per cent per annum with a
continuous compounding and e0·05 = 1.05127. Assume two time periods of one year each.
Using the Binomial Model, calculate:
(i)
The probability of price moving up and down;
(ii)
Expected pay offs at each nodes i.e. N1, N2 and N3 (round off upto 2 decimal
points).
(Exam Nov – 2020)
Solution:
Step 1: Given
S = 100
u=
108−1.08
100
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REVISION CLASS
DERIVATIVES
d=
95−0.95
100
R = 1.05127
(i)
Calculate the probability
P
=
e rt − d
u −d
=
1.5127 −0.95
1.08−0.95
= 0.78
(ii)
Value of Option
Node 2
Value =
(18.64 × 0.78) + (4.60 × 0.22)
1.05127
= ₹14.79
Intrinsic value
= (108-98)
= ₹10
Hence value of option at node 2 is ₹14.79 (Higher of two).
Node 3
Value =
(4.60 × 0.78) + (0 × 0.22)
1.05127
= ₹3.413
Intrinsic value
=0
= ₹10
Hence value of option at Node 3 = 3.41
Node 1
Value =
(14.79 × 0.78) + (3.41 × 0.22)
= ₹11.69
1.05127
Intrinsic value = (100-98) = ₹2
Hence value of call option is = 11.69
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REVISION CLASS
DERIVATIVES
Question – 12
Consider a two-year call option with a strike price of ₹ 50 on a stock the current price of
which is also ₹ 50. Assume that there are two-time periods of one year and in each year
the stock price can move up or down by equal percentage of 20%. The risk-free interest
rate is 6%. Using binominal option model, calculate the probability of price moving up and
down. Also draw a two-step binomial tree showing prices and payoffs at each node.
(SM New Syllabus & PM)
Solution:
1. Binominal tree
2. Calculation of Probability
P=
R-d
u-d
=
1.06-0.8
1.20-0.8
= 0.65
3. Value of option at each Node 2
Value =
(22 × 0.65) + (0 × 0.35)
1.06
= ₹ 13.49
Node 3
Value
=0
Node 1
Value =
(13.49 × 0.65) + (0 × 0.35)
1.06
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REVISION CLASS
DERIVATIVES
= 8.272
Question – 14
AB Ltd.'s equity shares are presently selling at a price of ₹500 each. An investor is
interested in purchasing AB Ltd.'s shares. The investor expects that there is a 70% chance
that the price will go up to ₹650 or a 30% chance that it will go down to ₹450, three
months from now. There is a call option on the shares of the firm that can be exercised
only at the end of three months at an exercise price of ₹550.
Calculate the following:
(i)
If the investor wants a perfect hedge, what combination of the share and option
should he select?
(ii)
Explain how the investor will be able to maintain identical position regardless of the
share price.
(iii)
If the risk-free rate of return is 5% for the three months period, what is the value of
the option at the beginning of the period?
(iv)
What is the expected return on the option?
(Exam Nov – 2019)
Solution:
E
= 550
S
= ₹500
uS
= ₹650
dS
= ₹450
(i) Delta of Call
=
Cu − Cd
uS −dS
=
100 − 0
650 − 450
= 0.5
For perfect hedge Inverts should write 1 call option Is buy 0.5 share today.
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REVISION CLASS
DERIVATIVES
(ii) Pay off
Price ₹650
Sell share (650 × 0.5)
Call Exercised
CI
= 325
= (100)
= 225
Price ₹450
Sell share (450 × 0.5)
Call Lapsed
CI
= 225
=0
= 225
(iii) Value of call
Cash Outflow
= Cash Inflow
(500 × 0.5 − Co) (1.05)
= 225
(250 − Co) (1.05)
= 225
Co
= ₹35.71
(iv) Expected Return of Option
Expected value of option
= (100 × 0.7) + (0 × 0.3)
= ₹70
Value of Option
= ₹35.71
Expected Return on option =
70−35.71
× 100 = 96.03%
35.71
(ii) Put Call Parity Theorem
Put call parity theorem is a strategy of combination of European call & put option at same
exercise price on same asset for same maturity period.
Equation of put call parity
So + Po
= Co + P.V, of EP
Where,
So
= Current market price
Po
= Value of put option/put premium
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Co
= Value of call option/call premium
This equation is derived with the help of following two parts.
Part I : Protective Put
Part II : Fiduciary Call
Example – 12
The following table provides the prices of options on equity shares of X Ltd. and Y Ltd. The
risk free interest is 9%. You as a financial planner are required to spot any mispricing in
the quotations of option premium and stock prices? Suppose, if you find any such
mispricing then how you can take advantage of this pricing position.
Share
Time to
Exercise
6 months
3 months
X Ltd.
Y Ltd.
Exercise
Price
100
80
Share Price
Call Price
Put Price
160
100
56
26
4
2
Solution:
Equation of Put call Parity
S0 + P0
X Ltd 160 + 4
164
= C0 + P.V. of EP
= 56 +
100
1.045
= 151.69
This is a mispricing and possibility of Arbitrage Cost of fiduciary call is less than cost of
protective put. Hence buy fiduciary call & sell protective pure.
Arbitrage Gain
= 164 – 151.69
= ₹12.31
Y Ltd 100 + 2
102
= 26 +
80
1.0225
= 104.24
Cost of protective put is less than cost of fiduciary call, hence buy protective put & sell
fiduciary Call.
Given = 104.24 – 102
Page 18
REVISION CLASS
DERIVATIVES
= 2.24
Question – 16
The following quotes are available for 3 months options in respect of a share of P Ltd.
which is currently traded at ₹ 310 :
Strike price
₹300
Call option
₹30
Put option
₹20
An investor devises a strategy of buying a call and selling the share and a put option.
(i)
Draw his profit/loss profile if it is given that the rate of interest is 10% per annum.
(ii)
What would be the position if the strategy adopted is selling a call and buying the
put and the share? (e0.025 = 1.0253; e0.25 = 1.2840)
Solution:
Strategy I buy call, sell share & Sell put
E = 300
Today’s Cash flows
Buy Call
= (30)
Short Sell share
= 310
Sell Put
= 20
Invest ₹300 @10% p.a. for 3 months 2.5%
300
700
Investment Amt 300 ×
+ 307.59
Buy share & Return to Lender
– 700
Call
+ 400
Put
0
Gain = 7.59
e0.025
200
+ 307.59
– 200
0
−100
7.59
Strategy II Sell Call, Buy share & Buy Put
Today’s Cash flows
Sell Call
+ 30
Buy Share
− 310
Buy Put
− 20
Borrow ₹300 @ 10% p.a. for 3 months.
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DERIVATIVES
Cash out flow
300
Maturity
Repayment
300 × 1.0253
Sell Share
700
200
− 307. 59
−307.59
+ 700
+ 200
Put
0
Call
+ 100
− 400
Loss
=
− 7.59
0
− 7.59
(iii) Black Sholes Model (BSM)
As per BSM, value of call option is calculated as under
Co
E
= So × n(d1) − rt × n(d2)
e
Where,
So
= Current Market Price
E
= Exercise Price
r
= Rate of Interest [ Always Continuously Compounding ]
n
= Normal Distribution Table (Z Table)
d1
= Delta of call or probability of stock price is more than exercise price
d2
= Probability of option exercise
S
σ2
Ln o + r +
t
E
2
d1
=
d2
= d1 − σ t
σ t
Question – 17
From the following data for certain stock, find the value of a call option:
Price of stock now
=
₹ 80
Exercise price
=
₹ 75
Page 20
REVISION CLASS
DERIVATIVES
Standard deviation of continuously compounded
annual return
=
0.40
Maturity period
=
6 months
Annual interest rate
=
12%
Given
Number of S.D. from Mean, (z)
0.25
Area of the left or right (one tail)
0.4013
0.30
0.3821
0.55
0.2912
0.60
0.2743
e0.12×0.5
= 1.062
In 1.0667
= 0.0646
(SM New Syllabus & PM)
Solution:
Given
Se
=
₹80
E
=
₹75
𝜎
=
0.40
t
=
0.50 Year
r
=
12%
Working Note 1: Calculation of d1
S
σ2
Ln o + r+
t
d1
E
=
σ t
Ln
d1
=
d1
=
d1
=
2
80
75
+ 0.12 +
(0.40)2
2
0.5
0.15 0.5
Ln 1.0667 + 0.10
0.2828
0.0646 + 0.10
0.2828
= 0.5820
Page 21
REVISION CLASS
DERIVATIVES
Working Note 2: Calculation of d2
d2
= d1 - σ t
= 0.5820 – 0.2828
= 0.2982
Working Note 3: n (d1 )
n (0.5820)
0.55
0.2912
0.60
0.2743
0.05
0.0169
0.0169
× 0.032
0.05
= 0.2912 −
= 0.2804
n(d1 )
= 1 − 0.2804 = 0.7196
Working Note 4: n (d2 )
n (0.2992)
0.25
0.4013
0.30
0.3821
0.05
0.0192
= 0.4013 −
00.0192
× 0.0492
0.05
= 0.3824
n (d2 ) = 1 − 0.3824
= 0.6176
Calculation of Value of Call Option
Co
E
= So × n d1 − rt × n d2
e
75
= ₹80 × 0.7196 – 0.12 ×0.5 × 0.6176
e
Page 22
REVISION CLASS
DERIVATIVES
= ₹57.50 – 43.6158
= ₹13.96
(3) Option Strategies
(i)
Straddles & Strangles
(ii)
Straps & Strips
(iii)
Bull & Bearish
(iv)
Butterfly
(i) Straddles & Strangles
Straddles:
−
An Investor expects that wide Volatility in price of underlying asset in future but he
is not sure about movement i.e. price goes up & goes down hence he creates
straddles strategy.
−
In straddles, we buy one call option & one put option at same strike price, on same
asset for same maturity period. (Long straddles)
−
If price will rise then we will exercise Call option & Put option will lapse.
−
If price will fall then we will exercise Put option & Call option will lapse.
Strangles:
−
An investor expects wide volatility in price of share but he is not sure about
direction i.e. price rise or price fall, hence he creates strangles strategy.
−
In strangles, we buy one call option & one put option at different strike price, on
same asset for same maturity period.
−
If price will rise then we will exercise call option & put option will lapse.
−
If price will fall then we will exercise put option & call option will lapse.
−
Cost of strangles strategy is less than cost of straddles strategy.
−
In strangles, Call option is bought at higher EP & Put option is lower EP.
Question – 23
Mr. P established the following spread on the Coastal Corporation’s stock:
Page 23
REVISION CLASS
DERIVATIVES
(i)
Purchased one 3-month call option with a premium of ₹ 6.5 and an Exercise price of
₹ 110.
(ii)
Purchased one 3-month put option with a premium of ₹ 10 and an Exercise price of
₹ 90.
Coastal Corporation’s stock is currently selling at ₹ 100. Determine profit or loss, if the
price of Coastal Corporation’s stock:
(i)
Remains at ₹ 100 after 3 months.
(ii)
Falls at ₹ 70 after 3 months.
(iii)
Rises to ₹ 138 after 3 months. Assume the size of option is 1,000 shares of Coastal
Corporation.
(RTP May – 2022)
Solution:
Cost of Strategy
= ₹ 6.50 + ₹ 10
= ₹ 16.50 + ₹ 1,000 Share
= ₹ 16,500
(i)
Price on Maturity ₹ 100
In this situation both call & put option will be lapsed loss to Mr. P is premium
amount i.e., 16,500.
(ii)
Price on Maturity ₹ 70
In this situation, Call option will lapse & put option be Exercised
(iii)
Gross pay off = ₹ 90 - ₹ 70
= ₹ 20
(X) No. of shares
= 1,000
Gross payoff
= ₹ 20,000
(-) Cost
= ₹ 16,500
Profit
= 3,500
Price on Maturity ₹ 138
In this situation, Put option will lapse & Call option be Exercise
Gross Profit (138-110)
= ₹ 28
Page 24
REVISION CLASS
DERIVATIVES
(X) No. of shares
= 1000
Gross payoff
= ₹ 28,000
(-) Cost of Strategy
= ₹ 16,500
Profit
= 11,500
Question – 29
A call and put exist on the same stock each of which is exercisable at ₹ 60. They now trade
for:
Market price of Stock or stock index
₹ 55
Market price of call
₹9
Market price of put
₹1
Calculate the expiration date cash flow, investment value, and net profit from:
(i)
Buy 1.0 call
(ii)
Write 1.0 call
(iii)
Buy 1.0 put
(iv)
Write 1.0 put
for expiration date stock prices of ₹ 50, ₹ 55, ₹ 60, ₹ 65, ₹ 70.
(Practice Manual)
Solution:
(i) Calculation of Cash flows
Stock Price
Buy 1 call
Write 1 call
Buy 1 put
Write 1 put
₹ 50
0
0
60
(60)
₹ 55
0
0
60
(60)
₹ 60
0
0
0
0
₹ 65
(60)
60
0
0
₹ 70
(60)
60
0
0
₹ 65
5
(5)
0
₹ 70
10
(10)
0
(ii) Calculation of Investment Value
(Grose Pay off)
Stock Price
Buy 1 call
Write 1 call
Buy 1 put
₹ 50
0
0
10
₹ 55
0
0
5
₹ 60
0
0
0
60
Page 25
REVISION CLASS
DERIVATIVES
Write 1 put
(10)
(5)
0
0
0
₹ 65
(4)
4
(1)
1
₹ 70
1
(1)
(1)
1
(iii) Calculation of Net Profit
60 Premium = 9
Stock Price
Buy 1 call
Write 1 call
Buy 1 put
Write 1 put
₹ 50
(9)
9
9
(9)
₹ 55
(9)
9
4
(4)
₹ 60
(9)
9
(1)
1
(4) Option Greeks
Price of option depends upon following factors.
(1) Stock price
(So)
(2) Exercise price
(E)
(3) Time
(t)
(4) Volatility
(σ)
(5) Rate of Interest (R)
Among these factors, exercise price is constant, remaining factors may change. Option
price will change due to change in these factors. We wish to carryout sensitivity analysis
i.e.
Rate of change in option price with respect to each factor, keeping other factors constant.
This rate of change have been assigned in Greek Letter.
(i) Delta:(a)
Delta means rate of change in option price with respect to stock price. Since call is
bullish & put is bearish hence call has positive delta & put has negative delta.
(b)
If call option is deeply out of the money then delta of call closer to zero. If call option
is deeply in the money then delta of call closer to 1.
(c)
Suppose delta of call 0.4 & Delta of put – 0.6 means.
 If means if price of stock goes by ₹ 1 then price of call option will go up by 40
paisa & price of put option will go down by 60 paisa .
 In Binomial
1 call is equivalent to 0.4 share long.
Page 26
REVISION CLASS
DERIVATIVES
1 put is equivalent to 0.6 share short
 In BSM
Delta = N (d1)
 Hedge Ratio
Delta call 0.4 = Write call & buy 0.4 shares.
(ii) Gamma:- Delta does not move at same rate hence rate of changes in delta with respect
to rate of change in stock price is called Gamma.
(iii) Theta:- Rate of change in option price with respect to rate & change in time is called
theta. Option price will go down due to passage of time.
(iv) Vega:- Rate of change in option price with respect to volatility is called vega. Price of
option will go up due to increase in volatility.
(v) RHO:- Rate of change in option price with respect to increase rate is called “Rho” If rate
of interest rises then price of call will go up & price of put will go down.
Part II : Forward & Future
(1)
(2)
(3)
Forward Contract
−
Forward contract is a contract between two parties to buy or sell an
underlying asset at predetermine price (forward Rate) in future delivery.
−
In forward contract forward buyer is obligated to buy & forward seller is
obligated to sell such underlying asset.
−
Forward contract is over the counter (OTC) contract.
Future Contract
Future contract is
−
Standardized forward contract
−
Traded at stock exchange
−
With margin requirement
−
No counter party default risk
There are Two parties in future contract
(a)
Future Buyer
Contract to buy
-
Upside betting
-
Long position
Page 27
REVISION CLASS
DERIVATIVES
(b)
(4)
Future Seller
Contract to sell
-
Downside betting
-
Short position
Forward contract V/S Future contract
Forward Contract
(i) Over the counter contract
Future Contract
(i) Exchange traded
(ii) Customized
(ii) Standardized
(iii) No margin requirement
(iii) Margin requirement
(iv) Counter party default risk
(iv) No counter party default risk
(v) Settlement only on maturity
(v) Daily settlement in margin balance
(Mark to Market settlement )
(vi) Less Liquidity
(vii) Less regulations
(viii) Generally used by hedgers
(5)
(vi)High liquidity
(vii)More regulations
(viii)Generally used by speculators .
Stock index future
−
Stock index future means future contract on stock index i.e. Nifty & Sensex
etc.
−
It could be on sector wise i.e. Bank nifty, IT index etc. Or it could be on
overall market i.e. Nifty, Senses etc.
−
It is settled only in cash, No physical delivery is possible. It is more liquid
than stock future.
−
It is difficult to manipulate.
NUMERICALS
(I)
Margin A/c
(II)
Valuation of future
(III)
Beta management or Hedging through future
(IV)
Commodity future
Page 28
REVISION CLASS
DERIVATIVES
(I) Margin A/c
There are three types of margin
(i)
Initial Margin:- Initial margin means margin amount is required at the time of
execution of contract
(ii)
Maintenance Margin:- Maintenance margin is minimum margin amount. If initial
margin is below maintenance margin then investor has to bring out extra margin.
(iii)
Variation Margin:- If initial margin is less than maintenance margin then investor
has to bring extra amount of margin & such extra amount is called variation
margin.
Important Notes
(i)
Margin amount can be withdrawn if margin money is more than initial margin. If
question is silent then assume no withdraws.
(ii)
Whenever contract is squared off then balance amount of margin is refunded .
(iii)
If initial margin is not given in question then it is calculated as under.
Initial Margin = µ + 3𝜎
µ = Average daily absolute change in price
𝜎 = Standard deviation in price
Question – 32
On 31/08/2021 Mr. R has taken a Long position of Two lots of Nifty Futures at 17,300.
One lot of Nifty future is 50 units.
Initial Margin required is 10% of Contract Value.
Maintenance Margin required is 80% of Initial Margin.
The closing price of 5 days are given below –
Date
01/09/2021
02/09/2021
03/09/2021
06/09/2021
07/09/2021
Closing Price of Nifty Future
17340
17180
16990
16900
17120
You are required to(i)
Prepare a statement showing the daily balances in the margin account & payment
on margin calls, if any.
Page 29
REVISION CLASS
DERIVATIVES
(ii)
Compute the Gain or Loss of Mr. R, if contract squared off on 07/09/2021.
(iii)
What would be the Gain or Loss if Mr. R, had taken the short position?
(Exam December - 2021)
Solution:
Upside betting 17,300
Contract Value
= 17,300 × 50 × 2
= 17,30,000
Initial Margin
= 17,30,000 × 10%
= 1,73,000
Maintenance Margin
= 17,30,000 × 80%
= ₹ 1,38,400
(i)
Margin A/c (long) Statement showing the daily balances in Margin A/c and margin
call if any,
Day
(ii)
31/08/21
Closing
Price
17300
Profit Loss
01/09/21
17340
(40 × 50 × 2) = 4,000
-
1,77,000
02/09/21
17180
(17180 − 17340) × 50 × 2 = -16000
-
1,61,000
03/09/21
16990
(16990 − 17180) × 50 × 2 = -19000
-
1,42,000
06/09/21
16900
(16900 − 16990) × 50 × 2 = -9000
4000
1,73,000
07/09/21
17120
(17120 − 16900)× 50 × 2 = 22000
-
Margin
A/c (₹)
-
Balance
1,73,000
1,95,000
Calculation of Profit/Loss
Ending margin
₹ 1,95,000
Less: Initial margin
₹ 1,73,000
Profit
₹ 22,000
Less: Margin call
₹ 40,000
Loss
₹ (18,000)
Page 30
REVISION CLASS
DERIVATIVES
Alternative:
(17120 - 17300) × 50 × 2 = (18,000)
(iii)
Margin A/c (short Position) 17300
Day
Closing
Price
17,300 -
31/08/21
Profit Loss
Margin
A/c (₹)
-
Balance
1,73,000
01/09/21
17,340 (17180 − 17340) 50 × 2 = 4000
-
1,69,000
02/09/21
17,180 (17340 − 17180)× 50 × 2 = - 16000
-
1,85,000
03/09/21
16,990 (17180 − 16990)× 50 × 2 = - 19000
-
2,04,000
06/09/21
16,900 (16900 − 16990)× 50 × 2 = - 9000
-
2,13,000
07/09/21
17,120 (16900 − 17120)× 50 × 2 = 22000
-
1,91,000
Calculation of Profit/Loss
Ending margin
1,91,000
Less: Initial margin
1,73,000
Profit
18,000
Question – 34
Sensex futures are traded at a multiple of 50. Consider the following quotations of Sensex
futures in the 10 trading days during February, 2009:
Day
High
Low
Closing
4-2-09
3306.40
3290.00
3296.50
5-2-09
3298.00
3262.50
3294.40
6-2-09
3256.20
3227.00
3230.40
7-2-09
3233.00
3201.50
3212.30
10-2-09
3281.50
3256.00
3267.50
11-2-09
3283.50
3260.00
3263.80
12-2-09
3315.00
3286.30
3292.00
14-2-09
3315.00
3257.10
3309.30
Page 31
REVISION CLASS
DERIVATIVES
17-2-09
3278.00
3249.50
3257.80
18-2-09
3118.00
3091.40
3102.60
Abhishek bought one sensex futures contract on February, 04. The average daily absolute
change in the value of contract is ₹ 10,000 and standard deviation of these changes is ₹
2,000. The maintenance margin is 75% of initial margin.
You are required to determine the daily balances in the margin account and payment on
margin calls, if any.
(SM New Syllabus & PM)
Solution:
Initial Margin
=μ+3σ
= 10,000 + (3 × 2000)
= ₹ 16,000
Maintenance Margin
= 16,000 × 75%
= ₹ 12,000
Margin A/c (Long)
Day
04/02/09
Closing
Price
3296.50 -
Profit Loss
Margin
A/c (₹)
-
Balance
05/02/09
3294.40 (3294.40 – 3296.50) × 50 = 105
-
15,895
06/02/09
3230.40 (3230.40 – 3294.40) × 50 = - 3200
-
12,695
07/02/09
3212.30 (3212.30 – 3230.40) × 50 = - 905
4210
16,000
10/02/09
3267.50 (3267.50 – 3212.30) × 50 = - 2760
-
18,760
11/02/09
3263.80 (3263.80 – 3267.50) × 50 = 185
-
18,575
12/02/09
3292.00 (3292.00 – 3263.8) × 50 = 1410
-
19,985
14/02/09
3309.30 (3309.30 – 3292.00) × 50 = 865
-
20,850
17/02/09
3257.80 (3257.80 – 3309.30) × 50 = -2575
-
18,275
18/02/09
3102.60 (3102.60 – 3257.80) × 50 = 7760
5485
16,000
16,000
Page 32
REVISION CLASS
DERIVATIVES
(II) Theoretical Future Price [Cost Of Carry Model]
As per cost of carry model, Theoretical future price or fair value of future is calculated as
under
F = Spot price + Interest saved – Dividend forgone
If
Actual future price > Value of future -: Future is overpriced
Actual future price < Value of future -: Future is Underpriced
Suppose
Spot price
= ₹ 500
Rate of Interest
= 10% p.a.
Period
= 1 year
Expected dividend = ₹ 40
(a)
Calculate theoretical future price
F
= Spot price + Interest saved – Dividend forgone
= 500 + 50 – 40
= 510
Or,
F
F
= S(1 + r)−D
= 500(1.10)−40 = ₹ 510
(b)
If actual price ₹550:- Since actual future price is more than theoretical future
hence future is overpriced.
(c)
If actual future price ₹505:- Since actual future price is less than theoretical
future price hence future is under price.
Example – 17
Spot Price = ₹ 500 (F.V. ₹ 100)
Period = 6 Months
Page 33
REVISION CLASS
DERIVATIVES
Dividend Rate = 20%
Rate of Interest = 10% p.a.
Calculate Theoretical Price of Future.
Solution:
Dividend Rate
= EV
Dividend Amount
= Spot Price or Market Price
D
= ₹ 100 × 20% = ₹ 20
F
= S(1+r)−D
= ₹ 500 (1.05) - 20
= ₹ 505
Example – 18
Spot Price
= ₹ 500
Period
= 6 Months
Dividend Yield
= 5% p.a.
Rate of Interest
= 10% p.a.
Calculate fair value of Future.
Solution:
Fair Value of Future
D
= 500 × 5% ×
6
12
= 12.50
F
= S (1 + r) - D
= ₹ 500 (1.05) – 12.50
= ₹ 512.50
Alternator
Page 34
REVISION CLASS
DERIVATIVES
= S [1 + (r−y)t]
F
= 500 [1 + (0.10−0.05) 6/12]
= 500 [1.025]
= ₹ 512.50
Example – 20
Consider the following:
Current value of index
-
1400
Dividend yield
-
6%
CCRRI
-
10%
Find the value of a 3 month forward contract.
Solution:
Alternator
F
= S × e(r-4)t
= 1400 × e(0.10-0.06)3/12
= 1400 × e0.01
= 1400 × 1.01005
F
= 1414.07
Question – 35
The following data relate to Anand Ltd.'s share price:
Current price per share
₹1,800
6 months future's price/share
₹1,950
Assuming it is possible to borrow money in the market for transactions in securities at
12% per annum, you are required:
(i)
to calculate the theoretical minimum price of a 6-months forward purchase; and
(ii)
to explain arbitrate opportunity.
(SM New Syllabus & PM)
Solution:
(i) Theoretical Future Price
F
= S (1 + r) – D
Page 35
REVISION CLASS
DERIVATIVES
= 1,800 (1.06) – 6 = ₹1,908
= ₹435
(ii) ARBITRAGE
Active
Since future is overpriced, hence sell future & buy spot process
Today
Borrow ₹1,800 & buy stock
Contract to sell such stock at future price ₹1,950
After 6 Months
Cash Inflows
Sell share at future price
=
₹1,950
₹1,800 (1.06)
=
₹1,908
Arbitrage Gain
=
₹42
Cash outflows
Repayment of Borrowing
Question – 40
On 31-8-2011, the value of stock index was ₹2,200. The risk free rate of return has been
8% per annum. The dividend yield on this Stock Index is as under:
Month
January
February
March
April
May
June
July
August
September
October
November
December
Dividend Paid p.a.
3%
4%
3%
3%
4%
3%
3%
4%
3%
3%
4%
3%
Assuming that interest is continuously compounded daily, find out the future price of
contract deliverable on 31-12-2011. Given: e0.01583 = 1.01593
(SM New Syllabus & PM)
Solution:
Page 36
REVISION CLASS
DERIVATIVES
(i)
Calculation of average dividend field
Avg. dividend Yield
=
3 +3 +4+3
4
= 3.25% P.a.
(ii)
Calculation of future price
F
= S × e(r-4)t
= ₹2,200 × e(0.08-0-0325)4/12
= ₹2,200 × e0.01583
= ₹2,200 × 1.01593
= ₹2,235.046
Question – 41
The NSE-50 Index futures are traded with rupee value being ₹100 per index point. On 15th
September, the index closed at 1195, and December futures (last trading day December
15) were trading at 1225. The historical dividend yield on the index has been 3% per
annum and the borrowing rate was 9.5% per annum.
(i)
Determine whether on September 15, the December futures were under-priced or
overpriced?
(ii)
What arbitrage transaction is possible to gain out this mispricing?
(iii)
Calculate the gains and losses if the index on 15thDecember closes at (a) 1260 (b)
1175.
Assume 365 days in a year for your calculations
(Exam November - 2019)
Solution:
(i) Calculation of theoretical
F
= S [1+(r−d)t ]
= 11.95 [1 + (0.095 − 0.03)91/365]
= 1,195 (1.0162)
= 1,214.37
= 1,214.37 × 1000
Page 37
REVISION CLASS
DERIVATIVES
= ₹1,21437
Actual future price = 1,225 × 100 = ₹1,22,500
Since Actual future price is more than theatrical future price, hence future is
overpriced.
(ii)
Since future is overpriced, hence possibility of arbitrage gains. In this situation we
Borrow & Buy stock at current market price & take short position on future at ₹
1,225.
(iii) Calculation of Arbitrage Gain
1260
1175
−122330
−122330
+126000
+117500
+893.79
+893.79
Profit/loss on future
(1,225 – 1,260) × 100
−3500
(1,225 − 1,175) × 100
+5000
Arbitrage Gain
=
Repayment of Borrowing
[1,195 (1 + 0.095 ×
91
365
)] × 100
Sell share
Dividend Income
(1,195 × 3% ×
91
365
) × 100
₹1,063.72
₹1,063.79
Question – 42
Suppose current price of an index is ₹13,800 and yield on index is 4.8% (p.a.). A 6 months
future contract on index is trading at ₹14,340.
Assuming that risk free rate of interest is 12%. Show Mr. X (an arbitrageur) can earn an
abnormal rate of return irrespective of outcome after 6 months . You can assume that after
6 months index closes at ₹10,200 and ₹15,600 and 50% of stock included in index shall
pay dividend in next 6 months. Also Calculate implied risk free rate.
Solution:
Calculation of theoretical
F
= S (1+ r) − D
= ₹13,800 (1.06) − 165.60
= ₹14,462.50
Actual future price = 1225 × 100 = ₹12,2500
Page 38
REVISION CLASS
DERIVATIVES
Action
D
Since future is underpriced, hence future & sell stock
= 13800 × 4.890 × 50
= 331.20 ×
6
12
= 165.60
Process
Today
− Short sell 1 share at 13,800 & Invest @ 12% P.a. for 6 months.
− Take long position at 14,340.
After 6 month
10200
15600
Investment (13800 × 1.06)
+1,4628
+1,4628
Buy share & return to Stock lender
−1,0200
−1,5600
Dividend Compensation
−165.60
−165.60
(10200 −14340)
−4,140
(15600 − 14340)
+1,260
₹122.40
₹12,240
Profit/loss on future
Arbitrage Gain
122.40
=
Implied RF Rate
= 12 + 1.77 = 13.77%
13800
× 100 ×
12
% of Return
6
= 1.77%
Question – 43
A future contract is available on R Ltd. that pays an annual dividend of ₹ 4 and whose
stock is currently priced at ₹ 125. Each future contract calls for delivery of 1,000 shares to
stock in one year, daily marking to market. The corporate treasury bill rate is 8%.
Required:
Given the above information, what should the price of one future contract be?
If the company stock price decreases by 6%, what will be the price of one futures contract?
As a result of the company stock price decrease, will an investor that has a long
position in one futures contract of R Ltd. realizes a gain or loss ? What will be the amount
of his gain or loss?
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REVISION CLASS
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(Ignore margin and taxation, if any)
(Exam Nov - 2019)
Solution:
(i) Calculation of Price of one future contract
F
=
S (1+ r) - D
=
₹ 125 (1.08) - 4
=
₹ 131
Price of one future Contract
=
1000 share × 131
=
₹ 131000.
(ii) Calculation of price of one future contract
Price decrease by 6% i.e., ₹ 125 × 0.94
F
=
₹ 117.50
=
117.50 (1.08) – 4
=
₹ 122.90
Price of one contract
=
₹ 122.90 × 1000 shares
=
122900.
(iii) Calculation of lose on long position
Loss
=
131000-122900
=
₹ 8100.
(III) Beta Management Or Hedging Through Stock Index Future
(1) What is Beta?
Beta is measurement of systematic risk which represent relationship between
change in stock return & change in market return.
Beta =
Change in stock ′s return
change in market return
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REVISION CLASS
DERIVATIVES
Suppose, change in stock return is = 20% & change in market return is 10%
Hence Beta of stock =
20%
10%
=2
Higher Beta means higher volatility i.e. higher risk.
(2) What is portfolio Beta?
−
Portfolio beta means weighted average beta of individual stocks.
−
Suppose, we invest in following stocks.
Stocks
Investment Amount
Beta
A
3,00,000
2
B
2,00,000
1.5
C
5,00,000
1
VP =
10,00,000
Method I:- Calculation of Beta of portfolio
BP =
3,00,000×2 + 2,00,000×1.5 +(5,00,000×1)
10,00,000
= 1.4
Method II:- Beta of Portfolio
Stocks
A
B
C
Amount
3,00,000
2,00,000
5,00,000
Weights
0.30
0.20
0.50
10,00,000
1.00
Beta
2
1.5
1
BP
W×B
0.6
0.3
0.5
1.4
Bp 1.4 means if index changes by 10% then value of portfolio will change by 14%.
(3) Beta Management or Hedging Through Stock Index Future
−
We know that beta is a relationship between stock & market.
−
Suppose we hold stock of RIL at ₹ 5,00,000(Portfolio = ₹ 5,00,000) & Beta = 1
& we expect that portfolio will rise but it may possible that market will fall.
We afraid from market falling & we want to hedge the risk of decrease in
value of portfolio.
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REVISION CLASS
DERIVATIVES
−
In order to hedge risk, we have to decrease beta of portfolio & we take short
position on stock index future ( Downside betting)
−
Suppose market will fall in future then we loose on long position of portfolio
but, make profit on short position of stock index future.
−
No. of contracts to be bought or sold of stock index future is calculated as
under
x=
Vp × BT − Bp
F × M × Bf
x
= No. of contracts
BT
= Target Beta (If not given in question, assume “0”)[perfect
hedge]
Bp
= Beta of portfolio
F
= Future price of stock index
m
= Multiplier (Lot size)
BF
= Beta of future (If not given in question , assume 1)
Question – 44
On April 1, 2015, an investor has a portfolio consisting of eight securities as shown below:
Security
A
B
C
D
E
F
G
H
Market Price
29.40
318.70
660.20
5.20
281.90
275.40
514.60
170.50
No. of Shares
400
800
150
300
400
750
300
900
Value
0.59
1.32
0.87
0.35
1.16
1.24
1.05
0.76
The cost of capital for the investor is 20% p.a. continuously compounded. The investor
fears a fall in the prices of the shares in the near future. Accordingly, he approaches you
for the advice to protect the interest of his portfolio.
You can make use of the following information:
(1)
The current NIFTY value is 8500.
(2)
NIFTY futures can be traded in units of 25 only.
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REVISION CLASS
DERIVATIVES
(3)
Futures for May are currently quoted at 8700 and Futures for June are being
quoted at 8850.
You are required to calculate:
(i)
The beta of his portfolio.
(ii)
The theoretical value of the futures contract for contracts expiring in May and June.
Given (e0.03 =1.03045, e0.04 = 1.04081, e0.05 =1.05127)
(iii)
The number of NIFTY contracts that he would have to sell if he desires to hedge
until June in each of the following cases:
(A)
His total portfolio
(B)
50% of his portfolio
(C)
120% of his portfolio
(SM & PM)
Solution:
(i) Calculation of Beta of Portfolio
Stocks
A
B
C
D
E
F
G
H
Market
Price
29.40
318.70
660.20
5.20
281.90
275.40
514.60
170.50
No. of
Shares
400
800
150
300
400
750
300
900
Value
11,760
2,54,960
99,030
1,560
1,12,760
2,06,550
1,54,380
1,53,450
9,94,450
Weight
0.01182
0.2564
0.0996
0.00157
0.1134
0.2077
0.15524
0.1543
Beta
W×B
0.59
0.0070
1.32
0.3384
0.87
0.0866
0.35
0.0005
1.16
0.1315
1.24
0.2575
1.05
0.1630
0.76
0.1173
B.P = 1.102
(ii) Calculation of theoretical future price
May future (2 months)
F
= Sert
= 8,500 ×
e0.20 × 2/12
= 8,500 × e0.033
= 8,500 × 1.03387
= 8,787.89
June future
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REVISION CLASS
DERIVATIVES
F
= 8,500 × e(0.20 × 3/12)
= 8,500 × 1.05127
= ₹8,935.79
(iii) Calculation of no. of contacts
No. of Contract =
VP × BT − Be
F × M × BE
(A) Total portfolio
No of contracts
=
9,94,450 × (0−1.102)
8,850 × 25 × 1
= 4.95 Contracts
5 contracts sold
(B) 50% of Portfolio
No of contracts
=
9,94,450 × 50% × (0−1.102)
8850 × 25 × 1
= 2.48 Contracts
2 contracts sold
(C) 120% of Portfolio
No of contracts
=
(994450 × 120%) × (0−1.102)
8850 × 25 ×1
= 5.94 Contracts
6 contracts sold.
Question – 45
Details about portfolio of shares of an investor is as below:
Shares
No. of shares (Iakh)
Price per share
Beta
A Ltd.
3.00
₹ 500
1.40
B Ltd.
4.00
₹ 750
1.20
C Ltd.
2.00
₹ 250
1.60
The investor thinks that the risk of portfolio is very high and wants to reduce the portfolio
beta to 0.91. He is considering two below mentioned alternative strategies:
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REVISION CLASS
DERIVATIVES
(i)
Dispose off a part of his existing portfolio to acquire risk free securities, or
(ii)
Take appropriate position on Nifty Futures which are currently traded at 8125 and
each Nifty points is worth ₹ 200.
You are required to determine:
(1)
Portfolio beta,
(2)
The value of risk free securities to be acquired,
(3)
The number of shares of each company to be disposed off,
(4)
The number of Nifty contracts to be bought/sold; and
(5)
The value of portfolio beta for 2% rise in Nifty.
(SM & PM)
Solution:
(i) Portfolio Beta
Stock
A Ltd.
B Ltd.
C Ltd.
No. of
shares
3.00
4.00
2.00
Price free
share
500
750
250
Value
Weight
1,500
30,000
5,00
5,000
0.3
0.6
0.1
Beta
1.40
1.20
1.60
B.P
=
W×S
0.42
0.72
0.16
1.30
(ii) The Value of Risk free Securities to be acquired
Disposed off existing securities & acquired RE Securities
Vp = 5,000
Bp = 1.30
Let the value of RF Securities be x
5000 × 1.30 – x −1.30 + (x × 0)
5000
0.91
=
4500
= 6,500 – 1.30x
x
=
x
= 1,500 L
6,500 – 4,500
1.30
Alternative
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REVISION CLASS
DERIVATIVES
B T 0.91
=
= 0.7
B p 1.30
Wp
=
WRF
= 0.3
Value of RF securities is ₹ 5,000 × 0.3 = ₹1,500 L.
(iii) No. of shares to be disposed off
Stock
A
B
C
Weight
0.3
0.6
0.1
Value
450
900
150
1,500
Price per
500
750
250
No. of shares
0.9 lacs
1.20 lacs
0.60 lacs
(iv) No. of Contracts to be bought or sold
No. of Contract
=
=
VP × BT − Be
F × M × BE
5000 ×(0.91−1.30)
8125 × 200 × 1
= 120 Contracts sold
(v) Value of portfolio Beta it nifty riser by 2%
If nifty rises by 2% than value of portfolio will rise by (2 × 1.30) = 2.6%
Value of portfolio
= 5000 + (5000 × 2.6%)
= ₹5,130
Loss on short position of Nifty (120 × 200 × 8,125) × 2% = (39)
VP
Bp
=
5294
=
1.82%
=
∆ portfolio Return ∆ Market
=
1.82%
2%
=
0.91.
Question – 51
Following information is available for consideration:
Page 46
REVISION CLASS
DERIVATIVES
BSE Index
25,000
Value of portfolio
₹50,50,000
Risk free interest rate
9% p.a.
Dividend yield on Index
6% p.a.
Beta of portfolio
1.5
We assume that a future contract on the BSE index with 4 months maturity is used to
hedge the value of portfolio over next 3 months. One future contract is for delivery of 50
times the index.
Based on the above information calculate:
(i)
Price of future contract.
(ii)
Gain on short futures position if index turns out to be 22,500 in 3 months.
Note: Daily compounding (exponential) formula is not required to be used.
(RTP May – 2022, Exam July – 2021)
Solution:
(i) Calculation of price of one future contract future Contract
F
= S [1 + (r−y)t]
= 25,000 (1 + (0.09 − 0.06) 4/12)
= 25,000 × 1.01
= ₹25,250
Price of one future contract
= 25,250 × 50
= ₹12,62,500
Calculation of 1 month future
F
= 25,000 (1 + (0.09 − 0.06) 1/12)
= 22,500 × 1.0025
= ₹22,556.25
Gain on short position of future
Page 47
REVISION CLASS
DERIVATIVES
Gain = (25,250 − 22,556.25) × 50 × 6
= ₹8,08,125
(ii) Gain on loss on short position of future
No. of Contract
=
VP × BT − Be
F × M × BE
=
5050000 × (0−1.50)
25250 × 50 × 1
= 6 Contracts sold
Gain
= (25,250 − 22,556-25) × 50 × 6
= ₹8,08,125
Question – 56
Shyam buys 10,000 shares of X Ltd., @ ₹25 per share and obtains a complete hedge of
shorting 400 Nifty at ₹1,100 each. He closes out his position at the closing price of the
next day when the share of X Ltd., has fallen by 4% and Nifty Future has dropped by
2.5%.
What is the overall profit or loss from this set of transaction?
(Exam January – 2021)
Solution:
Calculation of profit/loss
Loss on long position of X Ltd.
Loss on Short position of Nifty
(2,50,000 × 4%)
(4,40,000 × 2.5%)
Gain
(10000)
+(11000)
₹1,000
Question – 59
Which position on the index future gives a speculator, a complete hedge against the
following transactions:
(i)
The share of Right Limited is going to rise. He has a long position on the cash
market of ₹50 lakhs on the Right Limited. The beta of the Right Limited is 1.25.
(ii)
The share of Wrong Limited is going to depreciate. He has a short position on the
cash market of ₹25 lakhs on the Wrong Limited. The beta of the Wrong Limited is
0.90.
Page 48
REVISION CLASS
DERIVATIVES
(iii)
The share of Fair Limited is going to stagnant. He has a short position on the cash
market of ₹20 lakhs of the Fair Limited. The beta of the Fair Limited is 0.75.
(SM New Syllabus & PM)
Solution:
Statement showing position in future Market
Company
Right
Wrong
Fair
Position in cash
Market
Long
Short
short
Amount
Beta
50,00,000
25,00,000
20,00,000
1.25
0.90
0.75
Position in future
Market
62,50,000 Short
22,50,000 Long
15,00,000 Long
25,00,000 Short
Question – 60
Ram buys 10,000 shares of X Ltd. at a price of ₹22 per share whose beta value is 1.5 and
sells 5,000 shares of A Ltd. at a price of ₹40 per share having a beta value of 2. He obtains
a complete hedge by Nifty futures at ₹1,000 each. He closes out his position at the closing
price of the next day when the share of X Ltd. dropped by 2%, share of A Ltd. appreciated
by 3% and Nifty futures dropped by 1.5%.
What is the overall profit/loss to Ram?
(SM & PM)
Solution:
Statement showing position in future Market
Company
X Ltd.
A Ltd.
No. of Contracts
Position in
Cash Market
Long
Short
=
Amount
Beta
10,000 × 22 = 2,20,000
5,000 × 40 = 2,00,000
1.5
2
Position in
future market
3,30,000 Short
4,00,000 Long
70,000 Long
70,000
= 70 Long
1,000
Calculation of Overall project/long
Loss on long position of X Ltd
(2,20,000 × 2%)
(4,400)
Loss on short position of A Ltd
(2,00,000 × 3%)
(6,000)
Loss on Long position of Nifty
(70,000 × 1.5%)
(1,050)
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REVISION CLASS
DERIVATIVES
Loss
11,450
Question – 61
Mr. X is having a portfolio of shares worth ₹170 lakhs at current price and cash ₹30 lakhs.
The beta of share portfolio is 1.6. After 3 months the price of shares dropped by 3.2%.
Determine:
(i)
Current portfolio beta.
(ii)
Portfolio beta after 3 months if Mr. X on current date goes for long position on ₹200
lakhs Nifty futures.
(Exam July – 2021)
Solution:
(i) Portfolio Beta
Bp
=
(170 × 1.60) + (30 × 0)
200
= 1.36
(ii) Portfolio Beta after 3 Months Calculation % fall in nifty
=
1.6
=
X
=
% Change is share
% change is Nifty
3.2
X
3.2
1.6
= 2%
Calculation value of portfolio after 3 Months Value of share portfolio
= 170 – (170 × 3.2%)
= 164.56
Loss on long position on Nifty (200 × 2%)
Cash balance (30 − 4)
Vp after 3 Months
=4
= 26
= 190.56L
% change is portfolio
=
200−190.56
Beta of portfolio after 3 months
200
=
× 100 = 4.72%
4.72%
= 2.36.
2%
Question – 54
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REVISION CLASS
DERIVATIVES
Mr. SG sold five 4-Month Nifty Futures on 1st February 2020 for ₹ 9,00,000. At the time of
closing of trading on the last Thursday of May 2020 (expiry), Index turned out to be 2100.
The contract multiplier is 75.
Based on the above information calculate:
(i)
The price of one Future Contract on 1st February 2020.
(ii)
Approximate Nifty Sensex on 1 st February 2020 if the Price of Future Contract on
same date was theoretically correct. On the same day Risk Free Rate of Interest and
Dividend Yield on Index was 9% and 6% p.a. respectively.
(iii) The maximum Contango/Backwardation.
(iv) The pay-off of the transaction.
Note: Carry out calculation on month basis.
(RTP November - 2020)
Solution:
(i) Price of one future contract price of one contract
=
₹ 9,00,000
5
= ₹ 1,80,000
(ii) Calculation of Value of nifty as on 1/02/2020
F=
F
(iii)
₹ 1,80,000
= ₹ 2400
75
=
S [1 + (r-y)t]
2400 =
S (1 + (0.09-0.06) 4/12)
2400 =
S × 1.01
S
=
2400
1.01
=
₹ 2376.24
Maximum contango/Backwardation
Since Basis is negative, hence market is said to be contango, Minimum contango is
23.76
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REVISION CLASS
DERIVATIVES
(iv)
Pay off of the transaction
Gain on short position of Nifty
(2400-2100) × 5 × 75 = ₹ 112500
Question – 55
A Mutual Fund is holding the following assets in ₹ Crores :
Investments in diversified equity shares
90.00
Cash and Bank Balances
10.00
100.00
The Beta of the equity shares portfolio is 1.1. The index future is selling at 4300 level. The
Fund Manager apprehends that the index will fall at the most by 10%. How many index
futures he should short for perfect hedging? One index future consists of 50 units.
Substantiate your answer assuming the Fund Manager's apprehension will materialize.
(SM New Syllabus & PM)
Solution: (Home Work)
(IV) Commodity Future
Commodity future means future contra t on commodity like gold, steel, oil etc.
(i) Margin
(ii) Theoretical future price
(iii) Beta management
Theoretical Future Pricing of Commodity
As per cost of carry model, theoretical future price of Commodity is calculated as under.
F = (Spot price + PVSC − PVCY) (1 + r)
F
= Theoretical future price
PVSC = Present value of storage cost
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REVISION CLASS
DERIVATIVES
PVCY = Present value of convenience yield
Hedge Ratio or Hedging Through Future
Spot price of Commodity & future price is Commodity are positive correlated but not in
same rate. In this situation we have to find out the exact proportion & this is called “Hedge
Ratio”.
Hedge ratio is calculated by “Least Square Method”.
Hedge Ratio =
S.D. of Spot
× r Spot & Market
S.D. of Future
Question – 64
The following information is available about standard gold.
Spot Price (SP)
₹ 15,600 per 10 gms.
Future Price (FP)
₹ 17,100 for one year future contract
Risk free interest Rate (R)f
8.5%
Present Value of Storage Cost
₹ 900 per year
From the above information you are requested to calculate the Present Value of
Convenience yield (PVC) of the standard gold.
Solution:
F
17,100
1.085
PVC
= (Spot + PVSC) – PVC) (1 + r)
= 15,600 + 900 – PVC
= 739.63
Question – 66
A company is long on 10 MT of copper @ ₹ 474 per kg (spot) and intends to remain so for
the ensuing quarter. The standard deviation of changes of its spot and future prices are
4% and 6% respectively, having correlation coefficient of 0.75.
What is its hedge ratio? What is the amount of the copper future it should short to achieve
a perfect hedge?
(Practice Manual)
Solution:
(i) Calculation of Hedge Ratio
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REVISION CLASS
DERIVATIVES
Hedge Ratio =
=
S.D of shot
S.D of future
× r spot & future
4%
6%
= 0.5
Hedge Ratio 0.5 means it future changes by 1,070 then spot will change by 5%
(ii) Amount of copper future to be short for perfect hedge
Spot Market long position
= 10,000 kg × 474
= ₹47,40,000
Hedge Ratio
= 0.5
Amount of copper future to be short
= Exposure × Hedge ratio
= 47,40,000 × 0.5
= 23,70,000.
Question – 69
A Rice Trader has planned to sell 22000 kg of Rice after 3 months from now. The spot
price of the Rice is ₹ 60 per kg and 3 months Future on the same is trading at ₹59 per kg.
Size of the contract is 1000 kg. The price is expected to fall as low as ₹56 per kg, 3 months
hence.
Required:
(i)
To interpret the position of trader in the Cash Market.
(ii)
To advise the trader the trader should take in Future Market to mitigate its risk of
reduced profit.
(iii)
To demonstrate effective realized price for its sale if he decides to make use of future
market and after 3 months, spot price is ₹57 per kg and future contract price for
closing the contract is ₹58 per kg.
(RTP Nov – 2020 & MTP May – 2019)
Solution:
(i)
Traders wants to sell in future but he hall commodity now & Expects that price will
rise hence trader has long position on RICE in Cash Market.
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REVISION CLASS
DERIVATIVES
(ii)
Trader should take short position on Rice in future market to hedge the risk.
(iii)
Trader took short position on Rice at ₹ 59
After three months selling price of rice
= ₹57
Gain on short position of future (₹59 − ₹58)
= ₹1
Effective realized price
= ₹58
Question – 68
A call option on gold with exercise price ₹ 26,000 per ten gram and three months to expire
is being traded at a premium of ₹1,010 per ten gram. It is expected that in three months
time the spot price might change to ₹ 27,300 or 24,700 per ten gram. At present this
option is at-the-money and the rate of interest with simple compounding is 12% per
annum. Is the current premium for the option justified? Evaluate the option and
comments.
(Practice Manual)
Solution:
Calculation of value of call option (Risk Neutral Probability)
Step 1 : Given
E
=
26000
S
=
26000
U
=
d
=
24700
= 0.95
26000
R
=
1.03
27300
26000
= 1.05
Step 2 : Risk Neutral probability
P
=
R-d
= 1.05
u-d
=
1.03 – 0.95
= 0.8
1.05 – 0.95
Step 3 : Binomial Tree
Page 55
REVISION CLASS
DERIVATIVES
Step 4 :
Value of Call option
Co
=
=
=
Cup + Cd (1-p)
R
1300 ×0.8 + (0 ×0.2)
1.03
1009.71 or 1010
Since, premium amount is equal to value of option, hence option is correctly traded.
Page 56