Put-Call Parity ¤Portfolio 1 –Put option, U –Share of stock, P ¤Portfolio 2 –Call option, V –PV of exercise price, X

Put-Call Parity

¤ Portfolio 1– Put option, U– Share of stock, P

¤ Portfolio 2– Call option, V– PV of exercise price, X

Portfolio Payoffs at Expiration PT<X PT≥X

Port. 1 Port. 2 Port. 1 Port. 2

Stock PT PT

Put X-PT 0

Call 0 PT-X

Cash X X

Total X X PT PT

Put-Call Parity Relationship

¤ Portfolio payoffs are equal, so portfolio values also must be equal.

¤ Put + Stock = Call + PV of Exercise Price

U + P = V + Xe-rRFt

Option Trading Strategies

¤ Buying call options to achieve leverage

¤ The price of a call of one lot, say 100 shares is significantly lower than buying the 100 shares outright.

¤ Example: Stock XYZ sells at Rs50/share and a Rs50 call costs Rs5/share. The Investor can buy the call for Rs500 instead of the 100 shares for Rs5,000. If XYZ goes to Rs60, the value of the option is Rs1000.

¤ Return on option: Rs500/Rs500 = 100%

¤ Return on stock purchase: Rs1000/Rs5000 = 20%


¤ Buying call options to limit risk

¤ Investor dislikes the risk of buying XYZ and watching it go down in value. Therefore, Investor purchases XYZ 50 call at Rs5 and puts remaining money into risk-free securities. Hence, given the same Rs5,000, the Investor buys call and puts Rs4,500 into risk free securities.

¤ Example: If XYZ goes to Rs60, the investor can exercise the option to net Rs500 plus interest from risk free investment. If XYZ stays at Rs50 or falls below, the investor has lost his option premium which is partly offset by interest income.


¤ Buying call options to hedge short stock position

¤ Investor believes XYZ will decline. Investor sells XYZ short to obtain total profit potential but he is exposed to unlimited loss from stock price increase. The Investor buys a call to eliminate loss.


¤ Buying put options for leverage and limited risk

¤ Investor anticipates significant decrease in the stock price but does not have the margin money for a short sale, and does not want to be exposed to unlimited risk of stock price increases. Investor buys a put. Stock price must decline enough to break even.


¤ Buying put options to hedge against a possible

stock price decline

¤ Investor holds Infosys and is already sitting on paper profit. Investor believes Infosys will go higher and would like to participate in upside without risking a loss on paper profit. So he buys a put. If price goes up, the potential is only diminished by the cost of the put, whereas the paper profits are protected by the put and decreased only by the put price.

Protective Put Strategy

Market Price

Net Profit

Gain

Loss

0

OptionProfit

LongPosition


¤ Covered Call Strategy

¤ Investor owns 100 shares of XYZ (Rs50) and writes a call at Rs55 to earn a greater return than the stock alone. Investor earns Rs5.00 as call premium. Return is Rs5.00 plus any capital gains.

¤ If price goes above Rs55, the upside is limited to Rs10.00.

¤ Covered call also provides limited protection to writer against price decline. Price can decline to Rs45 (Rs50-Premium) before writer experiences paper loss.


¤ Naked Call Strategy

¤ Investor writes a call on XYZ at Rs55 and receives a premium income of Rs5 without owning the security

¤ If price remains below Rs55, the writer gains Rs5

¤ If price remains above Rs55 and below Rs60, the writer gains (Rs60- Price)

¤ If price goes above Rs60, the writer loses (Price- Rs60)

Naked Call Strategy

Upside potential is limited to the extent of premium received. Downside risk is unlimited.

Gain

Loss

0

Premium Received

Strike Price

Unlimited Loss

MarketPrice

Rs 60


¤ Buying/Writing an Option Straddle

¤ An Option Straddle is the purchase or the writing of both a put and a call on the same security.

¤ Buying a Straddle: Price of underlying security is expected to move SHARPLY up or down before option expiration date. Buy a put and a call. Say you pay for a put and a call premium of Rs3.00 each. If the stock moves from Rs50 to above Rs56 or below Rs44, a profit is made.


¤ Writing a Straddle: Price of the underlying

security is expected to stay at its current market value until the option expires. Write a put and write a call at Rs3.00 each and receive a total premium of Rs6.00.

¤ As long as the stock price remains between Rs44 and Rs56, the option straddle writer makes a profit.


¤ Bull Spread

¤ Buying a call and selling a call with a higher strike price

Example:

1. Buy call with Rs90 SP Premium paid= Rs5

2. Sell a call with Rs95 SP Premium received = Rs2

Profit Profile

85 90 95 100 105 110

P = Rs90

If stock price goes to Rs97, what is the net profit to the investor?

Determine profits from a range of Rs85 to Rs110 & profit profile.

93


¤ Bear Spread

¤ Buying a put and selling a put with a lower strike price

Example:

1. Buy put with Rs110 SP Premium paid= Rs5

2. Sell a put with Rs105 SP Premium received = Rs2

Profit Profile

85 90 95 100 105 110

P = Rs110

115

Determine net profits from a range of prices of Rs85-Rs115

107


¤ Butterfly Spread

¤ Butterfly spread is a neutral position that is a combination of both a bull and bear spread.

Example: Current Stock Price = Rs 60 July 50 call @ Rs12

July 60 call @ Rs 6July 70 call @ Rs 3

Butterfly spread:Buy 1 July 50 call: Rs(1200)Sell 2 July 60 calls: 1200Buy 1 July 70 call: (300)

Rs (300)

Profit Profile

Determine net profits from a range of Rs40-Rs80.

50 53 60 67 70


¤ Calendar Spread: Involves the sale of one

option and the simultaneous purchase of a more/less distant option, both with the same strike price

¤ Example: JAN. APR50’s JUL50’s OCT50’s XYZ: Rs5 Rs8 Rs10

Rs50


¤ Neutral Spread: Investor should have the initial intent of

closing the spread by the time the near-term option expires.¤ Assume the following:

APR50’s JUL50’s OCT50’s JAN. Rs5 Rs8 Rs10

APR. 0 5 8

¤ In January the investor sells the APR 50 call and buys the July 50. His spread cost is 3.

¤ In April the price is unchanged and the 3 month call (July) should be worth 5. The spread between the April 50 and the July 50 has now widened to 5. Since the spread cost is 3, a 2 pt. profit exists. Investor should now close his long position by selling his July 50 call and reaping a 2 pt. profit.


¤ Bullish Calendar Spread: Investor sells the near-

term call and buys a longer-term call when the underlying stock is some distance below the SP of the calls.

¤ Feature of low investment and large potential profit.

Example: XYZ: Rs45 in Jan.

Sell April 50 for Rs1Buy July 50 for Rs1.5


¤ Investor wants 2 things to happen:

1. Near-term call expires worthless2. Stock price must rise by the time July call expires

¤ Assume price goes to 52 b/w April & July. Investor nets 1.5 pts. How?

Binomial Pricing Model

What is the fair value of a call (or put) today?

¤The Binomial Option Pricing Model (BOP) can be used to determine the fair value of an option.

¤ The assumption is that the underlying asset will attain one of two possible known prices at the end of each of the finite number of periods (given its price at the start of each period).


Consider the following example:

¤The stock of GTL today (t=0) is Rs100. You analyze the firm and conclude that one year from now (t=1) the stock will sell for either Rs125 (a rise of 25%) or Rs80 (a drop of 20%).

¤The risk free rate is 8% compounded continuously.


Consider a call option on GTL…

¤Let us say that the call’s exercise price is Rs100 and that the expiration date is one year from now.

¤One year from now, the call will have a value of either Rs25 (if GTL sells at Rs125) or Rs0 (if GTL sells at Rs80).


Share of GTL

Call on GTL

Rs100

????

Rs125

Rs80

Rs25

Rs0

t=0 t=1

Up State

Up State

Down State

Down State

Binomial Pricing ModelThree investments are of interest to us:

stock option risk free bond

Payoffs and Prices of Instruments:

Security Payoff: Payoff: Current

Up state Down state Price

Stock Rs125.00 Rs 80.00 Rs100.00Bond 108.33 108.33 100.00Call 25.00 0.00 ???

BOP: Replicating Portfolios

The Call Option on GTL can be valued by finding a portfolio that replicates the payoff of the Call in all states of nature.

The replicating portfolio will look as follows: A position in GTL stock A position in a risk free bond

The replicating portfolio will generate the same cash flows as a call and hence they must have the same value… otherwise there will be an arbitrage opportunity The portfolio’s cost will be the fair value of the option.

BOP: Replicating Portfolios

Consider a portfolio with ‘S’ shares of GTL and ‘B’ risk free bonds. What are the payoffs of such a portfolio?

In the Up State: Rs125S + Rs108B = Rs25In the Down State: Rs80S + Rs108 B = Rs0

Solving the two equations simultaneouslyS = .5556B = -.4103

Replicating Portfolios

What is the meaning of the numbers we have just obtained?

The investor can replicate the payoffs from the call by short selling Rs41.03 of the risk free bond and buying 0.5556 shares of GTL stock.

The payoffs will confirm this…

Replicating Portfolio

Portfolio Part Up State Down State

GTL Stock 0.5556 x Rs125=Rs69.45

0.5556 x Rs80=Rs44.45

Risk-free Bond -Rs41.03 x 1.083

=-Rs44.45

-Rs41.03 x 1.083

=-Rs44.45

Net Payoff Rs25 Rs0

Replicating Portfolios

Cost of building the replicating portfolio

– Rs55.56 must be spent to purchase .5556 shares of GTL at $100 per share

– Rs41.03 income is provided by the bonds

– Total cost is:Rs55.56 - Rs41.03 = Rs14.53

Documents

Put-Call Parity ¤Portfolio 1 –Put option, U –Share of stock, P ¤Portfolio 2 –Call option, V –PV of exercise price, X