Derivatives – Options, Futures

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From an investment perspective

Why derivatives

• Risk –return framework expanded – ability to modify the risk and expected return

characteristics of existing investment portfolios. – That is, options and futures allow investors to hedge

(or even increase) the risk of a collection of stocks in ways that go far beyond the diversification results

• Replication of cash flow patterns that already exist in other forms, arbitrage if two otherwise identical series of cash flows do not carry the same current price

Impact of Futures• Generally, a dollar-for-dollar relationship exists between the

changes in the price of the underlying security and the price of the corresponding futures contract.

• In effect, being long (short) in futures is identical to subtracting (adding) cash from (to) the portfolio.

• Long futures positions have the effect of increasing the exposure of the portfolio to the asset; Long positions in futures on the portfolio’s underlying asset increase the portfolio’s exposure (or sensitivity) to price changes of the asset

• Shorting futures decreases the portfolio’s exposure - effect of decreasing the portfolio’s sensitivity to the underlying asset.

• Long is always the buyer – here with a time lag. Short is always the seller – again with a time lag

Forward ContractDefinition: A forward contract is a commitment to purchase at a future date a given amount of a

commodity or an asset at a price agreed on today.

The price fixed now for future exchange is the forward price.The party with a “long position” will be the buyer of the underlying asset or commodity.

Features of forward contracts:• custom tailored• traded over the counter (not on exchanges)• no money changes hands until maturity• non-trivial counter-party risk.

Example. Consider a 3-month forward contract for 1,000 tons of soybean at a forward price of Rs. 16500/ton. The long side is committed to buy 1,000 tons of soybean from the short side in three months at the price of Rs.16500/ton.

Futures ContractForward contracts have two limitations:(a) illiquidity(b) counter-party risk.

Futures contracts are designed to address these two limitations.Definition: A futures contract is an exchange-traded, standardized, forward-like contract that is

marked to the market daily. This contract can be used to establish a long (or short) position in the underlying asset.

Features of futures contracts:• Standardized contracts:(1) underlying commodity or asset(2) quantity(3) maturity.• Exchange traded• Guaranteed by the clearing house — no counter-party risk• Gains/losses settled daily• Margin account required as collateral to cover losses

Futures TerminologySpot price: The price at which an asset trades in the spot market.

Futures price: The price at which the futures contract trades in the futures market.

Contract cycle: The period over which a contract trades. The index futures contracts on the NSE have one- month, two-months and three months expiry cycles which expire on the last Thursday of the month. Thus a January expiration contract expires on the last Thursday of January and a February expiration contract ceases trading on the last Thursday of February. On the Friday following the last Thursday, a new contract having a three- month expiry is introduced for trading.

Expiry date: It is the date specified in the futures contract. This is the last day on which the contract will be traded, at the end of which it will cease to exist.

Contract size: The amount of asset that has to be delivered under one contract. Also called as lot size.

Basis: In the context of financial futures, basis can be defined as the futures price minus the spot price. There will be a different basis for each delivery month for each contract. In a normal market, basis will be positive. This reflect that futures prices normally exceed spot prices.

Cost of carry: The relationship between futures prices and spot prices can be summarized in terms of what is known as the cost of carry. This measures the storage cost plus the interest that is paid to finance the asset less the income earned on the asset.

Initial margin: The amount that must be deposited in the margin account at the time a futures contract is first entered into is known as Initial margin.

Marking-to-market: In the futures market, at the end of each trading day, the margin account is adjusted to reflect the investor's

gain or loss depending upon the futures closing price. This is called marking-to-market.

Maintenance margin: This is somewhat lower than the initial margin. This is set to ensure that the balance in the margin account never becomes negative. If the balance in the margin account falls below the maintenance margin, the investor receives a margin call and is expected to top up the margin account to the initial margin level before trading commences on the next day.

Participants in Derivates markets

• Based on the applications that derivatives are put to, these investors can be broadly classified into three groups:

• · Hedgers• · Speculators, and• · Arbitrageurs

Hedgers• These investors have a position (i.e., have bought

stocks) in the underlying market but are worried about a potential loss arising out of a change in the asset price in the future. Hedgers try to avoid price risk through holding a position in the derivatives market. For Example: Investor has a target buy price for Wipro @ 250 in three months time. However, the investor is worried may increase above Rs.250 and hence is keen to hedge this risk of price rise. Assuming that there is a futures contract that is trading at Rs. 250 he can lock in the price purchase price now.

Possible payoff of long hedging

Expiry day spot price

Rs. 300 Rs.250 Rs.200

Investor Payoff Rs. 50 Rs. 0 Rs.-50

Cost of investing Rs.250 Rs.250 Rs.250

Commodity hedging• Hedging on Commodities

• If I have 100 tonnes of coconut Oil, I want to lock in the price. I am Long on the commodity. A short position in a forward contract based on the same commodity would provide the desired negative price correlation.

• By virtue of holding a short forward position against the long position in the commodity, the investor has entered into a short hedge.

SpeculatorsA Speculator is one who bets on the derivatives market based on his views on the potential movement of the underlying stock price. Speculators take large, calculated risks as they trade based on anticipated future price movements. They hope to make quick, large gains;

For Example: Currently ICICI Bank Ltd (ICICI) is trading at, say, Rs. 500 in the cash market and also at Rs.500 in the futures market (assumed values for the example only). A speculator feels that post the RBI’s policy announcement, the share price of ICICI will go up. The speculator can buy the stock in the spot market or in the derivatives market. If the derivatives contract size of ICICI is 1000 and if the speculator buys one futures contract of ICICI, he is buying ICICI futures worth Rs 500 X 1000 = Rs. 5,00,000. For this he will have to pay a margin of say 20% of the contract value to the exchange. The margin that the speculator needs to pay to the exchange is 20% of Rs. 5,00,000 = Rs. 1,00,000. This Rs. 1,00,000 is his total investment for the futures contract. If the speculator would have invested Rs. 1,00,000 in the spot market, he could purchase only 1,00,000 / 500 = 200 shares. Let us assume that post RBI announcement price of ICICI share moves to Rs. 520. With one lakh investment each in the futures and the cash market, the profits would be: (520 – 500) X 1,000 = Rs. 20,000 in case of futures market and(520 – 500) X 200 = Rs. 4000 in the case of cash markThe opposite holds with wrong calculations!

Arbitrageurs Arbitrageurs attempt to profit from pricing inefficiencies in the

market by making simultaneous trades that offset each other and capture a risk-free profit. An arbitrageur may also seek to make profit in case there is price discrepancy between the stock price in the cash and the derivatives markets.For example, if on 1st August the SBI share is trading at Rs. 1780 in the cash market and the futures contract of SBI is trading at Rs. 1790 (fair value), the arbitrageur would buy the SBI shares (i.e. make an investment of Rs. 1780) in the spot market and sell the same number of SBI futures contracts and book a profit of Rs. 10.

Expiry Date Price 2000 1780 1500

Payoff in Spot 220 0 -280

Payoff in Futures -210 10 290

Total 10 10 10

Spot future parity

• The future value discounted is equal to present value F0

T = S0(1 + cost of carry)T

• Works fine in case there is no dividend, or cost of storage. Else need to adjust for these.– Cost of carry = storage cost + foregone interest − income from

holding• The spot price of XYZ today is $50, and the annual risk-free rate

of return is Rf = 5%. It is known that XYZ does not pay dividends during the next year. What is the forward price of XYZ for delivery 1 year from today?– ‘Fair’ Forward price for delivery 1 year from today = F0

1 = 50(1 + 0.05) = $52.50

Risk Less Arbitrage Strategy: Spot futures Parity relationship

Arbitrage StrategyAction Initial Cash

FlowCash flow in 1 year

Borrow So funds So -So (1+rf)

Buy Stock for So -So St+D

Enter short Futures position

0 Fo-St

With Div 0 Fo – So(1+rf)+D

Cost of Carry RelationshipFo = So (1+rf) – D For Multi period Fo = So ((1+rf)-D)t

For continuous compounding rate

• Where:• F = Futures price• S = Spot price of the underlying asset• r = Cost of financing (using continuously

compounded interest rate)• T = Time till expiration in years• e = 2.71828• D = dividend

Real World Fair Value CalculationCost of carry model where futures fair value FVt should be FVt = St(1+i)T . where St is the spot price for the same time stamp and i is the cost of carry and T is the time period. For estimate i and T we use the normal overnight borrowing rate in India (3.5% p.a or 1.035*(1/365) -1 = 0.009425%) T = days left for expiry. For instance if the spot price of a stock is Rs. 786 and the number of days before expiry are 25 days then the fair value of the futures price should be 786*(1+0.009425)^25 = 787.85. Arbitragers will have different thresholds of reaction : say more than 1% deviation

Settlement ProcedureInitial Margin + Mark to Market MarginLet us say, Initial Futures Price = Rs. 1000; Initial Margin requirement = Rs. 500;

Maintenance Margin Requirement = Rs. 300; Contract size = 10 (that is, one futures contract has 10 shares of XYZ. How the end of day margin balance of the holder of (i) a long position of a contract and (ii) a short position of a contract, varies with the changes in settlement price from day to day is given below.

Contract Specification for S&P CNX Nifty

Contract Specification for Stock Futures

Index Arbitrage

If Futures Price > Spot Price - Short futures and buy stocks (Negative Arbitrage)

If Futures Price< Spot price – Long Futures and sell stock (Positive Arbitrage)

Stock current price is $100, one year rf is 6%, One year future price is 104. How would you arbitrage?

Not easy : you need to perfectly hedge!There are some benefits of program trading. How about short sales restrictions?

Hedging using Futures• Basis and basis riskPerfect hedge does not always exist– The asset we are trying to hedge may not be

exactly the same as the asset underlying the futures.

– The time at which we sell the asset (which could be random)might not be exactly be the same as the delivery date of the futures.

Basis Risk - IntuitionAt time of origination of contract (initial basis)

B0,T = S0 – F0,T

At time of Closure of contract (cover basis)Bt,T = St – Ft,T

Price risk is St- S0

Basis risk is Bt,T – B0,T = (St – Ft,T ) – (S0 – F0,T )

The investor is swapping the price risk for the basis risk. So pure price risk should be greater than basis risk for the future/forward market to be efficient

Intuition

Notice, however, that only the cover basis is unknown at Date 0,and so real exposure is to the correlation between future changes in the spot and forward contract prices. If these movements are highly correlated, the basis risk will be quite small

Minimum Variance hedge

• Profit from the hedge

Calculations on MV hedge

Using Index futures for Hedging• Let us consider an example. Suppose you own a portfolio of Rs 3 million which has a

beta of 0.9. How would I hedge this portfolio? By selling Rs 2.7 million of index futures.

• The portfolio beta is computed as the weighted average of the stock betas. Suppose the above portfolio is composed of Rs 1 million in ITC, which has a beta of 1.2 and Rs 2 million in Hindustan lever, which has a beta of 0.8, the portfolio beta in this case is equal to ( 1*1.2 + 2*0.8)/3 or approximately 0.9. Now to obtain a complete hedge which would remove the hidden index exposure, I would have to take a short position of portfolio value times portfolio beta which as we mentioned is approximately Rs 2.7 million.

• So if the Nifty is at 1500 and the market lot on NSE's futures market is 200, each market lot of Nifty would cost 3,00,000. Hence I would have to sell 9 market lots to obtain a position:

• Long Portfolio: Rs 3,000,000 Short S&P CNX Nifty: Rs 2,700,000

• This position will essentially be immune to any fluctuation of the index. If the index goes up, the portfolio gains and the futures lose. If the index goes down, the portfolio loses but the futures gain. In either case, the investor is hedged against market fluctuations.

Risk-Minimization HedgingSoybeans Case

• Suppose that you are a soybean meal processor. You hold soybeans for regular use in your business. From this inventory, you purchase soybeans and sell soybeans to your customers.

• You have an ongoing inventory of 1,000,000 bushel of soybeans. Suppose the cash price of soybeans is currently Rs.7.19 . As such, the inventory is currently valued at Rs.7,190,000.

• You wish to minimize the price risk associated with holding the inventory. You realize that if the price of soybeans were to drop substantially, you would experience a loss in value on the soybeans that you have in inventory. Such a loss would be devastating to your profits. You wish to hedge the price risk of the inventory that you hold.

• Because the inventory is an ongoing asset, the processor does not have a specified horizon. As such, the processor wishes to engage in a risk-minimizing hedge.

• Step 1: estimate the price change and percentage price changes for the soybean using the previous 60 days daily data of soybean prices.

Day Cash (S) Future (F) Chg (S) Chg (F)1 890327 7552 890328 749.5 1 -5.53 890329 747.5 1 -24 890330 725 1 -22.55 890331 723 1 -26 890403 697.5 72 -25.57 890404 702 1 4.58 890405 702 1 09 890406 701 1 -1

10 890407 699 1 -211 890410 707 3 812 890411 708 1 113 890412 715.5 1 7.514 890413 716 1 0.515 890414 722.5 1 6.516 890417 736 3 13.517 890418 738.5 1 2.518 890419 737.5 1 -119 890420 741 1 3.520 890421 754.5 1 13.5

. . . . .

59 890616 717.5 1 060 890619 733 3 15.5

Where :Chg (S) = change in cash priceChg (F) = change in futures price

• Step 2: run a regression analysis in Excel (recall that to do this in Excel you need to go to “Tools”, “Data Analysis”, “Regression”. The Y variable will be the change in the commodity price. The X variable will be the change in the futures price.

SUMMARY OUTPUT

Regression StatisticsMultiple R 0.11726695R Square 0.013751537Adjusted R Square -0.003551067Standard Error 15.62544125Observations 59

ANOVAdf SS MS F Significance F

Regression 1 194.0458 194.0458 0.794767 0.376411Residual 57 13916.8 244.1544Total 58 14110.85

Coefficients Standard Error t Stat P-value Lower 95%Upper 95%Lower 95.0%Upper 95.0%Intercept 4.879820628 2.035745 2.397068 0.019828 0.803311 8.95633 0.803311 8.95633X Variable 1 -0.185935589 0.208566 -0.891497 0.376411 -0.603581 0.23171 -0.603581 0.23171

• Step 3: interpret the results.• From the above regression, we have:

0103.0

5159.0

Results: in order to complete the risk-minimizing hedge, we should sell 0.5159 bushel of soybean futures contracts for each bushel that we have in inventory. Since we have 1,000,000 bushel of soybeans in inventory, we should sell:

0.5159 X 1,000,000 = 515,900 bushel of soybeans in the futures market.

4758.02 R

• Soybeans are traded in 5,000 bushel futures contracts, we should sell:

Quantity Contract

Hedged Be ToAmount # Contracts

You should sell 103 Soybean Contracts.

18.1035000

900,515# Contracts

Bond Hedges

• From the modified duration, we find that

• Optimal Hedge ratio is given by

Bond Hedges

• Find Yield Beta by correlating the 10 year yield to the 15 year yield. If it follows expectations hypothesis, the yield beta = 1.

• Find Modified Durations of A and B and current price (Modified Duration and prices are 6.4036 years and 87.71 for Security A and 8.2009 years and 117.12 for Security B)

A T-Bond/T-Note (NOB) Futures Spread

• speculators in the bond market will have a clear view on a change in the overall shape of the yield curve but be less certain as to the actual direction in future rate movements.

• One way to mitigate this unwanted risk while investing (based on your view) is to go both long and short in contracts representing different points on the yield curve.

• This is known as the Treasury “Notes over Bond” spread (or “NOB” spread) strategy. rate movements

• Mid feb prices for a june expiry

• You expect a flattening yield curve – the existing 27 bp yield difference will disappear

• Strategy– Go long in one Treasury bond futures.– Go short in one Treasury note futures– Note over Bond (NOB futures)– Make use of Liquidity premiums disappearing over time

Contract Settlement price Implied Yield

20 yr, 6% T bond 103.0625 5.74

10 yr 6% T note 104.0625 5.47

• Profit from strategy

OPTIONS

Economic Benefits of Options

• Help bring about a more efficient allocation and management of risk

• Save transactions costs• Permit investment strategies that would not

be possible otherwise.

• 1. Suppose we are given a financial derivative. What is the “fair" price for such a contract? [Pricing]

• 2. Suppose we are managing a large portfolio. How can we control the exposure to financial risks? [Hedging]

• 3. Suppose that you have to achieve some financial targets with your portfolio (e.g., think of yourself being the manager of a pension fund. How can I minimize the risk and the costs of missing the target? [Risk Management]

Underlying assets

• Options are written on almost anything one can think of. I.e., possible underlyings can be such things as stocks, foreign exchange rates, interest rates, gold, pork bellies, et cetera. Here we discuss primarily options on stocks.

Option contracts• Options

– Instruments that grant their owners (holders) the right, but not the obligation, to buy or sell an underlying asset at a specific price (or exercise or strike price), either on a specific date or any time up to a specific date (or expiration date).

• Options fall into two broad categories

– Call options: Give holder the right (but not the obligation) to buy an underlying asset at a specified price on or before a specified date.

– Put options: Give holder the right (but not the obligation) to sell an underlying asset at a specified price on or before a specified date.

• Options can be either sold (written) or purchased, thus giving rise to four possible basic option positions

– Long call position (holder of a call option)

– Short call position (writer or seller of a call option)

– Long put position (holder of a put option)

– Short put position (writer or seller of a put option)

• Options also referred to: American and European options

– American options: can be exercised any time before expiration date.

– European options: can be exercised only at expiration date.

• Other key definitions used in option trading

– Exercise (strike) price: Pre-determined price at which the underlying asset may be purchased (in the case of a call) or sold to a seller or writer (in the case of a put).

– Expiration date: Last date at which an option can be exercised.

– Option premium: Price paid by the option buyer to the seller of the option, whether put or call.

– In-the-money option: The option has intrinsic value, and would be exercised if it were expiring

– Out-of-the-money option: The option has no intrinsic value, would not be exercised if expiring

• If not expiring, could still have value since it could later become in-the-money

– At-the-money option: If the spot market price of the underlying asset is equal to the exercise price.

Example: Call optionsSuppose you own a call option (long call position) with an exercise (strike) price of Rs.30.

– If the current stock price is Rs.40 (in-the-money):• Your option has an intrinsic value of Rs.10. • You have the right to buy at Rs.30, and you can exercise

your right and then sell the stock in the market for Rs.40.

– If the current stock price is Rs.20 (out-of-the-money):• Your option has no intrinsic value.• You would not exercise your right to buy the stock for

Rs.30 as you can buy the stock for Rs.20 in the stock market.

Example: Put optionsSuppose you own a put option (long put position) with an exercise (strike) price of Rs.30.

– If the current stock price is Rs.20 (in-the-money):• Your option has an intrinsic value of Rs.10.• You have the right to sell at Rs.30, so you can buy the

stock at Rs.20 and then exercise and sell for Rs.30.

– If the current stock price is Rs.40 (out-of-the-money):• Your option has no intrinsic value.• You would not exercise your right to sell the stock for

Rs.30 as you can sell the stock for Rs.40 in the stock market.

Long call position (buying a call option)

• Position taken in the expectation that price will rise.

• Example: • For a call buyer with

– an exercise price = Rs.70

– option premium (option price) paid by the call buyer = Rs.6.13

• The following diagram shows different total dollar profits for buying a call option with a strike price of Rs.70 and a premium of Rs.6.13

Long call position (Buying call option)

40 50 60 70 80 90 100

(1,000)

Exercise price = Rs.70Call premium = Rs.6.13

Stock Price at Expiration

Profit from Strategy

Limited loss

Call premium

Break-even price

unlimited profit

Example (Cont)• For a call buyer with an exercise price = Rs.70.

– If current share price > exercise price (Rs.80); call option is in the money.

– If current share price = exercise price (Rs.70); call option is at the money

– If current share price < exercise price (Rs.60); call option is out of the money

• Call premium (option price) paid by the call buyer = Rs.6.13• Breakeven point = Exercise price + Premium

= Rs.70 + Rs.6.13 = Rs.76.13Profit realised, if the share price rises above Rs.76.13

• Profit = Spot rate – (Exercise price + Premium) e.g. spot rate = Rs.80

Profit = 80 – (70+6.13)Note payoff profit - Profits on an option strategy include option payoffs and the premium paid for the option

Short call position (selling a call option)

• Position taken in the expectation that price will remain steady or decline

Example: • For a call seller (writer) with

– an exercise price = Rs.70– option premium (option price) earned by the call seller =

Rs.6.13

• The following diagram shows different total dollar profits for selling a call option with a strike price of Rs.70 and a premium of Rs.6.13

Short call position (selling a call option)

40 50 60 70 80 90 100

(1,000)

(2,000)

(1,500)

Profit limited to call premium

Call premium

Unlimited loss

Long put position (buying a put option)

• Position taken in the expectation that price will decline.

Example: • For a put buyer with

– an exercise price = Rs.70– option premium (option price) paid by the put buyer =

Rs.6.13

• The following diagram shows different total dollar profits for buying a put option with a strike price of Rs.70 and a premium of Rs.6.13

Long put position (buying a put option)

40 50 60 70 80 90 100

(1,000)

Exercise price = Rs.70Put premium = Rs.6.13

Profit up to Rs.6,387 [100x(Rs.70-Rs.6.13)]

Limited lossBreak-even price

Put premium

Example (Cont)• For a put buyer with an exercise price = Rs.70.

– If current share price < exercise price (Rs.60); put option is in the money.

– If current share price = exercise price (Rs.70); put option is at the money

– If current share price > exercise price (Rs.80); put option is out of the money

• Put premium (option price) paid by the put buyer = Rs.6.13 (thus maximum loss Rs.6.13 up to exercise price of Rs.70)

• Breakeven point = Exercise price - Premium

= Rs.70 – Rs.6.13 = Rs.63.87

Profit realised, if the share price drop below Rs.63.87

• Profit = Exercise price – (Spot rate + Premium)

e.g. spot price = Rs.60

Profit = 70 – (60+6.13)

Short put position (selling a put option)

• The put writer bets that the price will not decline greatly – collects premium income with no payoff

• The payoff for the buyer is the amount owed by the writer (payoff loss limited to the strike price since the stock’s value cannot fall below zero)

Example: • For a put seller with

– an exercise price = Rs.70– option premium (option price) earned by the put seller =

Rs.6.13

• The following diagram shows different total dollar profits for selling a put option with a strike price of Rs.70 and a premium of Rs.6.13

Short put position (selling a put option)

40 50 60 70 80 90 100

(1,000)

(2,000)

(1,500)

Call premium

Limited profit (put premium)

Loss up to Rs.6,387 [100x(Rs.70-Rs.6.13)]

Option Strategies

• Single Option and Stock– Covered Call– Protective puts

• Spreads : multiple options of same or different types– Bull spread– Bear Spread– Box Spread– Butterfly spread– Calendar Spreads– Diagonal Spread

• Combinations– Straddle– Strip , Strap– Strangle

• Focus not on replicating all payoffs, but general understanding of these strategies

Technical Trading using options

• Contrary-opinion technicians use put options, which give the holder the right to sell stock at a specified price for a given time period, as signals of a bearish attitude.

• A higher put/call ratio indicates a pervasive bearish attitude, which technicians consider a bullish indicator.

• This ratio fluctuates between .60 and .40, and it has typically been substantially less than 1 because investors tend to be bullish and avoid selling short or buying puts.

• The current decision rule states that a put/call ratio above .60—sixty puts are traded for every 100 calls—is considered bullish, while a relatively low put/call ratio of .40 or less is considered a bearish sign

Protective Puts

( C ) protective Put strategy : the investment strategy involves buying a put option ona stock and the stock itself. (D) a short position in a put option is combined with a short position in the stock. This

is the reverse of a protective put.

Protective Puts

Strike Price X = Rs. 100; Stock Price S = Rs. 97 (at the expiration date)

Value of Put = X-St = 100 – 97 = 3Protective Put means holding stock and put options. If the price of the stock moves down you have protection on value loss. Used as a simple portfolio insurance strategy

ST <= X ST > X

Stock ST ST

+Put X-ST 0

Total X ST

Covered Calls

(A) the portfolio consists of a long position in a stock plus a short position in a call option. This is known as writing a covered call. The long stock position "covers" or protects the investor from the payoff on the short call that becomes necessary if there is a sharp rise in the stock price.

(B) A short position in a stock is combined with a long position in a call option. This is the reverse of writing a covered call

Covered Calls

Purchase of a stock and simultaneous sale of a call on the stock (the sales is “covered” as you own the stock). Writing without owning stock is “naked” position. Fund Manager has a target sell price (for an owned stock) say: Rs. 110. And the fund owns 1000 shares of the stock. The current price is Rs. 100. Fund manager writes a call at Rs. 110 say for Rs. 5. If the price goes up fund manager gets Rs. 110 and if the price goes down then the fund manager gets Rs. 5000. When the fund manager owns lot of stocks and is clear with the target sell prices then it is a simple disciplined strategy to boost income.

ST <= X ST > X

Stock ST ST

+Written Call -0 -(ST-X)

Total ST X

Using Options instead of stocks• Suppose you have a choice of two investment strategies. The first is to

invest Rs.100 in a stock. The second strategy involves investing Rs.90 in 6 month T-bills and Rs.10 in 6 month calls. If the call is priced at Rs.5, then you are able to buy 2 calls.

• The payoffs for this strategy are outlined below.

Put Call ParityConsider a combination of Call plus bond (where the value of the bond is equal to the strike price of the stock) payoff

The Payoff is similar to Protective put! Hence:C+X/(1+Rf)T = So +PP = C + PV(k) - S

ST <= X ST > X

Call 0 St-X

Bond X X

Total X ST

Synthetic T bills

Put Call Parity - Example

Stock Price Rs.110

Call Price (1 –year expiration, X =Rs. 105) Rs.17

Put Price (1 –year expiration, X =Rs. 105) Rs.5

Risk Free interest rate 5%

Put Call Parity

Is Parity Violated?17+105/(1.05) = 110 +5 = 2 YES!Arbitrage Strategy Position Cash Flow

NowST< 105 ST>105

Buy Stock -110 ST ST

Borrow (equals to the exercise price = 105/1.05

100 -105 -105

Sell Call 17 0 -(ST-105)

Buy Put -5 105-ST 0

Total 2 0 0

Pricing of Options: Arbitrage BoundsOptions are priced based on arbitrage principle

(as they are redundant securities)For example: Suppose that an American call option on Mahindra Satyam with an exercise price of Rs.70 is selling for Rs.3 and that the Mahindra Satyam stock price is Rs.75. An arbitrage can be executed by buying the option and exercising it immediately, yielding an immediate profit of Rs.2 per share. Hence the lower bound for the option can be written as:CA >= max(0, S-X) here A represents American Option

Arbitrage BoundsLower Bound on an American Put OptionPA >= max(0, X-S)

Lower Bound on a European Call OptionCE >= max(0, S-X(1+r)-T)

Lower Bound on a European Put OptionPE >= max[0, X(1+r)-T - S]

However, Exercise Price is set based on an unknown future stock price! Hence, pricing is not simple. We have relay on probability of future prices.

Option Valuation: A simple example

• A stock is currently priced at Rs.40 per share.• In 1 month, the stock price may

– go up by 25%, or– go down by 12.5%.

A simple example

• Stock price dynamics:

Rs.40x(1+.25) = Rs.50

Rs.40x(1-.125) = Rs.35

t = now t = now + 1 month

up state

down state

Call option

• A call option on this stock has a strike price of Rs.45

t=0 t=1

Stock Price=Rs.40;

Call Value=Rs.c

Stock Price=Rs.50;

Call Value=Rs.5

Stock Price=Rs.35;

Call Value=Rs.0

A replicating portfolio

• Consider a portfolio containing D shares of the stock and $B invested in risk-free bonds.– The present value (price) of this portfolio is DS + B

= Rs.40 D + B

Portfolio value

t=0 t=1

Rs.50 D + (1+r/12)B

Rs.35 D + (1+r/12)B

Rs.40 D + B

up state

down state

A replicating portfolio

• This portfolio will replicate the option if we can find a D and a B such that

Rs.50 D + (1+r/12) B = Rs.5

Rs.35 D + (1+r/12) B = Rs.0

Portfolio payoff = Option payoff

Up state

Down state

The replicating portfolio

• Solution:– D = 1/3 – B = -35/(3(1+r/12)).

• Eg, if r = 5%, then the portfolio contains– 1/3 share of stock (current value Rs.40/3 =

Rs.13.33)– partially financed by borrowing Rs.35/(3x1.00417)

= Rs.11.62

The replicating portfolio

• Since the replicating portfolio has the same payoff in all states as the call, the two must also have the same price.

• The present value (price) of the replicating portfolio is Rs.13.33 – Rs.11.62 = Rs.1.71.

• Therefore, c = Rs.1.71

A general (1-period) formula

Cu CdSu Sd

BSuCd SdCu

1 r Su Sd

p r du d

c S BpCu 1 p Cd

Two Periods

Suppose two price changes are possible during the life of the option

At each change point, the stock may go up by Ru

% or down by Rd%

Two-Period Stock Price Dynamics

• For example, suppose that in each of two periods, a stocks price may rise by 3.25% or fall by 2.5%

• The stock is currently trading at Rs.47• At the end of two periods it may be worth as

much as Rs.50.10 or as little as Rs.44.68

Two-Period Stock Price Dynamics

Rs. 47

Rs.48.53

Rs.45.83

Rs.50.10

Rs.47.31

Rs.44.68

Terminal Call Values

Cuu =Rs.5.10

Cud =Rs.2.31

Cdd =Rs.0

At expiration, a call with a strike price of Rs.45 will be worth:

Two Periods

The two-period Binomial model formula for a European call is

C p2CUU 2p(1 p)CUD (1 p)2CDD

Black-Scholes (1973) Model (BSM)• Using stochastic calculus and the heat exchange equation from

physics, Black and Scholes developed the following model.

)5.0()/(1

)()()(

TrXSnd

dNeXdSNC rT

05.0)(/35183/2812/

/35183/2812/

difedNdd

C0 = the call price;

S = current market price of underlying ordinary shares;

X = exercise price of call option;

T = time to expiration (fraction of year);

r = current annualised market interest rate for prime commercial paper;

= standard deviation of annual return on underlying asset (volatility)

N(d1) = cumulative density function of d1 as defined earlier

e = base of natural logarithms (approx 2.71828);

N(d2) = cumulative density function of d2 as defined earlier; ln(S/X) = natural log of ( C/X)

Example:

From the following information, calculate the price of call option

S = Rs.40

X = Rs.40

r = 9%

T = 1 year

)5.0()/(1

)()()(

TrXSnd

dNeXdSNC rT

15.0)1(3.045.0

45.0)1(3.0

1))3.0(5.009.0()40/40(1

6736.0)( 1 dN

5596.0)( 2 dN

49.6$)5596.0)((40)6736.0)(40( )1(09.00 eC

Options StrategiesStrategy 1: Vertical Spread (So = Rs. 40 and LOT SIZE 100)

Cost ST = 46 ST = 38

Buy Call ITM @40 -200 600 0

Sell Call OTM @45 +100 100 0

Net Position@ ST -100 00 -100

Strategy 2: Long Strangle(So = Rs. 40 and LOT SIZE 100)

Buy Put OTM @35 -100 0 0

Buy Call OTM @45 -100 500 0

Options StrategiesStrategy 3: Long Straddle (So = Rs. 40 and LOT SIZE 100)

Buy Put ATM @40 -100 0 0

Buy Call ATM @40 -100 1000 0

Strategy 4: STRAP(So = Rs. 40 and LOT SIZE 100)

Buy 2 Calls ATM @40

-400 0 2000

Buy Put ATM @40 -200 1000 0

Net Position@ ST -600 400 1400

Options StrategiesStrategy 5: STRIP (So = Rs. 40 and LOT SIZE 100)

Buy 2 Puts ATM @40

-400 0 2000

Buy 1Call ATM @40 -200 1000 0

Net Position@ ST -600 800 1400

Strategy 6: Butterfly Spread(So = Rs. 40 and LOT SIZE 100)

Buy Call ITM @30 -1100

Sell 2 calls ATM @40

Buy Call OTM @50 -100

Net Position@ ST -400

Derivatives – Options, Futures

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