futures and options for cma exam

8/13/2019 Futures and Options for CMA Exam

http://slidepdf.com/reader/full/futures-and-options-for-cma-exam 1/47

www.singaracademy.com

Derivatives 6.1

1. DERIVATIES

Question 1: What are derivatives?

Answer: Derivatives are financial instruments whose values depend on the value of the underlying assets

Question 2: What is forwards contract?

Answer: Forward Contract is a simple derivative. It is an agreement to buy or sell an asset at a certain

future time for a certain price. A forward contract is traded in the over-the-counter market – usually

between a financial institution and of its client. Forward contracts are widely used in foreign exchange.

Question 3: What is future contract?

Answer: Future Contract like a forward contract, a future contract is an agreement between two parties to

buy or sell an asset at a certain time in the future for a certain price. Unlike forward contracts, Future

contracts are normally traded on an exchange with pre-standardized lot.

At the expiry of the contract, the future-contract price tends to future spot price of the underlying asset.

Otherwise there is no profit or loss on settlement date.

A futures contract has

a) The date on which the contract is being executed

b) The name of the underlying asset

c) The quantity of the asset

d) The contract price

e) The period of the contract.

Question 4: Differentiate between Futures and Forwards.

Answer:

Futures Forwards

1 Trade in organized exchange OTC

2 Contract term Standardized Customized

3 Liquidity More Less

4 Margin payment Requires Nil

5 Settlement Daily At the end of period

6 Risk of default Taken by clearing corporation Borne the client.




Derivatives 6.2

Question 5: How to determine theoretical futures price?

Answer:

Formula Financial futures Commodity futures

Spot price ××× ×××

Add Cost of carry ××× ×××

Cost of finance × × or

S(− ) – S××× ×××

Storage cost S × Rate of Storage × Not Applicable ×××

Insurance Cost Not Applicable ×××

Others ××× ×××

Less Returns ××× ×××

Future value of dividend or ( × ) ××× Not Applicable

Future Price [Fair Value] ××× ×××

Where: S = Spot rate, r = rate of interest, t = time, e = constant for continuous compounding, d = rate of

dividend, FV = Face Value of share, D = Dividend per share, dy = dividend yield, cy = Convenience Yield

Question 6: What is an index future?

Answer: An index future is a derivative whose value is dependent on the value of the underlying asset

(e.g. BSE Sensex, S&P CNX Nifty). In index futures, an investor buys and sells a basket of securities

comprising an index in their relative weights.

Practical Problems

Question 1: The following quotes were observed by Mr. X on Mar 11, 2005 in the Economic Times.

Contracts Open High Low CloseOpen

Interest

Traded

quantity

Number of

Contracts

1 SBI MAR 05 FUT 735 740 735 738 433 138000 92

2 NIFTY MAY 05 FUT 2800 2830 2800 2830 1016 102400 512

Required: Explain the details that are displayed against the futures.

Answer:

Column Particulars Meaning – Row 2

1 Contracts SBI – stock future expires on Mar 2005

2 Open Day‟s open-rate of SBI-stock future

3 High Day‟s high-rate of SBI-stock future

1 Cost of finance is calculated using simple interest rate [ × × ] or continuous compound interest rate [ ]2 Return is in absolute numbers for shares and in dividend yield (dy) % for index future




Derivatives 6.3

4 Low Day‟s low-rate of SBI-stock future

5 Close Day‟s close-rate of SBI-stock future

6 Open Interest Pending future contract

7 Traded quantity Number of shares traded in the day

8 Number of contracts Number of contracts traded in the day

Note:

Number of shares per contract [Lot size] =

= 1,38,000

92 = 1,500

Positions outstanding = Open interest × Lot size = 433 × 1,500

= 6,49,500 shares

2 – Details of Nifty Index is also as similar as the above

FUTURES PRICING & ARBITRAGE

Question 2: Calculate the price of 3-month RIL futures and find the chance of arbitrage if any, if RIL

(Face Value ₹10) quotes at ₹520 on NSE, and the 3 month futures price quotes at ₹542, and the one month

borrowing rate is given as 15% p.a. and the expected annual dividend yield is Nil p.a. payable before

expiry.

Answer:

Formula Calculation ₹

Spot price 520.00

Add Cost of finance × × 520 ×15

100×

3

12 19.50

Future Price [Fair Value] 539.50

Analysis: The fair value of Futures price [₹539.50] is lesser than the quote in the exchange [₹542]. Hence

the Futures are overvalued in the market. Hence Arbitrageurs would buy stocks in cash market and sell the

Futures.

Question 3: Calculate the price of 3-month M&M futures and find the chance of arbitrage if any, if M&M

(Face Value ₹10) quotes at ₹520 on NSE, and the 3 month futures price quotes at ₹532, and the one month

borrowing rate is given as 15% p.a. and the expected annual rate of dividend is 25% p.a. payable before

expiry.

Answer:


Spot price 520.00

Add Cost of finance × × 520 × 15100 × 3

12 19.50




Derivatives 6.4

Less Dividend FV× 10 ×25

100 2.50


Analysis: The fair value of Futures price [₹537] is lesser than the quote in the exchange [₹532]. Hence the

Futures are undervalued in the market. Hence Arbitrageurs would sell stocks in cash market and buy the

Futures.

[CA FINAL]

Question 4: The following data relates to ABC Ltd‟s share prices:

Current price per share ₹180

Price per share in the 6 months futures market ₹195

It is possible to borrow money in the market for securities transactions at the rate of 12% p.a.

Required:

a. Calculate the theoretical minimum price of a 6-month futures contract

b. Explain if any arbitrage opportunities exist

Answer:

(a) Formula Calculation ₹

Spot price 180.00


100 ×6

12 10.80


(b) Analysis: The fair value of Futures price [₹190.80] is lesser than the quote in the exchange [₹195].

Hence the Futures are overvalued in the market. Hence Arbitrageurs would buy stocks in cash market and

sell the Futures.

Cash-flow

Borrow for 6 months at an interest rate of 12% 180.00

Less Buy share spot in cash market with the borrowings (180).00

Add Sell Future 195.00

Less Repay the borrowing together with interest 180 +180×0.12×0.5 (190.80)

Net Cash Flow [Arbitragers surplus per share] 4.20

Note: We have ignored transaction costs like commission, margin, etc.

Question 5: Consider a 3 month expiry futures contract on a non-dividend paying stock. The underlying

stock is available for ₹70 (Face Value ₹10). With continuously compounded Risk free rate (CCRRI) of 8% p.a.




Derivatives 6.5

1. Find the price of futures.

2. If the stock pays a dividend yield of 5%, find the price of the futures.

3. If the stock pays a dividend of ₹1.50 in 3 months time, find the price of the futures.

4. If the stock pays a dividend of ₹1.50 in 1 month‟s time, find the price of the futures.

5. If the stock pays a dividend of ₹2 today, find the price of the futures.

6. If the stock pays a dividend of 10% today, find the price of the futures.

Answer:

Formula 1 2 3 4 5 6

Spot price 70.00 70.00 70.00 70.00 70.00 70.00

Add Cost of finance S(− ) – S 1.41 0.53 1.41 1.41 1.41 1.41

Less Future value of dividend D or (FV× ) Nil Nil 1.50 1.52 2.04 1.02

Future Price 71.41 70.53 69.91 69.89 69.37 70.39

Question 6: A stock index currently stands at ₹3500. The risk free interest rate is 8% per annum and the

dividend yield on the index is 4% per annum. Expiry is in 4 months. What should the Index futures price

be if it has continuously compounded rate?

Answer:

Formula Calculation Index

Spot price 3,500.00

Add Cost of finance S(− ) – S 3,5000.08−0.040.33 – 3,500 46.97

Future Price 3,546.97

Question 7: Assume that the risk-free interest rate is 9% per annum with continuous compounding and

that the dividend yield on a stock index varies throughout the year. In August and November dividends are

paid at a rate of 5% per annum, in September at a rate of 6% per annum and in October at a rate of 2% per

annum. Suppose the value of an index on July end is 3000. What is the futures price for a contract

deliverable on November end?

Answer: Formula for finding the future value = S e(r-dy)t

The August Futures Price = 3000×e. - .

3010.017

The September Futures Price = 3010.017×e. - .

3017.551

The October Futures Price = 3017.551×e. - .

3035.205

The November Future Price = 3035.205×e. - .

3045.339

Therefore the November‟s Futures Price as at July end 3045.339

Question 8: Current value of stock index is 4500 and the annualized dividend yield is 4%. A three month

futures contract on the SENSEX can be purchased for a price of ₹4600. The risk free rate of return is 10%.

Can the investor earn abnormal risk free rate of return by resorting to Arbitrage?




Derivatives 6.6

Assume that 50% of the stocks in the index will pay dividends during the next three months.

Ignore transaction costs, margin requirements and taxes.

Answer: The fair value of Index Futures contract is

Formula Calculation Index

Spot price 4,500.00

Add Cost of finance S(− ) – S 4,5000.1−0.04×50%0.25 – 4,500 90.90

Future Price 4,590.90

The value of current index future is ₹4600 is overpriced. The arbitrageur can exploit the opportunity by

buying in cash market and sell the future.

Cash-flow

Borrow for 3 months at an interest rate of 10% 4,500.00

Less Buy index in cash market with the borrowings 4,500.00

Add Sell Future 4,600.00

Less Repay the borrowing together with net interest 4,5000.1−0.04×50%0.25 4,590.90

Net Cash Flow [Arbitragers surplus per lot] 9.10

Question 9: You have entered into a sale of one gold futures contract in Multi Commodity Exchange, to

sell 1 kg of gold at ₹960,000 per kg. The contract now still has 6 months to expiry. Gold is trading now at

₹9350 per 10 gms (1 Futures Contract = 1 Kg.) and the six-month storage and risk-free rate are 0.2% & 5%

with continuous compounding, respectively. What is the fair value of the contract?

[Assume storage cost is paid in advance]

Answer:

Formula ₹

Spot price 9,35,000

Add Cost of carry

Cost of finance S – S 9,35,000(0.05×0.5) – 9,35,000 23,667

Future value storage cost S × Rate of Storage × 9,35,000×0.002×(0.05×0.5) 1,917

Future Price 9,60,584

Question 10: The current price of wheat is ₹900 per bushel. The storage costs are ₹145 per bushel per year

payable in advance. Assuming that interest rates are 10% per annum with continuous compounding for all

maturities and this year an expected convenience yield of 2% is observed. Calculate one year futures price

of wheat.




Derivatives 6.7

Answer:


Spot price 900.00

Add Cost of finance S(− ) – S 900(0.1−0.02)1 – 900 74.95

Add Future Value of Storage cost Storage Cost × (−) 145(0.1−0.02)×1 157.08

Future Price [Fair Value] 1,132.03

Question 11: The current price of Cotton is ₹400 per bale. The storage costs are ₹100 per bale per year

payable in arrears. Assuming that, the interest rates are 10% per annum with continuous compounding for

all maturities. Calculate one year futures price of 500 bales of Cotton.

Answer:


Spot price 2,00,000.00

Add Cost of carry

Cost of finance S(− ) – S 2,00,0000.1×1 – 2,00,000 21,031.89

Future value of Storage cost Storage Cost 50,000.00

Future Price [Fair Value] 2,71,031.89

Question 12: Consider a 6 month gold futures contract of 100 grams. If the spot price is ₹480 per gram

and that it costs ₹3 per gram for the period to store gold and that the cost is incurred at the end of the

period.

a. If the continuously compounded Risk free rate (CCRRI) of 10% per annum, compute futures price.

b. If futures are available at ₹520 per gram what action would be suggested? If futures are available at

₹490 per gram what action would be suggested?

Answer:

(a) Formula Calculation ₹

Spot price 48,000.00

Add Cost of finance S – S 48,0000.1×0.5 – 48,000 2,460.75

Add Future value of Storage cost Storage Cost 3×100 300.00

Future Price [Fair Value] 50,760.75

(b)

Actual Value of Futures = ₹52,000 Fair Value < Actual Value Sell Futures, Buy Gold in Spot Market

Actual Value of Futures = ₹49,000 Fair Value > Actual Value Buy Futures, Sell Gold in Spot Market

Question 7: Explain the concept of margin in F&O.

Answer: Initial margin

1. Meaning: Initial margin, a sum of money, is deposited by both the buyer and the seller.

2. Purpose: To assure the execution of the contract.




Derivatives 6.8

3. When to deposit: At the time of entering into the futures contract.

4. How much: Minimum margins are set by the Exchange and are usually about 10% of the total value

of the contract.

5. How to calculate: An initial margin is calculated based on the concept of Value-at-Risk (VAR).

6. Why: The initial margin deposit is large enough to cover a one-day‟s loss that can be encountered in99% of the days.

Maintenance Margin: Maintenance Margin is the minimum margin required to hold a position.

Maintenance margin should be sufficient to support the daily settlement process called “mark -to-market”,

where by losses that have already occurred are collected. Initial margin, on the other hand seeks to

safeguard against potential losses on outstanding positions. Maintenance margin is the margin required to

be kept by the investor in the equity account equal to more than the specified percentage of the amount

kept as initial margin. Normally this is 75% to 80% of the initial margin. In case this requirement is not

met, the investor is advised to deposit cash to make up for the shortfall. If the investor does not respond,

then the broker would close out the investor‟s position by entering a reverse trade in the investor‟s account.

If the customer selects to liquidate open positions in order to meet a maintenance margin call, such

liquidations are completed immediately. Any profits over the margin requirement can be withdrawn or

used for other futures contracts.

Variation Margin: Variation margin is simply the running profit or loss on positions paid out or received

on a daily basis. If a margin call is made and the money is deposited by the trader / investor, to bring the

account to the level of initial margin, then the amount that is deposited is called as the variation margin.

Variation margin is the amount needed to restore the initial margin once a margin call has been issued. The

variation margin may change depending on how far the margin account has fallen below the maintenance

margin level.

MARGINS

Question 13: Nifty Index is currently quoting at 1329.78. Each lot is 250 units. X purchases a March

contract at 1364. He has been asked to pay 10% initial margin. What is the amount of initial margin? Nifty

futures rise to 1370. What is the percentage gain?

Answer:

Particulars Formula Calculation ₹

The initial margin deposited Value of Contract × Initial Margin% 1,364×250×0.10 34,100

Profit or Loss (Latest price – earlier price) ×Units (1,370 – 1,364)×250 1,500

Profit % %

1,500

34,100% 4.4%

Question 14: Suppose that „X‟ bought 1 contract of Andhra Bank Futures (each underlying 2300 equity

shares) for ₹62.80 per share. The initial margin is 50%. The maintenance margin is 40%. Suppose that the

stock price drops to ₹50 per share.

a. Does X need to put additional fund to his account?




Derivatives 6.10

Tandon

[Long 5]

Tilak

[Short 5]Aug Details

2 Initial Margin Paid 12,000 12,000

+/- Profit / (Loss) [mark-to-market] (818 – 840)×250 shares (5,500) 5,500

Closing Balance Balance before Margin Call 6,500 17,500

Deposit / (Drawings) 5,500 (2,750)

Closing Balance Balance after Margin Call 12,000 14,750

2 Opening Balance From previous day 12,000 14,750

+/- Profit / (Loss) [mark-to-market] (866 – 818)×250 shares 12,000 (12,000)

Closing Balance Balance before Withdrawals 24,000 2,750

Deposit / (Drawings) 6,000 9,250

Closing Balance Balance after Margin Call 18,000 12,000

4 Profit / (Loss) [mark-to-market] (830 – 866)×250 shares (9,000) 9,000

Closing Balance Balance before Margin Call 9,000 21,000

Deposit / (Drawings) 3,000 4,500

Closing Balance 12,000 16,500

5 Profit / (Loss) [mark-to-market] (846 – 830)×250 shares 4,000 4,000

Closing Balance Balance before withdrawals 16,000 12,000

Deposit / (Drawings) Half of (16,000 – 12,000) 2,000 250.00Closing Balance 14,000 12,250.00

Net gain or loss Closing Balance – Initial Margin – Variation margin Paid + Profit Withdrawn

Mr. Tandon‟s gain 14000 – 12000 – [3000 + 5500] + [2000 + 6000] 1,500

Mr. Tilak‟s loss 12250 – 12000 – [9250] + [250 + 4500 + 2750] -1,500

Net position of the two 0

Question 16: Nifty Index is currently quoting at 1300. Each lot is 250. Mr. X purchases a March contractat 1300. He has been asked to pay 10% initial margin. What is the amount of initial margin? To what level

Nifty futures should rise to get a percentage gain of 5%.

Answer:

Particulars Formula Calculation ₹

1 The initial margin Value of Contract × Initial Margin% 1,300×250×0.10 32,500

2 Nifty rise to gain 5% Price +Deposit × ROI

1,300 +32,500×0.05

250 1,306.50

Question 17: A futures contract is available on a company that pays an annual dividend of ₹5 and whose

stock is currently priced at ₹200. Each futures contract calls for delivery of 1,000 shares of stock in one




Derivatives 6.11

year, daily marking to market, an initial margin of 10% and a maintenance margin of 5%. The current

Treasury bill rate is 8%.

a. Given the above information, what should the price of one futures contract be?

b. If the company stock decreases by 7%, what will be, if any, the change in the futures price?

c. As a result of the company stock decrease, will an investor that has long position in one futurescontract of this company realize a gain or a loss? Why? What will be the amount of this gain or loss?

d. What must the initial balance in the margin account be? Following the stock decrease, what will be, if

any, the change in the margin account? Will the investor need to top up the margin account? If yes, by

how much and why?

e. Given the company stock decrease, what is the percentage return on the investor‟s position? Is it

higher, equal or lower than the 7% company stock decrease? Why?

Answer:

Formula Calculation (a) ₹ (b) ₹

7% price fall

Spot price 200.00 186.00


100× 1 16.00 14.88

Less Dividend 5.00 5.00

Future Price [Fair Value] 211.00 195.88

(c) Loss because of price fall 1,000 (211 – 195.88) 15,120

(d) Initial Margin required for 1000

shares

Value of shares×Initial

Margin% 211×

1,000×

10% 21,100

Balance after loss 5,980

Maintenance Margin required

for 1000 shares

Value of shares

×Maintenance Margin%211×1,000×5% 10,550

Balance after loss is less than maintenance margin hence he has to invest to bring his

balance equal to initial margin else his position will be closed out by the broker15,120

(e) Return percent %

−15,120

21,100% -71.7%

The loss is 10 times higher than the actual decrease in the stock price. The 10-to-1 ratio of percentage

changes reflects the leverage inherent in the futures contract position.

Question: What is the Hedging?

Answer: Hedging is taking an equal and opposite position in another market so that loss that may arise in

one market would be compensated by a gain in another market. The extent of hedging (hedge ratio) is

determined by the beta of a security. If the beta is greater than one (i.e. hedge ratio is greater than one) then

the position hedged would be higher than the underlying position and would be proportionate to the beta of

the security.




Derivatives 6.12

Question: What is the Hedge Ratio?

Answer: The hedge ratio is the ratio between future position and underlying asset position. The hedge

ratio allows the hedger to determine the number of contracts that must be employed in order to minimize

the risk of the combined cash-futures position.

Hedge Ratio =

As explained earlier, for a perfect hedge, in case of a stock / portfolio position hedged with an index

futures position, the hedge ratio is the beta of stock or portfolio. Else, if a stock is hedged using the same

stock futures position, the hedge ratio is one. If the hedger wants to hedge his stock / portfolio position

partially, then the hedge ratio would be less than one. On the hand if his future hedge position is more than

that of his current position, we say that the hedge ratio is more than one.

FUTURES – HEDGING

Question 18: Identify the hedging strategies that would be required using the index futures under

following circumstances:

Stock Position Beta Number of Shares Price Hedge Needed

RIL Long 1.2 1000 500 Full

Satyam Long 0.8 1000 350 Full

RIL Short 1.2 1000 500 Full

Satyam Short 0.8 1000 500 80%

Infosys Long 1.0 1000 1700 120%

Answer: Future position = Beta × Number of Shares × Price × % Hedging required

Stock Original

Position

Beta No. of

Shares

Price Hedge

Needed

Hedge

Position

Future position Futures

Strategy

[lacs]

RIL Long 1.2 1000 500 Full Short 1.2×1000×500×1.0 Short 6

Satyam Long 0.8 1000 350 Full Short 0.8×1000×350×1.0 Short 2.8

RIL Short 1.2 1000 500 Full Long 1.2×1000×500×1.0 Long 6

Satyam Short 0.8 1000 500 80% Long 0.8×1000×500×0.8 Long 3.2

Infosys Long 1.0 1000 1700 120% Short 1.0×1000×1700×1.2 Short 20.4

[CA FINAL]

Question 19: Which position on the Index future gives a speculator a complete hedge against the

following transactions?

1. The share of Right Ltd. is going to rise. He has a long position on the cash market on ₹50 lakhs on the

Right Ltd. The beta of the Right Ltd. is 1.25.




Derivatives 6.13

2. The share of Wrong Ltd. is going to depreciate. He has a short position on the cash market of ₹25 lacs

on the Wrong Ltd. The beta of the wrong Ltd. is 0.9

3. The share of Fair Ltd is going to stagnate. He has a short position on the cash market of ₹20 lacs of

Fair Ltd. The beta of the Fair Ltd. is 0.75

Answer: Future position = Beta × Value of InvestmentStock Original

Position

Beta Value Hedge

Position

Future

position

Futures

Strategy [lacs]

Right Ltd Long 1.25 50 Short 1.25×50 Short 62.5

Wrong Ltd Short 0.9 25 Long 0.9×25 Long 22.5

Fair Ltd Short 0.75 20 Long [Optional] 0.75×20 Long 15

[CA FINAL]

Question 20: Ram buys 10,000 shares of X Ltd. at ₹22 and obtains a complete hedge of shorting 400 NIFTIES at ₹1,100 each. He closes out his position at the closing price of the next day at which point the

share of X Ltd. has dropped 2% and the Nifty future has dropped 1.5%. What is the overall profit or loss of

this set of transaction?

Answer: The gain or loss incurred by Ram can be estimated as follows:

Value of Bought Shares Value of Short Futures

Today‟s Valuation 22×10,000 ₹2.2 Lakhs 400×1100 ₹4.4 lakhs

Next Day‟s Valuation* 21.56×10,000 ₹2.156 lakhs 400×1083.5 ₹4.334 lakhs

Gain / (Loss) (0.044) lakhs 0.066 lakhs

*When share price drops by 2% and Futures drop by 1.5%

Net Gain = 0.066 lakhs – 0.044 lakhs = 0.022 lakhs = ₹2200

ALTER THE BETA OF PORTFOLIO

Question 21: A portfolio manager manages a large portfolio of ₹300 million with a beta of 1.6 (90% of

total portfolio consists of stock and rest 10% is cash) He expects the market to be volatile in the near future

and contemplates to reduce his portfolio beta to 1.0. How he can accomplish his goal using stock index

futures? (Assume the current index to be quoting at 1000 with a market lot of 100).

Answer:

Equity 0.9 ₹270 million

Cash 0.1 ₹30 million

Total portfolio 1.0 ₹300 million

The fund manager has to sell index futures to reduce the beta. He would sell „N‟ contracts so that thefollowing equation is satisfied:




Derivatives 6.14

(270 million×1.6) – (1000×100×N) = 300 million×1.0

Solving the above equation gives N = 1320

The fund manager would sell 1320 contracts i.e. 1320×1000×100 = ₹132 million of Index futures and

accomplish his goal of reducing the beta to 1.0

Question 22: A portfolio manager manages a large portfolio of ₹200 million with a beta of 1.0 (80% of

total portfolio consists of stock and rest 20% is cash) He expects the market to rally in the near future and

contemplates to increase his portfolio beta to 1.5. How he can accomplish his goal using stock index

futures? (Assume the current index to be quoting at 1000 with a market lot of 100).

Answer:

Equity 0.8 ₹160 million

Cash 0.2 ₹40 million

Total portfolio 1.0 ₹200 million

The fund manager has to buy index futures to increase the beta. He would Buy „N‟ contracts so that the

following equation is satisfied:

(160 million×1.6) + (1000×100×N) = 200 million×1.5

Solving the above equation gives N = 1400

The fund manager would buy 1400 contracts i.e. 1400×1000×140 = ₹140 million of Index futures and

accomplish his goal of increasing the beta to 1.5

Question 23: Let us consider a portfolio held by Mr. X.

Portfolio Characteristics Portfolio Beta (β) Total Cost in Lakhs Market Value in Lakhs

20 Stocks 0.983 1520.87 1767.59

Mr. X believes that the portfolio performance since the day it was constructed has been good. He has

clocked a return of 16.22% over a three month period. However, now he is getting worries whether the out

performance would continue next month. He wants to protect his portfolio by using 1 month S&P CNX

Nifty Futures which is quoting at 2189.05, Each Nifty contract is 200 units. Explain the use of hedging by

Index Futures. Also explain how much overall gain / loss he would witness (a) if the market rises by 10%

(b) if the market falls by 10%.

Answer: As given Mr. X has decided to use S&P CNX Nifty futures for hedging. One Nifty contract

represents ₹200×2189.05 = ₹437,810. Mr. X has to first have the beta of the portfolio. Which we know: βp

= 0.983. Now, we calculate the number of futures he has to sell. This is equal to

0.983×1767.59×100000/437810 = 397 contracts approximately.

Thus, Mr. X will have to sell 397 contracts of 1 month Nifty futures to hedge the portfolio equivalent to

2189.05×200×397 = ₹1738 lakhs.

Let us estimate what happens when the nifty futures moves up or down. We also know that the index too

would track similar movement. [We omit interest cost on initial margin, since no information is provided]




Derivatives 6.15

Portfolio Value Nifty Futures * Gain / Loss

Market up

10%

1767.59 × 1.1 =

1944.35 lakhs

397×200×2189.05×1.1

= 1911.92 lakhs

(1944.35 – 1767.59) – (1911.92 –

1738) = 2.84 lakhs

Market down

10%

1767.59 × 0.9

= 1590.83 lakhs

397×200×2189.05×0.9

= 1564.30 lakhs

(1590.83 – 1767.59) – (1564.30 –

1738) = -3.06 lakhs

It can be seen that in the first case, when the market rose, X made a small profit and incurred a little loss of

(3.06 / 1767.59) = 0.17% only when the market fell. As against a 10% loss which he would have incurred

had we not hedged, X incurred a minimal acceptable loss. Obviously the gains were also clipped to (2.84 /

1767.59) = 0.16% as against 10% rise of market. Thus hedging does not aim to make profits or reduce

losses; it aims to lock a portfolio value i.e. it aims to reduce the uncertainty that may arise while managing

the portfolio.

Question 24: Suppose you own a grove of Apple trees. The harvest is still two months away but you are

concerned about price risk. You want to guarantee that you will receive ₹10.00 per KG in two months

regardless of what the spot price is at that time. You are selling 25,000 Kgs.

a. If you short sell apples at ₹10 per Kg., show the economics of a short transaction in the forward

market if the spot price on delivery date is ₹7.50 per Kg, ₹10.00 per Kg, or ₹12.50 per Kg.

b. What would have happened to you if you had not entered the hedge and each scenario is equally

likely?

c. What is the variability of your receipts after the hedge is in place?

Answer:

a)

Apple Grower‟s Transaction ₹7.50 / Kg ₹10.00 / Kg ₹12.50 / Kg

Proceeds from sale of Apple ₹187,500 ₹250,000 ₹312,500

Cash flow from futures contract +₹62,500 ₹0 -₹62,500

Total receipts ₹250,000 ₹250,000 ₹250,000

b) You would have had a 1/3 chance each of paying out ₹187500 (less than expected), ₹250000 (the same

as expected) or ₹312500 (more than expected) i.e. ₹250000

c) No variability. Receipts are always equal to ₹250000.

Question 25: Suppose in six months‟ time the cost of 1 Kg of Mentha oil will either be ₹480 or ₹520. The

current futures price is ₹500 per Kg.

How can two parties (seller & buyer) use the Mentha oil futures market to reduce their risks and lock in a

price of ₹500 per Kg.? Assume each contract is for 360 Kgs and they each need to hedge 720 Kgs.

Can you say that each party has been made better off? Why or why not?




Derivatives 6.16

Answer: (a)

Formula Buyer Formula Seller

Mentha Oil [S6 / kg] 480 520 480 ₹520 / Kg.

Gain (Loss) (S6 – X)×720 kg (14,400) 14,400 (X – S6)×720 kg 14,400 (14,400)

Net Value 3,60,000 3,60,000 3,60,000 3,60,000

S6 = Spot price at the end of 6th

month

(b) Even though it appears that in each scenario one party has benefited at the expense of the other, both

have really benefited because both parties were able to lock in a price of ₹500 per Kg. and eliminate all

risk.

Question 26: Suppose you are a CFO of Hotels ITC and you purchase a large quantity of coffee each

month. You are concerned about the price of coffee one month from now. You want to guarantee that youwill not pay more than ₹100 per Kg. of “Coffee A” for 15,000 Kgs. You do not want to pay for insurance

but you do want to lock in a current price of ₹100 per Kg for 15,000 Kgs.

a. Show the economics of a futures transaction if the spot price on the delivery date is ₹75, ₹100, or

₹125.

b. What is the variability of Hotels ITC‟s total outlays under the futures contract?

c. If at the time of delivery coffee is ₹75 per Kg, should you have forgone entering into the futures

contract? Why or why not?

Answer:

(a) CFO would sell “Coffee A” futures entailing 15,000 Kgs. at the prevailing price of ₹100 / Kg

CFO Hotels‟ Transaction ₹75/Kg ₹100/Kg ₹125/Kg

Cost of coffee purchased from supplier ₹1125000 ₹1500000 ₹1875000

Cash flow from futures contract +375000 ₹0 (₹375000)

Total outlay ₹1500000 ₹1500000 ₹1500000

a. Outlays are fixed at ₹1500000.

b. Regardless of the outcome of the price of coffee at the delivery date, the Treasurer did the righttransaction if he wanted to lock in a price of ₹100 per Kg. Although he gave up any opportunity to pay

a lower price, he also guaranteed that he would never pay more than ₹100 per Kg. A hedge transaction

is only useful if one does not know the future price of some item, hence the need to hedge the risk of

uncertainty.




Derivatives 6.17

OPTIONS BASICS

Question 1: Write a note on options.

Answer: There are two types of options, calls and puts.

Call option: gives the buyer of the call option the right, but not the obligation, to buy shares of the

underlying security or index at a specific price for a specified time. The seller of a call option (writer) has

the obligation to sell the underlying share if the buyer exercises the option.

Put option: gives the buyer of the put option the right, but not the obligation, to sell shares of the

underlying security or index at a specific price for a specified time. The seller of a put option (writer) has

the obligation to buy the underlying share if the buyer exercises the option.

Question 2: Explain At-the-money option, In-the-money option and Out-of-the money option.

Answer:

Term Meaning Call option Put option

In-the-money Profit position in option Market price > Strike price Market Price < Strike Price

At-the-money Break-even in option Market Price = Strike Price Market Price = Strike Price

Out-of-the-money Loss position in option Market Price < Strike Price Market price > Strike price

Question 3: Explain intrinsic value and time value

Answer:

Intrinsic Value = Gain when the option is in-the-money or 0 when the option is out-of-the-money

Extrinsic Value or Time Value = Premium – Intrinsic Value

Question 4: Write a note on long and short.

Answer: “Long” is buy position and “short” is sell position

Long Call & Short Call: Long call is buy position of call and Short call is sell position of call

Long Put & Short Put: Long put is buy position of put and Short put is sell position of put

Question 5: What are the variables of a call option?

Answer:

1. Price of the underlying asset

2. Exercise price

3. Variability of return

4. Time left to expiration

5. Risk-free interest




Derivatives 6.18

PRACTICAL PROBLEMS

Question 1: The following quotes were observed by Hari on March 11, 2005 in the Economic Times.


Interest

Traded

quantity

Number of

ContractsUnderlying

CE-1950-

March 2005191.05 205 191.05 204.90 41,000 1,600 8 NIFTY

PE-2100-

March 200519.50 26 18.65 19.9 26,47,000 13,69,000 6845 NIFTY

Explain what these quotes indicate.

Answer:

Column Particulars Meaning – Row 21 Contracts Call European of NIFTY with strike price ₹1,950 and

expires on March 2005

2 Open Day‟s open-premium-rate of NIFTY call European

3 High Day‟s high-premium-rate of NIFTY call European

4 Low Day‟s low-premium-rate of NIFTY call European

5 Close Day‟s close-premium-rate of NIFTY call European

6 Open Interest Pending contracts

7 Traded quantity Number of NIFTY index traded in the day

8 Number of contracts Number of contracts traded in the day

Each Nifty is 1600 / 8 = 200 units of the underlying.

Question 2: The following quotes were observed by Sanjay on Mar 11, 2005 in the Economic Times.


Interest

Traded

quantity

Number of

ContractsUnderlying

CA-370-Mar2005

7.35 9.60 7.35 8.75 649,500 138,000 92 ACC

PA-135-Mar

20053.00 3.10 1.75 1.90 203200 102,400 64 MTNL

Explain what these quotes indicate.

Answer: CA – Call American, PA – Put American. Other than CA and PA, the explanations for the other

columns are the same as earlier problem.

Question 3: For the following identify the nature of the option Mr. X holds:

a) The option gives him the right to purchase equity shares of Satyam at ₹725 on or before March 28,

2006.




Derivatives 6.19

b) The option gives him the right to sell equity shares of Sesa Goa at ₹1025 on or before March 28, 2006.

Answer:

Nature of option

(a) Option to purchase Satyam at ₹725 Call option

(b) Option to sell of Sesa Goa ₹1,025 Put option

Question 4: Mr. Ramesh purchases the following European Call options on Reliance. He also purchases

the following European put options on ACC. What decision he would take on expiry, if Reliance (RIL)

closes at ₹835 and ACC closes at ₹565? Ignore premium paid.

1. RIL 830 Call

2. RIL 840 Call

3. ACC 510 Put

4. ACC 580 PutAnswer:

X Position Profit

(a) Call Option to purchase RIL ₹830 ₹835 Exercise ₹5

(b) Call Option to purchase RIL ₹840 ₹835 Do not exercise ₹0

(c) Put option to sell ACC ₹510 ₹565 Do not exercise ₹0

(d) Put option to sell ACC ₹580 ₹565 Exercise ₹15

Question 5: Identify which of the following options is In-The-Money (ITM), At-The-Money (ATM) orOut-of-The-Money (OTM) for the buyer of option. Which of these options would be exercised? Treat each

case individually.

1. RIL 840 CALL when the price on expiry is ₹855



4. ACC 510 PUT when the price on expiry is ₹510



Answer:

X ITM / OTM / ATM Position Profit=

Max [S – X, 0]

1 RIL Call Option ₹840 ₹855 In the money Exercise ₹15

2 RIL Call Option ₹830 ₹840 In the money Exercise ₹10

3 RIL Call Option ₹800 ₹765 Out of the money Lapse ₹0

4 ACC Put Option ₹510 ₹510 At the money Lapse ₹0

5 ACC Put Option ₹520 ₹500 In the money Exercise ₹20




Derivatives 6.20

6 ACC Put Option ₹540 ₹555 Out of the money Lapse ₹0

For Call Option: ITM: S > X, OTM: S < X & ATM: S = X

For Put Option: ITM: S < X, OTM: S > X & ATM: S = X

Question 6: For each of the following options, find out the Intrinsic Value and the Time Value. The

premium paid by the buyer is given in brackets. Details of options purchased are given. Treat each case

individually.

1. HLL 180 PUT (₹9)

2. L&T 1510 PUT (₹7)

3. ACC 540 PUT (₹39)

4. HLL 205 CALL (₹2)

5. L&T 1500 CALL (₹12)

6. RIL 800 CALL (₹37)

On the day of expiry the prices of stocks were: HLL ₹200; L&T ₹1510; RIL ₹825 & ACC ₹515.

Answer:

X Premium

PaidPosition

Intrinsic

Value1

Time

Value2

1 HLL Put Option ₹180 ₹200 ₹9 OTM Lapse ₹0 ₹9

2 L&T Put Option ₹1,510 ₹1,510 ₹7 ATM Lapse ₹0 ₹7

3 ACC Put Option ₹540 ₹515 ₹39 ITM Exercise ₹25 ₹124 HLL Call Option ₹205 ₹200 ₹2 OTM Lapse ₹0 ₹2

5 L&T Call Option ₹1,500 ₹1,510 ₹12 ITM Exercise ₹10 ₹2

6 RIL Call Option ₹800 ₹825 ₹37 ITM Exercise ₹25 ₹12

Question 7: The call option of „X‟ with a ₹25 strike price is available. The following table contains

historical values for this option at different stock prices:

Stock Price Call Option Price

₹25 ₹3.00

₹30 ₹7.50

₹35 ₹12.00

₹40 ₹16.50

₹45 ₹21.00

₹50 ₹25.50

1 Intrinsic Value [Profit]= Max [S – X, 0]

2 Time Value = Premium – Intrinsic Value




Derivatives 6.21

Create a table which shows:

a) Stock price b) Strike price,

c) Intrinsic value, d) Option price,

e) The time value.

Answer: Call Option

X Premium

PaidPosition

Intrinsic

Value1

Time

Value2

1 ₹25 ₹25 ₹3.00 ATM Lapse ₹0 ₹3

2 ₹25 ₹30 ₹7.50 ITM Exercise ₹5 ₹2.50

3 ₹25 ₹35 ₹12.00 ITM Exercise ₹10 ₹2

4 ₹25 ₹40 ₹16.50 ITM Exercise ₹15 ₹1.50

5 ₹25 ₹45 ₹21.00 ITM Exercise₹20 ₹1

6 ₹25 ₹50 ₹25.50 ITM Exercise ₹25 ₹0.50

Question 8: What is the Intrinsic Value and time value in the following cases?

Price of Stock (S) ₹ Strike Price (X) ₹ Premium ₹ Nature of Option

25 25 13.00 Call

30 35 17.50 Put

45 25 22.00 Put

20 25 16.50 Call45 55 12.00 Put

50 50 20.50 Put

Answer:

OptionX

Premium

PaidPosition

Intrinsic

Value3

Time

Value4

1 Call 25 25 13.00 ATM Lapse 0 13.00

2 Put 35 30 17.50 ITM Exercise 5 12.50

3 Put 25 45 22.00 OTM Lapse 0 22.00

4 Call 25 20 16.50 OTM Lapse 0 16.50

5 Put 55 45 12.00 ITM Exercise 10 2.00

6 Put 50 50 20.50 ATM Lapse 0 20.50

[CA FINAL]

Question 9: An investor has the following position on options of CIPLA.

1 Intrinsic Value [Profit]= Max [S – X, 0]

2 Time Value = Premium – Intrinsic Value3 Intrinsic Value [Profit]= Max [S – X, 0]

4 Time Value = Premium – Intrinsic Value




Derivatives 6.23

3 short puts

6 long puts

5 short calls

4 long puts and 4 shares

30 short calls and 3 shares.

Answer:

Position Details Net Delta

4 Long Calls Delta of Call Positive, Delta of Long Position Positive +4×+1 = +4

3 short puts Delta of Put Negative, Delta of Short Position Negative – 3× – 1 = +3

6 long puts Delta of Put Negative, Delta of Long Position Positive +6× – 1 = – 6

5 short calls Delta of Call Positive, Delta of Short Position Negative – 5×+1 = – 5

4 long puts

& 4 shares

Delta of: Put Negative; Long Position Positive;

Underlying Positive

– 4 +4 = 0

30 short calls

and 3 shares

Delta of: Call Positive; Short Position Negative;

Underlying Positive

– 30+3= – 27

Question 2: Calculate the Net Delta of the following individual contracts. Also calculate the Net delta if

these contracts form a single portfolio.

1. Short 2 Calls with a delta of 0.5

2. Long 1 Puts with a delta of 0.3

3. Long 5 Calls with a delta of 0.2

Answer: We know that Delta of Long Calls & Short Puts is positive and that of Short Calls and Long Puts

are negative.

Position Calculation Net Delta

1 Short 2 Calls with a delta of 0.5 Delta – 2×0.5 – 1.0

2 Long 1 Puts with a delta of 0.3 Delta – 1×0.3 – 0.3

3 Long 5 Calls with a delta of 0.2 Delta +5×0.2 +1.0

4 Portfolio [Total] – 0.3

Question 3: Mr. A decides to purchase Two RIL 840 call options which have a delta of 0.75 each. He also

plans to simultaneously hedge by buying Four RIL 900 Put options which has a delta of 0.375. What is the

net delta of each position? What is the net delta of overall position? Is he fully hedged?

Answer:

Position Calculation Net Delta

1 Delta of RIL Call position (+2)×(+0.75) +1.5

2 Delta of RIL Put position (+4)× – 0.75 – 1.5

3 Net Delta of entire position +1.5 – 1.5 0Since net delta of the entire position is zero, Mr. A is perfectly hedged.




Derivatives 6.25

Answer:

Position Individual Position

Delta Gamma Vega Delta Gamma Vega

Short 2000 X 1.00 0.00 0.00 – 2000 0 0

Short 100 X May 25 calls 0.89 0.01 0.02 – 8900 – 100 – 200

Long 50 X May 30 Calls 0.76 0.03 0.05 +3800 +150 +250

Long 10 X August 30 Calls 0.74 0.02 0.07 +740 +20 +70

Total – 6360 +70 +120

The Gamma of the current position is +70. To make it neutral we short 70 more X May 25 calls. In that

case the new position Gamma would be = -170×100×0.01 + 50×100×0.03 + 10×100×0.02 = 0. Now our

entire position is gamma neutral. But by selling 70 more May 25 Calls, our position delta would have

changed. The delta of this new position would be = -2000 – 170×100×0.89 + 50×100×0.76 + 10×100×0.74= -12590. The earlier position delta has increased significantly to – 12590. This can be made neutral only

by going long 12590 shares. Since we are already short 2000 shares, the net will be long 10590 shares of

the underlying. Thus the new position would be depicted as under.

Position Individual Position

Delta Gamma Vega Delta Gamma Vega

Long 10590 X 1.00 0.00 0.00 +10590 0 0

Short 170 X May 25 calls 0.89 0.01 0.02 – 15130 – 170 – 340

Long 50 X May 30 Calls 0.76 0.03 0.05 +3800 +150 +250

Long 10 X August 30 Calls 0.74 0.02 0.07 +740 +20 +70

Total 0 0 – 20

We can see that after the portfolio has been converted to delta and gamma neutral, the position Vega is just

– 20. This implies that an increase in implied volatility by 1% our profits would reduce by ₹20 on the

overall position. Obviously, we would make ₹20 for every fall in implied volatility of 1%.

Question 7: X holds 100 contracts each of the following options. Each contract has 100 shares of the

underlying. The theta of the options is as follows:

Option Theta

July 30 call – 0.03

July 30 put – 0.03

(a) What is the position theta?

(b) How much X will lose or gain per day?

(c) How much a seller of this position will lose or gain per day?




Derivatives 6.26

Answer:

(a) Calculation of position theta:

Option Position Theta Position Theta

Long 100 July 30 call – 0.03 – 300

Long 100 July 30 put – 0.03 – 300

– 600

(b) Theta is always given as a negative number. A long position holder would witness time decay. In this

case, X‟s overall position would lose ₹600 per day.

(c) Theta (time decay of an option) always favors‟ the seller. Hence the seller of this position would gain

₹600 daily from this position.

Question 8: We have stock P whose price if ₹480. With three months to expiration, we have the following calls available. April 500 call and April 600 call. Each contract has 100 shares of the underlying. Given the

following data

Option Delta Gamma Vega

April 500 Call 0.373 0.009 0.006

April 600 Call 0.095 0.002 0.004

A spreader desires to make a profit of approximately ₹50 for each one percentage decrease in volatility.

Contract a strategy using mathematical approach assuming that he wants his position delta and gamma

neutral. i.e., how many options should be spread to achieve the desired result?

Answer:

Our aim is first to make the portfolio gamma neutral. Then make it delta neutral and finally ensure that

position Vega is – 50. This would ensure that the entire position would give us ₹5000 for every decrease of

1% in implied volatility.

Let X represent the number of April 500 Calls we buy and y represent the number of April 600 calls we

buy. First we make the portfolio gamma neutral i.e. ensure that the weighted average of gamma of both

these calls is zero.

0.009x + 0.002y = 0 [1]

We construct a second equation of Vega to give us the desired ₹50 i.e. o.5 (50/100) on every contract of

100 shares.

0.006x + 0.004y = -0.5 [2]

Multiplying equation [1] by 0.006 & equation [2] by 0.009 and then subtracting [2] from [1], we get

(0.0000Q0.000036) y = -0.0045

Therefore y = -187.5

Solving these two simultaneously we get:

Y = -188 and X= +42




Derivatives 6.27

This means that we need to sell 188 contracts of April 600 Calls & we need to buy 42 contracts of April

500 calls to make ₹50 for every fall in 1%. This would however change our net delta, which we need to

neutralize. The delta of the position would be calculated as follows:

-188×100×0.095 + 42×100×0.373 = -219

Since this is equivalent to short 219 shares, we make it neutral by going long 219 shares.

Valuation of Options

Question: What are the methods of valuation of option?

Binomial Model for Option Valuation: the value of a call option as per the binomial model is equal to

the hedge portfolio (consisting of equity and borrowing) that has a payoff identical to that of the call

option.

Assumptions for the binomial model:

The stock, currently selling for S, can take two possible values next year, uS or dS

An amount of B can be borrowed or lent at a rate of „r‟ the risk -free rate.

The value of r is greater than d but smaller than u. This condition ensures that there is no risk-free

arbitrage opportunity.

The exercise price is E

The value of call option

If the stock price goes up to uS [Cu]= Max (uS – E, 0)

If the stock price goes down to dS [Cd]= Max (dS – E, 0)

Portfolio of ∆ shares and B rupees of borrowing

(1) Stock price rises, Cu = ∆uS – RB

(2) Stock price falls, Cd = ∆dS – RB

From (1) and (2),

∆ =−(−)

=

B =−

(1+)(−)

Since the portfolio (consisting of ∆shares and B debt) has the same payoff as that of a call option, the value

of the call option is

C = (∆S – B)

Risk-Neutral Valuation: Current value of call option is present value of the expected future value of the

option

Question 1: Calculate the value of call option from the following data

S 200 u 1.4,

E 220 d 0.9

r 10%




Derivatives 6.28

Answer: The value of call option under option equivalent method [one-step binomial model]

Formula Calculation

1 Cu Max (uS – E, 0) Max (1.4 × 200 – 220, 0) 60

2 Cd Max (dS – E, 0) Max (0.9 × 200 – 220, 0) 0

3 ∆ −(−)

60−0

200(1.4−0.9) 0.6

4 B −(1+)(−)

0.9(60−0)

1.1(1.4−0.9) 98.18

5 C ∆S – B 0.6×200 – 98.18 21.82

The value of call option under Risk-Neutral Valuation

Particulars Formula Calculation

Expected return Pr ×(u – 1)+(1 – Pr)(d – 1) = 0.1

Probability of rise [Pr] From the above formula 0.4

Future value of call option Pr ×Cu + (1 – Pr)Cd 0.4×60 + (1 – 0.4)0 ₹24

Current value of call option 1+

24

1+0.1 ₹21.82

Question: Explain Black Scholes Model of pricing an option.

Answer: Fischer Black and Myron Scholes published a paper in 1973 for the pricing of options and

corporate liabilities that is now known as the Black-Scholes model. It is the standard method of pricing a

European call option.

Variables used in the model are

stock price

expiration date

risk-free return

standard deviation (volatility) of the stock‟s return.

Black-Scholes formula for pricing call option = C = S N(d1) – Xe-rT

N(d2)

d1 = In(

) + (r +2

2) T

σ

d2 = d1 – σ

Where

Variables Meaning

C Price of the Call option

S S pot price of the underlying stock

X Option eXercise price

r R isk-free interest rate

T Time to expiration

N() area under the Normal curve




Derivatives 6.29

A comparison between binomial model and black-scholes model

Option Position Binomial Model Black-Scholes Model

1 Buy Call Option Borrow B

Buy ∆ shares

Borrow Xe-r

N(d2)

Buy N(d1) shares

2 Sell Call Option Lend B

Sell ∆ shares

Lend Xe-r

N(d2)

Sell N(d1) shares

3 Buy Put Option Lend B

Sell ∆ shares

Lend Xe-r

(1 – N(d2))

Sell 1 – N(d1) shares

4 Sell Put Option Borrow B

Buy ∆ shares

Borrow Xe-r

(1 – N(d2))

Buy (1 – N(d1)) shares

Adjustment for Dividends

Short-term options, Adjusted stock price = S = S – 1+

Long-term options, where dividend yield (y) expected is stable

Black-Scholes formula for pricing call option = C = S− N(d1) – Xe-rT

N(d2)

d1 = In(

) + (r−y +2

) T

σ

d2 = d1 – σ

Question 2: Calculate the value of a put option from the following date

S 60 t 3 months

E 50 σ 0.4

R 8%

Answer:

Formula Calculation

1 d1 In(

) + (r +2

) T

σ In(

60

50) +(0.08 +

0.042

2)0.25

0.4 0.25

1.1115

2 d2 d1 – σ 1.1115 – 0.4 0.25

0.9115

3 C S N(d1) – Xe-r

N(d2) 60 N(1.1115) – 50e- . × .

N(0.9115) ₹11.87

4 P 0 − 0 + 11.87 − 60 +

50

0.08×0.25 ₹0.88

PUT CALL PARITY

Question 3: What would be the price of a call, if Value of a Put=₹5, Strike Price = ₹100, Current price =

₹100, Rate of interest = 6%, Time Period = 2 months.

Answer:




Derivatives 6.30

Formula Calculation

0 0 + 0 − 100 + 5 − 100

0.06×2

12

6

Question 4: The common share of a company is selling at ₹90. A 26 week call is selling at ₹8. The call‟s exercise price is ₹100. The risk free rate is 10% p.a. What should be the price of a 26 week put of ₹100?

Answer:

Formula Calculation

0 0 − 0 + 8 − 90 +

100

0.1×.5

6

Question 5: Mr. Narendra holds an American put option on Delta Airlines a non-dividend paying stock.

The strike price of the put is ₹40, and Delta Airlines stock is currently selling for ₹35 per share. The

current market price of the put is ₹4.50. Is this option correctly priced? If not, should Mr. Narendra buy orsell the option in order to take advantage of the mispricing?

Answer: the option pricing is mispriced, that leads to arbitrage gain as follows

Strategy Cash Flow

1 Buy put option –₹4.50

2 Buy stock –₹35.00

3 Exercise put option +₹40.00

Arbitrage Profit +₹0.50

Therefore, Mr. Narendra should buy the option for ₹4.50, buy the stock for ₹35, and immediately exercise

the put option to receive its strike price of ₹40. This strategy yields a risk less, arbitrage profit of ₹0.50

(=₹5 – ₹4.50)

Question 6: GESCO has both European call and put options traded on NSE. Both options have same

exercise price of ₹40 and both expire in one year. GESCO does not pay any dividends. The call and the put

are currently selling for ₹8 & ₹2 respectively. The risk free rate of interest is 10% p.a. What should the

stock price of GESCO trade in order to prevent arbitrage?

Answer:

Formula Calculation

0 0 = 0 − 0 + 2 = 8 − 0 +

40

0.1 42.36

[CS FINAL]

Question 7: The following quotes are available for 3-months options in respect of a share currently traded

at ₹31:

Strike price ₹30

Call option ₹3




Derivatives 6.31

Put options ₹2

An investor devises a strategy of buying a call and selling the share and a put option. What is his profit /

loss profile if it is given that the rate of interest is 10% per annum? What would be the position if the

strategy adopted is selling a call and buying the put and the share?Answer:

Formula Calculation

0 + 0 = 0 + 31 + 2 = 3 +

30

0.1×0.25

33 = 32.27 LHS ≠ RHS,

Hence arbitrage exist

Arbitrage strategy: Buying a call & Selling a put & spot leading a profit of LHS – RHS [33 – 32.27=0.73]

Cash Flow

< E, = E > E

= 25 = 30 = 35

Buy a call – 3 – 3 – 3

Sell a put and spot 33 33 33

Net Investment @ 10% 30 30 30

Withdraw investment 30.75 30.75 30.75

Call [Exercise | Lapse] Lapse Lapse – 30

Put [Exercise | Lapse] – 5 Lapse Lapse

Buy stock to cover short – 25 – 30 0

Net Flow 0.75 0.75 0.75

Similar strategy if developed by selling the call and buying the share and put would result in an initial out

flow of 0.73, and hence not advisable.

Risk-neutral approach

Question 8: We provided with the following information:Stock price = ₹88; Risk free rate = 3%; In 3 months‟ time the stock could either go up to ₹95 or down to

₹82. The strike price is ₹90. Compute the value of put option using risk neutral probability.

Answer:




Future value of call option Pr ×Cu + (1 – Pr)Cd 0.5123×0 + (1 – 0.513)8 ₹3.896

Current value of call option

1+ 3.896

1+0.0075 ₹3.867




Derivatives 6.32

Question 9: We are provided with the following information:

Stock price = ₹88; Risk free rate = 3%; In 3 months‟ time the stock could either go up to ₹95 or down to

₹82. The strike price is ₹90. Compute the value of call option using risk neutral probability. Using the

answer of the previous problem, verify whether Put Call parity holds.

Answer:Particulars Formula Calculation



Future value of call option Pr ×Cu + (1 – Pr)Cd 0.5123×5+(1 – 0.513)0 ₹2.5615


2.5615

1+0.0075 ₹2.54

Formula Calculation

0 + 0 = 0 + 88 + 3.87 = 2.54 +

90

0.0075

91.87 = 91.87 LHS = RHS,

Put-call parity exists

Question 10: We have a stock which is quoted at ₹80. Its beta is 1.2. In 2 months time the stock market

can either go up by 20% or fall by 10% from the current price. Mr. X wishes to find the price of call option

of strike price = 80, on this stock using risk less hedge approach. The risk free rate is given as 12%.

Explain how hedge can be created using these call options if Mr. X holds 1,000 shares of the stock. It is

given that each call option underlies 1,000 shares of the stock.

Answer:

„u‟ = market u beta = 20% 1.2 = 24% [1 + 24% = 1.24]

„d‟ = market d beta = 10% 1.2 = 12% [1 – 12% = 0.88]

Formula Calculation

1 Cu Max (uS – E, 0) Max (1.24×80 – 80, 0) 19.2

2 Cd Max (dS – E, 0) Max (0.88×80 – 80, 0) 0

3 ∆ −

(

−)

19.2−0

80(1.24

−0.88)

0.667

4 B −(1+)(−)

0.88×19.2−0

(1+0.12

12×2)(1.24−0.88)

46.01

5 C ∆S – B 0.667×80 – 46.01 7.35

Two-step binomial model

Question 11: Consider a stock which is quoted at ₹84. A put option on this available at a strike price of

₹87.50. The stock can take values of ₹89 or ₹79 in 3 months. If it takes a value of ₹89, it can go to either

₹94 or ₹84 in another 3 months. And if it takes the value of ₹79 after 3 months, it can go to either ₹84 or

₹74 in another 3 months. The stock is not expected to pay any dividend. It is given that the risk free rate is

4%. Find the price of the put option using binomial model.




Derivatives 6.33

Answer:

Formula Calculation Pu Calculation Pd

u 94

89 1.0562 84

79 1.0633

d 84

89 0.9438 74

79 0.9367

1 Pu Max (uS – E, 0) Max (87.5 – 1.0562×89, 0) 0 Max (87.5 – 1.0633×79, 0) 3.5

2 Pd Max (dS – E, 0) Max (87.5 – 0.9438×89, 0) 3.5018 Max (87.5 – 0.9367×79, 0) 13.5

3 ∆ −(−)

0−3.5018

891.0562−0.9438 -0.3501 0−0

79(1.0633−0.9367) -1

4 B −(1+)(−)

0.9438×0−0.3501

(1+0.01)(1.0562−0.9438) 32.57 0.9367×0−0

1+0.011.0633−0.9367 86.625

5 P ∆S – B 0.3501× 89 – 32.57 1.42 1× 79 – 86.625 7.625

Formula Calculation

u 89

84 1.0595

d 79

84 0.9405

1 ∆ −(−)

1.42−7.625

89−79 -0.6205

2 B −(1+)(−)

0.9405×3.7873−0

1+0.011.0595−0.9405 56.08

3 P ∆S – B -0.6205× 84 – 56.08 3.97

Question 12: Consider a stock which is quoted at ₹84. A call option on this available at a strike price of

₹87.50. The stock can take values of ₹89 or ₹79 in 3 months. If it takes a value of ₹89, it can go to either

₹94 or ₹84 in another 3 months. And if it takes the value of ₹79 after 3 months, it can go to either ₹84 or

₹74 in another 3 months. The stock is not expected to pay any dividend. It is given that the risk free rate is

4%. Find the price of the call option using binomial model. Using the value of put option; verify the put

call parity theorem.

Answer:

Formula Calculation Cu Calculation Cd

u 9489

1.0562 8479

1.0633

d 84

89 0.9438 74

79 0.9367

1 Cu Max (uS – E, 0) Max (1.0562×89 – 87.5, 0) 6.5018 Max (1.0633×79 – 87.5, 0) 0

2 Cd Max (dS – E, 0) Max (0.9438×89 – 87.5, 0) 0 Max (0.9367×79 – 87.5, 0) 0

3 ∆ −(−)

6.5018−0

891.0562−0.9438 0.6499 0−0

79(1.0633−0.9367) 0

4 B −(1+)(−)

0.9438×6.5018−0

(1+0.01)(1.0562−0.9438) 54.0538 0.9367×0−0

1+0.011.0633−0.9367 0

5 C ∆S –

B 0.6499× 89 –

54.0538 3.7873 0 0




Derivatives 6.34

Formula Calculation

u 89

84 1.0595

d 79

84 0.9405

1 ∆ −(−) 3.7873 −0

89−79 0.3787

2 B −(1+)(−)

0.9405×3.7873−0

1+0.011.0595−0.9405 29.6339

3 C ∆S – B 0.3787× 84 – 29.6339 2.1769

Formula Calculation

0 + 0 = 0 + 84 + 3.97 = 2.1769 +

87.50

0.04×0.5

87.97 = 87.97 LHS = RHS,

Put-call parity exists

[CS FINAL]

Question 13: The current market price of the equity shares of Bharat Bank Ltd. is ₹190 per share. It may

be either ₹250 or ₹140 after a year. A call option with a strike price of ₹180 (time 1 year) is available. The

rate of interest applicable to the investor is 9%. Rahul wants to create a replicating portfolio in order to

maintain his pay off on the call option for 100 shares. Find out (i) hedge ratio; (ii) amount of borrowing;

(iii) fair value of the call; and (iv) his cash flow position after a year.

Answer:

Formula Calculation Cu

u 250

190 1.3158

d 140

190 0.7368

1 Cu Max (uS – E, 0) Max (1.3158×190 – 180, 0) 70.002

2 Cd Max (dS – E, 0) Max (0.7389×190 – 180, 0) 0

3 ∆ −(−)

70.002−0

1901.3158−0.7368 0.6363

4 B −(1+)(−)

0.7368×70.002−0

(1+0.09)(1.3158−0.7368) 81.725

5 C ∆S –

B 0.6363× 190 –

81.725 39.172

Question 14: Suppose Ann‟s stock price is currently ₹25, In the next six months it will either fall to ₹15 or

rise to ₹40. What is the current value of a six -month call option with an exercise price of ₹20? The six -

month risk-free interest rate is 5% (periodic rate). [Use risk-neutral valuation]

Answer:

u =40

25 = 1.6, d =

15

25 = 0.6




Derivatives 6.35



Probability of rise [Pr] From the above formula

Future value of call option Pr ×Cu + (1 – Pr)Cd


Question 15: Kim is interested in buying a European call option on Nalco a non-dividend paying stock,

with strike price of ₹110 and one year until expiration. Currently Nalco sells for ₹100 per share. In one

year Kim knows that Nalco would either trade at ₹120 or ₹80 per share. Kim is able to borrow and lend at

risk free rate of 2.5% per annum, (assume simple interest). How much Kim should pay for this call option?

Use risk neutral argument.

Answer:



Probability of rise [Pr] From the above formula

Future value of call option Pr ×Cu + (1 – Pr)Cd


Question 16: What is the value of the following call option according to the Black Scholes Option Pricing

Model?

Stock Price ₹27.00

Exercise Price ₹25.00

Time to Expiration 6 Months

Risk-Free Rate 6.0%

Stock Return Variance 0.11

Answer: Black Scholes Model

Formula Calculation

1 d1 In(

) + (r +2

2) T

σ In(

27

25) +(0.06 +

0.11

2)0.5

0.3317 0.5

0.5736

2 d2 d1 – σ 0.5736 – 0.3317 0.5

0.3391

3 C S N(d1) – Xe-r

N(d2) 27 N(0.5736) – 25e- . × .

N(0.3391) ₹4.0036

4 P 0 − 0 + 4.0036 − 27 +

25

0.06×0.5 ₹1.2754

Question 17: The share of APAR Ltd. is currently priced at ₹415 and call option exercisable in 3 months‟

time has an exercise rate of ₹400. Risk free interest is 5% p.a. and standard deviation (volatility) of share price is 22%.




Derivatives 6.36

(a) Based on the assumption that APAR Ltd. is not going to declare any dividend over the next three

months, is the option worth buying for ₹25?

(b) Calculate value of aforesaid call option based on Block Schole‟s valuation model if the current price is

considered as ₹380.

(c)

What would be the worth of put option if a current price is considered ₹380? (d) If APAR ltd. share pr ice at present is taken as ₹408 and a dividend of ₹10 is expected to be paid in the

two months‟ time, then calculate value of the all options.

Answer: (a)

Formula Calculation

1 d1 In(

) + (r +2

) T

σ In(

415

400) +(0.05 +

0.222

2)0.25

0.22 0.25

0.5033

2 d2 d1 – σ 0.5033 – 0.22 0.25

0.3933

3 C S N(d1) – Xe-r

N(d2) 415 N(0.5033) – 400e- . × .

N(0.3966) ₹27.58

4 P 0 − 0 + 11.87 − 60 +

50

0.08×0.25 ₹0.88

Since market price of ₹25 is less than ₹27.58 (Black Scholes Valuation model). This indicates that the

option is under priced, hence worth buying.

(b) If the current price is taken as ₹380 the computations are as follows:

Formula Calculation

1 d1 In(

) + (r +

2

) T

σ In(

380

400) +(0.05 +

0.222

2 )

0.25

0.22 0.25

-0.2976

2 d2 d1 – σ -0.2976 – 0.22 0.25

-0.4077

3 C S N(d1) – Xe-r

N(d2) 380 N(-0.2976) – 400e- . × .

N(-0.4077) ₹7.10

(c) P 0 − 0 + 7.10 − 380 +

400

0.05×0.25 ₹22.16

(d) Since dividend is expected to be paid in two months time we have to adjust the share price and then use

Block Schole‟s model to value the option.

Adjusted S = S – Present Value of Dividend = 408 – 10 (

−0.05×

2

12

)

= 398.08

Formula Calculation

1 d1 In(

) + (r +2

2) T

σ In(

398.08

400) +(0.05 +

0.222

2)0.25

0.22 0.25

0.125

2 d2 d1 – σ -0.2976 – 0.22 0.25

0.015

3 C S N(d1) – Xe-r

N(d2) 398.08 N(0.125) – 400e- . × .

N(0.015) ₹18.98

(c) P 0 − 0 + 18.98 − 398.08 +

400

0.05×0.25




Derivatives 6.37

OPTION STRATEGIES

Question: What is option spreads?

Answer: Option spread means taking position in two or more options of the same type (i.e. calls or puts)

on the same underlying assets.

1. Vertical spread is an option spread, which has different strike prices but the same expiration date .

2. Horizontal spread is the spread, which has different expiration dates but the same strike price .

This spread is also called time spread or calendar spread.

3. Diagonal spread is the spread in which two legs of the spread have different strike prices and

different expiration dates. This position has features of both vertical and horizontal spreads and so may

be called a hybrid product.

Question: Write a note on option strategies.

Answer:

1. Bull call spread: A bull call spread involves the purchase and sale of call options at different

exercise prices but with the same expiry date. The purchased calls should have a lower exercise

price than the written calls.

2. Bull put spread: A bull put spread involves the purchase and sale of put options at different

exercise prices but with the same expiry date. The purchase puts should have a lower exercise price

than the written puts.

3. Bear call spread: A bear call spread involves the purchase and sale of call option at different exercise

prices and the same expiry date. But the purchased calls have a higher exercise price than the written

calls.

4. Bear put spread: A bear put spread involves the purchase and sale of put option at different exercise

prices and the same expiry date. But this time purchased puts have a higher exercise price than the

written puts.

Question: Write a note on straddle and strangle.

Answer: Straddle & Strangle: Straddle & Strangle are strategies tailor made for volatile situations.

Long straddle: Purchase a call option and a put option with the same exercise price.

Short straddle: Sell a call option and a put option with the same exercise price.

Long strangle: Purchase a call and a put with different exercise prices

Short strangle: Sell a call and a put with different exercise prices

Question: Explain Butterfly Spread.

Answer: A Butterfly Spread is an option strategy combining a bull and bear spread. It uses three strike

prices. The lower two strike prices are used in the bull spread, and the higher strike price in the bear

spread. Both puts and calls can be used. A butterfly spread consists of either all calls or all puts and all

options expire at the same time.

Long butterfly spread: A long butterfly spread can be created by buying one option at each of the outsideexercise prices and selling two options at the inside exercise price.




Derivatives 6.38

Short butterfly spread: A short butterfly spread can be created by selling one option at each of the

outside exercise prices and buying two options at the inside exercise price.

The butterfly spread is a neutral options strategy position used when the underlying security is not too

volatile by expiration. Both risk and profit are limited and commission costs are high. The maximum profit

is realized if the stock price expires at the strike price.

General concept of using options and futures

Bullish Perspective Bearish Perspective

Futures Buy Futures Sell Futures

Call options Buy options Sell options

Put options Sell options Buy options

Strategies with Individual Stock OptionProtective Put: protect against potential losses beyond a level [invest in stock and purchase put]

Payoff

≤X ≥X

Stock

Add Put X- 0

Total X

Covered Call: Invest in stock and short call in the same stock

Payoff

≤X ≥X

Stock

Add Call 0 -( − )

Total X

Payoff of a Straddle

≤X

≥X

Payoff of a call 0 ( − )

Add Call ( − ) 0

Total ( − ) ( − )

Payoff of a Spread

< < < >

Payoff of a call, X = 0 ( − ) ( − )

Add Payoff of call, X = -0 -0 -( − )

Total 0 ( − ) ( 2 − 1)




Derivatives 6.39

Collar: limits the value of a portfolio held within two bounds by buying put and writing call at relatively

same X for both options. The premium paid for long put is offset with premium received for short call

Question 1: Mr. Eswar sold 10 BILT put options and bought 5 BILT call options. Both options have same

exercise price of ₹80 and the same expiration date. Draw the payoff diagram with respect to various pricesBILT may take at expiration.

Answer: X = ₹80

S Payoff p.c. c. 50 55 60 65 70 75 80 85 90 95 100 105 110

CE Max(S-X, 0) 5 0 0 0 0 0 0 0 25 50 75 100 125 150

PE Max (X-S, 0) 10 -300 -250 -200 -150 -100 -50 0 0 0 0 0 0 0

Total -300 -250 -200 -150 -100 -50 0 25 50 75 100 125 150

Question 2: Suggest what strategies an investor could adopt on Reliance Industries in the options marketin each of the following, if:

(a) Investor is strongly bullish.

(b) Investor believes the bullish trend would continue but is not very bullish.

(c) Investor believes that the chance of market going up is more than the chance of market going down.

(d) Investor believes that the chance of market going up is more than the chance of market going down

and wants to earn income.

(e) What is common in all the above strategies?

Answer:

(a) It is without doubt, that when an investor is bullish, he would buy a call option on Reliance Industries.

His loss is limited to premium paid. It is generally adopted when the option is undervalued and

volatility is increasing.

(b) When an investor believes the bullish trend would continue but is not very bullish, he may sale a put

option on Reliance Industries. Selling a put is a neutral-bullish position. Here his profit is limited to

premium received. It is generally adopted when the option‟s volatility is increasing.

(c) When an investor believes that the chance of market going up is more than the chance of market going

down, he may buy Call & Sell call of higher strike price, or Reliance industries. This is a buying a Bull

Call Spread strategy. This transaction would provide a range bound payoff, both on the upside and the

downside and maximum loss is limited to the net debit of the position.(d) When an investor believes that the chance of market going up is more than the chance of market going

down and wants to earn income, he would sell Put & but Put of lower strike price, of Reliance

Industries. This is a selling Bear Put Spread strategy. In this case the loss is limited to strike price

difference – premium received. This would be used when the overall position derives a good income.

(e) All the strategies explained above are adopted when the view on the stock / market is bullish.

Question 3: Suggest what strategies an investor could adopt on Reliance Industries in the options market

in each of the following if:

(a) Investor is strongly bearish.

(b) Investor believes the bearish trend would continue but is not very bearish.




Derivatives 6.40

(c) Investor believes that the chance of market going down is more than the chance of market going up.

(d) Investor believes that the chance of market going down is more than the chance of market going up

and wants to earn income.

(e) What is common in all the above strategies?

Answer:

(a) It is without doubt, that when an investor is bearish, he would buy a put option on Reliance Industries.

His loss limited to premium paid. It is generally adopted when the option is undervalued and volatility

is increasing.

(b) When an investor believes the bearish trend would continue but is not very bearish, he may sell a call

option on Reliance Industries. Selling a call is a neutral-bearish position. Here his profit limited to

premium received. It is generally adopted when the option is overvalued and market trend is flat to

bearish.

(c) When an investor believes that the chance of market going down is more than the chance of market

going up, he may buy Put & Sell of higher strike price, of Reliance Industries. This is a buying BearPut Spread strategy. This transaction would provide a range bound payoff, both on the upside and the

downside and maximum loss is limited to the net debit of the position.

(d) When an investor believes that the chance of market going down is more than the chance of market

going up and wants to earn income, he would sell Call & buy Call of higher strike price, of Reliance

Industries. This is selling Bear Call Spread strategy. In this case the loss is limited to strike price

difference – credit.

(e) All the strategies explained above are adopted when the view on the stock / market is bearish.

Bull Call Spread

Question 4: X is moderately bullish on the market and wants to capitalize on a modest advance in price of

the L&T. He is not very bullish on L&T. He has a discomfort with the cost of purchasing and holding the

long call alone. On 1st November, the share price of L&T is 204. Suggest a suitable strategy if call options

on L&T with strike prices of ₹200 & ₹220 are available for ₹16 & ₹8 respectively. Explain with the help

of payoff table and diagram, what strategy he would adopt.

Answer: X = ₹200 and ₹220, Apply Bull Call Spread [purchase call at lower X, and sell call at higher X]

X P S 150 170 190 200 208 210 216 220 240 260

CE B 200 16 Max (S-X, 0) -16 -16 -16 -16 -8 -6 0 4 24 44

CE S 220 8 Max (S-X, 0) 8 8 8 8 8 8 8 8 -12 -32

Payoff -8 -8 -8 -8 0 2 8 12 12 12

Bull Put Spread

Question 5: X is moderately bearish on the market and wants to capitalize on a modest decrease in price of

the L&T. He is not very bearish on L&T. He has a discomfort with the cost of purchasing and holding the

long put alone. He needs a small income on the spread. On 1 November, the share price of L&T is 204.

Suggest a suitable strategy if put options on L&T with strike prices of ₹200 & ₹220 are available for ₹7 &

₹18 respectively. Explain with the help of payoff table and diagram, what strategy he would adopt.




Derivatives 6.41

Answer: X = ₹200 and ₹220, Apply Bull Put Spread [purchase put at lower X, and sell put at higher X]

X P S 150 170 190 200 208 209 210 216 220 240 260

PE B 200 7 Max (X-S, 0) 43 23 3 -7 -7 -7 -7 -7 -7 -7 -7

PE S 220 14 Max (X-S, 0) -52 -32 -12 -2 6 7 8 14 18 18 18

Payoff -9 -9 -9 -9 -1 0 1 7 11 11 11

Bear Call Spread

Question 6: Tata Tea is trading at ₹228. X an investor is moderately bearish on the stock and wants to

create a spread using calls that would earn him little income. Call options on Tata Tea are available with

strike prices of 240 & 220 priced at ₹9 & ₹20 respectively. Explain with the h elp of payoff table for all

prices, what strategy he would adopt.

Answer: X = ₹240 and ₹220, Apply Bear Call Spread [purchase call at higher X, and sell call at lower X]

X P S 190 200 210 220 230 231 240 250 260 270

CE B 240 9 Max (X-S, 0) -9 -9 -9 -9 -9 -9 -9 1 11 21

CE S 220 20 Max (X-S, 0) 20 20 20 20 10 9 0 -10 -20 -30

Payoff 11 11 11 11 11 0 -9 -9 -9 -9

Bear Put Spread

Question 7: Tata Tea is trading at ₹228. X an investor is moderately bullish on the stock and wants to

create a spread using puts. Put options on Tata Tea are available with strike prices of 240 & 220 priced at

₹16 & ₹7 respectively. Explain with the help of payoff table for all prices, what strategy he would adopt.

Answer: X = ₹240 and ₹220, Apply Bear Put Spread [purchase put at higher X, and sell put at lower X]

X P S 190 200 210 220 230 231 240 250 260 270

PE B 240 16 Max (X-S, 0) 34 24 14 4 -6 -7 -16 -16 -16 -16

PE S 220 7 Max (X-S, 0) -23 -13 -3 7 7 7 7 7 7 7

Payoff 11 11 11 11 1 0 -9 -9 -9 -9

Question 8: Over the coming year the common stock of Dabur, will either halve to ₹50 from its current

level of ₹100, or rise to ₹200. The 1-year risk-free interest rate is 5%. What is the delta of a one-year call

option on Dabur stock with a strike price of ₹170? How an investor can hedge 1000 shares of Dabur which

he holds, if each call option underlies 100 shares of Dabur?

Answer:

Formula Calculation

The delta is [∆]

30−0

200−50 0.2

Short Call ∆ × No. of Shares 0.2×1,000 200 shares (or)

2 call options




Derivatives 6.42

Question 9: We have put options on a stock available with strike prices of ₹30 and ₹35. While the ₹30 Put

costs ₹2, the 35 put costs ₹5. Explain how can we form a bull spread and a bear spread? Also tabulate the

values at expiry for various values of stock ST.

Answer: X = ₹30 and ₹35, Apply Bull Put Spread [purchase put at lower X, and sell put at higher X]X P S 35+ 30-35 30-

PE B 30 2 Max (X-S, 0) -2 -2 Max (30-S, 0) – 2

PE S 35 5 Max (X-S, 0) 5 Max (S-35, 0) + 5 Max (S-35, 0) + 5

Payoff 3 S-32 -2

X = ₹30 and ₹35, Apply Bear Put Spread [purchase put at higher X, and sell put at lower X]

X P S 35+ 30-35 30-

PE S 30 2 Max (X-S, 0)

PE B 35 5 Max (X-S, 0)

Payoff

Butterfly Spread

Question 10: We have put options on a stock available with strike prices of ₹55, ₹60 and ₹65 and they

cost ₹3, ₹5 & ₹8 respectively. Explain how we can form a butterfly spread using these puts. Also tabulate

the values at expiry for various values of stock ST.

Answer: X = ₹55, ₹60 and ₹65, Butterfly Spread [purchase put at the lowest and highest X each one, and

sell 2 puts at mid X]

X P S 50 55 56 60 64 65 70

PE B 55 3 Max (X-S, 0) 2 -3 -3 -3 -3 -3 -3

PE B 65 8 Max (X-S, 0) 7 2 1 -3 -7 -8 -8

2PE S 60 5 Max (X-S, 0) -10 0 2 10 10 10 10

Payoff -1 -1 0 4 0 -1 -1

The butterfly leads to a loss then the final stock price is greater than ₹64 or less then ₹56.

Butterfly Spread

Question 11: Construct a Butterfly Spread using XYZ November 90 call (priced at ₹6.50), XYZ

November 100 calls (priced at ₹3.50) and XYZ November 110 call (priced at ₹2). Draw the payoff

diagram for range or prices at expiry (70 – 130). What specific consideration if anyone needs to take

before setting up this spread?

Answer: X = ₹90, ₹100 and ₹110, Butterfly Spread [purchase calls at the lowest and highest X each one,

and sell 2 calls at mid X]




Derivatives 6.43

X P S 70 80 90 95 100 105 110 120 130

CE B 90 6.5 Max (S-X, 0) -6.5 -6.5 -6.5 -1.5 3.5 8.5 13.5 23.5 33.5

CE B 110 2 Max (S-X, 0) -2 -2 -2 -2 -2 -2 -2 8 18

2CE S 100 3.5 Max (X-S, 0) 7 7 7 7 7 -3 -13 -33 53

Payoff -1.5 -1.5 -1.5 3.5 8.5 3.5 -1.5 -1.5 -1.5

Maximum loss in the butterfly spread is 1.5

Maximum gain in the butterfly spread is 8.5

This strategy involves enter in to 4 options, which leads to high cost of commission. The profit from the

spread may be eroded by the cost of the commission. Hence care should be taken before the construction

of butterfly spread.

Long Straddle

Question 12: We have call and put options on a stock available with a strike price of ₹60. While the call

options costs ₹6, the put options cost ₹4. Explain how we can form a straddle. Also tabulate the values at

expiry for various values of stock ST.

Answer: X = ₹60, Straddle [purchase call and put at the same X]

X P S 45 50 55 60 65 70 75

CE B 60 6 Max (X-S, 0) -6 -6 -6 -6 -1 4 9

PE B 60 4 Max (X-S, 0) 11 6 1 -4 -4 -4 -4

Payoff5 0 -5 -10 -5 0 5

There is a loss with this strategy if the final stock price is between ₹50 and ₹70.

Long Straddle

Question 13: Assume you can buy or sell either the call or the put options, with a strik e price of ₹35. The

call option has a premium of ₹3, and the put option has a premium of ₹2. Which of these option contracts

can be used to form a long straddle? What is the payoff if the stock price closes at ₹38 on the op tion

expiration date? What is the payoff if the stock price closes at ₹28 on the option expiration date?

Answer: X = ₹35, Long Straddle [purchase call and put at the same X]

X P S 38 28

CE B 35 3 Max (X-S, 0) 0 -3

PE B 35 2 Max (X-S, 0) -2 5

Payoff -2 2

Short Straddle

Question 14: Assume you can buy or sell either the call or the put options, with a strike price of ₹35. The

call option has a premium of ₹3, and the put option has a premium of ₹2. Which of these option contracts




Derivatives 6.45

(a) The investor it appears has hedged his long future position by selling calls. Any rise in futures would

be offset by writing calls (though not fully). He bought the call to square off the written call.

(b) The strategy was to hedge that may arise out of future position by writing calls. The strategy may not

be termed logical owing to the meager compensation provided by the written calls.

Question 16: Mr. Ashok holds 10000 shares of IOB bought at ₹35. He is of the opinion that his portfolio

needs protection on the downside. He has the following options short listed:

(a) To write cover ed calls at a strike price of ₹45 (January Expiry) which are priced at ₹3 per share (each

contract underlies 1000 shares of IOB).

(b) To buy protective puts at a strike price of ₹35 (January Expiry) which are priced at ₹3 per share (each

contract underlies 1000 shares of IOB).

(c) To establish a “Collar” with these call & put.

(d) Which of these would be advised to Mr. Ashok? How you rank them?

Answer:

(a) By writing covered call options, Ashok collects premium income of ₹30000. If the price of the stock in

January is less than or equal to ₹45, he will have his stock plus the premium income. The stock will be

called away from him if its price exceeds ₹45. The payoff structure is:

Stock price Portfolio value

Less than ₹45 10,000 times stock price + ₹30000

More than ₹45 ₹450000 + ₹30000 = ₹480000

(b) By buying put options with a ₹35 exercise price, Ashok will be paying ₹30000 in premiums to insure a

minimum level of ₹35×10,000 – ₹30000 = ₹320000. This strategy allows for upside gain, but exposes

Ashok to the possibility of a moderate loss equal to the cost of the puts. The payoff structure is:


Less than ₹35 ₹350000 – ₹30000 = ₹320000

More than ₹35 10,000 times stock price - ₹30000

(c) A collar can be established by holding shares of an underlying stock, purchasing a protective put and

writing a covered call on that stock. In other words, one collar equals one long put and one written call

along with owing in 100 shares of the underlying stock. The primary concern in employing a collar is

protection of profits accrued from underlying shares rather than increasing returns on the upside. In the

present case the net cost of the collar is zero. (This is because, the income received by writing a call

will be used to pay premium of the put option). The value of the portfolio will be as follows:


Less than ₹35 ₹350000




Derivatives 6.46

Between ₹35 and ₹45 10,000 times stock price

More than ₹45 ₹450000

If the stock price is less than or equal to ₹35, the collar preserves the ₹350000 in principal. If the price

exceeds ₹45 Ashok gains up to a cap of ₹450000. In between, his proceeds equal 10000 times the

stock price.

(d) The best strategy in this case would be (c) since it satisfies the two requirements of preserving the

₹350000 in principal while offering a chance of getting ₹450000. Strategy (a) seems ruled out since it

leaves Ashok exposed to the risk of substantial loss of principal. The ranking would be (c) (b) and (a),

in that order.

Protective Put

Question 17: Ram and Shyam purchase an Index at 1200. However, they decide to seek downside

protection by buying put option of different strike prices. Whereas, Ram prefers at the money Put option

costing ₹60, Shyam buys In-the-Money Put option with a strike price of 1170, costing ₹45. Compare and

contrast their profits of the respective protective puts they have purchased.

Answer:

Ram‟s strategy

Initial cost Payoff

S ≤ 1200 S > 1200

Stock Index 1200 S SPut option (X = 1200) 60 1200 – S 0

Total 1260 1200 S

Profit = Payoff – 1260 1200 – 1260 = -60 S – 1260

Break Even Point = 1260

Shyam‟s strategy

Initial cost Payoff

S ≤ 1170 S > 1170

Stock Index 1200 S S

Put option (X = 1170) 60 1170 – S 0

Total 1245 1170 S

Profit = Payoff – 1245 1170-1245= -75 S – 1245

Break Even Point = 1245

Shyam does better when the stock price is high, but worse when the stock price is low. Both Ram &Shyam

incur same losses of ₹60, at the break -even point of S = ₹1185. Shyam‟s strategy has greater systematicrisk. Profits are more sensitive to the value of the stock index.




[Value of a Warrant]

Question 18:

Consider the following data

Number of share outstanding 80 million

Current stock price ₹ 80

Ratio of warrants issued to the number of outstanding share 0.05

Exercise price ₹ 84

Standard deviation of continuously compounded annual returns 0.30

Time to expiration of warrants 3 months

Risk-free interest rate per year 8 %

What is the value of a warrant? Ignore the complication arising from dividends and / or dilution.

Answer: The value of the warrant (call option using Black-Scholes) is calculated below:

Formula Calculation

1 d1 In(

) + (r +2

2)T

σ In80

84+0.08 +

0.32

2

0.25

0.3 0.25

-0.117

2 d2 d1 – σ -0.117 – 0.3 0.25

-0.267

3 C S N(d1) – Xe-r

N(d2) 80 N(-0.117) – 84e- . × .

N(-0.267) ₹3.77

futures and options for cma exam

Documents