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Options

Option Basics Option strategies Put-call parity Binomial option pricing Black-Scholes Model

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Buy - Long Sell - Short Call Put Key Elements

Exercise or Strike Price Premium or Price Maturity or Expiration

1. Option Basics: Terminologies

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In the Money - exercise of the option would be profitable.

Call: market price>exercise price

Put: exercise price>market price

Out of the Money - exercise of the option would not be profitable.

Call: market price<exercise price

Put: exercise price<market price

At the Money - exercise price and asset price are equal.

Market and Exercise Price Relationship

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Examples

Tell if it is in or out of the money For a put option on a stock, the current stock

price=$50 while the exercise price is $47. For a call option on a stock, the current stock price

is 55 while exercise price is 45.

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American - the option can be exercised at any time before expiration or maturity.

European - the option can only be exercised on the expiration or maturity date.

Should we early exercise American options?

American vs. European Options

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Stock Options Index Options Futures Options Foreign Currency Options Interest Rate Options

Different Types of Options

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Notation

Stock Price = ST Exercise Price = X

Payoff to Call Holder

(ST - X) if ST >X

0 if ST < X

Profit to Call Holder

Payoff - Purchase Price

Payoffs and Profits at Expiration - Calls

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Payoff to Call Writer

- (ST - X) if ST >X

0 if ST < X

Profit to Call Writer

Payoff + Premium

(pages 678 – 680)

Payoffs and Profits at Expiration - Calls

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Payoff and Profit to Call Option at Expiration

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Payoff and Profit to Call Writers at Expiration

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Payoffs to Put Holder

0 if ST > X

(X - ST) if ST < X

Profit to Put Holder

Payoff - Premium

Payoffs and Profits at Expiration - Puts

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Payoffs to Put Writer

0 if ST > X

-(X - ST) if ST < X

Profits to Put Writer

Payoff + Premium

Payoffs and Profits at Expiration - Puts

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Payoff and Profit to Put Option at Expiration

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Straddle (Same Exercise Price) – page 685

Long Call and Long Put

Spreads - A combination of two or more call options or put options on the same asset with differing exercise prices or times to expiration -- page 686.

Vertical or money spread:

Same maturity

Different exercise price

Horizontal or time spread:

Different maturity dates

2. Option Strategies

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Value of a Protective Put Position at Expiration

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Protective Put vs Stock (at-the-money)

Page 684

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Value of a Covered Call Position at Expiration

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Covered call example

Go to yahoo.finance.com, click on “investing”. Select the Citi Corp. (ticker symbol: C). Choose “options”. Suppose you are interested in the July 2013 options (i.e., option expiration date is July 13, 2013) and decided to buy 100 shares of Citi Corp on 4/24/2013. In the meantime, you shorted 1 contract (100 shares) of Citi’s stock option with an exercise price of $____/share.

  What is the profit from the portfolio at the expiration date when the price of

Citi on July 13, 2013 is $35?

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Intrinsic value - profit that could be made if the option was immediately exercised. Call: stock price - exercise price Put: exercise price - stock price

Time value - the difference between the option price and the intrinsic value.

Option Values

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Figure 21.1 Call Option Value before Expiration

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Restrictions on Option Value: Call

Value cannot be negative Value cannot exceed the stock value Value of the call must be greater than the value of

levered equityC > S0 - ( X + D ) / ( 1 + Rf )T

C > S0 - PV ( X ) - PV ( D )

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If the prices are not equal arbitrage will be possible

Put-call Parity

0(1 )Tf

XC S P

r

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Stock Price = 110 Call Price = 17

Put Price = 5 Risk Free = 5%

Maturity = 1 yr X = 105

117 > 115

Since the leveraged equity is less expensive, acquire the low cost alternative and sell the high cost alternative

Put Call Parity - Disequilibrium Example

0(1 )Tf

XC S P

r

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100

120

90

Stock Price

C

10

0

Call Option Value X = 110

4. Binomial Option Pricing: Text Example

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Alternative Portfolio

Buy 1 share of stock at $100

Borrow $81.82 (10% Rate)

Net outlay $18.18

Payoff

Value of Stock 90 120

Repay loan - 90 - 90

Net Payoff 0 30

18.18

30

0

Payoff Structureis exactly 3 timesthe Call

Binomial Option Pricing: Text Example

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18.18

30

0

C

30

0

3C = $18.18C = $6.06

Binomial Option Pricing: Text Example

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Generalizing the Two-State Approach

See page 725, get hedge ratio, then compute call price.

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Co = SoN(d1) - Xe-rTN(d2)

d1 = [ln(So/X) + (r + 2/2)T] / (T1/2)

d2 = d1 + (T1/2)

where

Co = Current call option value.

So = Current stock price

N(d) = probability that a random draw from a normal dist. will be less than d.

5. Black-Scholes Option Valuation

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X = Exercise price

e = 2.71828, the base of the natural log

r = Risk-free interest rate (annualizes continuously compounded with the same maturity as the option)

T = time to maturity of the option in years

ln = Natural log function

Standard deviation of annualized cont. compounded rate of return on the stock

Black-Scholes Option Valuation

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So = 100 X = 95

r = .10 T = .25 (quarter)

= .50

d1 = [ln(100/95) + (.10+(5 2/2))] / (5.251/2)

= .43

d2 = .43 + ((5.251/2)

= .18

Call Option Example

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Co = SoN(d1) - Xe-rTN(d2)

Co = 100 X .6664 - 95 e- .10 X .25 X .5714

Co = 13.70

Implied Volatility

Using Black-Scholes and the actual price of the option, solve for volatility.

Is the implied volatility consistent with the stock?

Call Option Value

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Black-Scholes Model with Dividends

The call option formula applies to stocks that pay dividends.

One approach is to replace the stock price with a dividend adjusted stock price.Replace S0 with S0 - PV (Dividends)

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Put Value Using Black-Scholes

P = Xe-rT [1-N(d2)] - S0 [1-N(d1)]

Using the sample call data

S = 100 r = .10 X = 95 g = .5 T = .25

95e-10x.25(1-.5714)-100(1-.6664) = 6.35

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Determinants of Call Option Values

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Hedging: Hedge ratio or delta The number of stocks required to hedge against the price

risk of holding one option.

Call = N (d1)

Put = N (d1) - 1

Option Elasticity

Percentage change in the option’s value given a 1% change in the value of the underlying stock.

Using the Black-Scholes Formula

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Figure 21.9 Call Option Value and Hedge Ratio

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Buying Puts - results in downside protection with unlimited upside potential.

Limitations Tracking errors if indexes are used for the puts. Maturity of puts may be too short. Hedge ratios or deltas change as stock values

change.

Portfolio Insurance

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Solution

Synthetic Put Selling a proportion of shares equal to the put

option’s delta and placing the proceeds in risk-free T-bills (page 763-764)

Additional problem Delta changes with stock prices Solution: delta hedging But increase market volatilities October 19, 1987

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Hedge Ratios Change with Stock Prices

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Hedging On Mispriced Options

Option value is positively related to volatility: If an investor believes that the volatility that is

implied in an option’s price is too low, a profitable trade is possible.

Profit must be hedged against a decline in the value of the stock.

Performance depends on option price relative to the implied volatility.

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Hedging and Delta

The appropriate hedge will depend on the delta.Recall the delta is the change in the value of the option

relative to the change in the value of the stock. Example (page 767):

Implied volatility = 33% Investor believes volatility should = 35% Option maturity = 60 days Put price P = $4.495 Exercise price and stock price = $90 Risk-free rate r = 4% Delta = -.453