McGraw-Hill/Irwin Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter...

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McGraw-Hill/Irwin Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter 21 Chapter 21 Option Valuation Option Valuation

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21-3 Time Value of Options: Call Option value X Stock Price Value of Call Intrinsic Value Time value

Transcript of McGraw-Hill/Irwin Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter...

Page 1: McGraw-Hill/Irwin Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter 21 Option Valuation.

McGraw-Hill/Irwin Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved.

Chapter 21Chapter 21

Option ValuationOption Valuation

Page 2: McGraw-Hill/Irwin Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter 21 Option Valuation.

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Intrinsic value - profit that could be made if the option was immediately exercised.

Call: stock price - exercise pricePut: exercise price - stock price

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

Option Values

Page 3: McGraw-Hill/Irwin Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter 21 Option Valuation.

21-3

Time Value of Options: Call

Option value

XStock Price

Value of Call Intrinsic Value

Time value

Page 4: McGraw-Hill/Irwin Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter 21 Option Valuation.

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Factor Effect on valueStock price increasesExercise price decreasesVolatility of stock price increasesTime to expiration increasesInterest rate increasesDividend Rate decreases

Factors Influencing Option Values: Calls

Page 5: McGraw-Hill/Irwin Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter 21 Option Valuation.

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Restrictions on Option Value: Call

Value cannot be negativeValue cannot exceed the stock valueValue of the call must be greater than the value of levered equityC > S0 - ( X + D ) / ( 1 + Rf )T

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

Page 6: McGraw-Hill/Irwin Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter 21 Option Valuation.

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Allowable Range for Call

Call Value

S0

PV (X) + PV (D)

Upper

boun

d = S 0

Lower Bound

= S0 - PV (X) - PV (D)

Page 7: McGraw-Hill/Irwin Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter 21 Option Valuation.

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100

200

50

Stock Price

C

75

0

Call Option Value X = 125

Binomial Option Pricing: Text Example

Page 8: McGraw-Hill/Irwin Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter 21 Option Valuation.

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Alternative PortfolioBuy 1 share of stock at $100Borrow $46.30 (8% Rate)Net outlay $53.70PayoffValue of Stock 50 200Repay loan - 50 -50Net Payoff 0 150

53.70

150

0Payoff Structureis exactly 2 timesthe Call

Binomial Option Pricing: Text Example

Page 9: McGraw-Hill/Irwin Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter 21 Option Valuation.

21-9

53.70

150

0

C

75

0

2C = $53.70C = $26.85

Binomial Option Pricing: Text Example

Page 10: McGraw-Hill/Irwin Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter 21 Option Valuation.

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Alternative Portfolio - one share of stock and 2 calls written (X = 125)

Portfolio is perfectly hedgedStock Value 50 200Call Obligation 0 -150Net payoff 50 50

Hence 100 - 2C = 46.30 or C = 26.85

Replication of Payoffs and Option Values

Page 11: McGraw-Hill/Irwin Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter 21 Option Valuation.

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Generalizing the Two-State Approach

Assume that we can break the year into two six-month segments.

In each six-month segment the stock could increase by 10% or decrease by 5%.

Assume the stock is initially selling at 100.Possible outcomes:

Increase by 10% twiceDecrease by 5% twiceIncrease once and decrease once (2 paths).

Page 12: McGraw-Hill/Irwin Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter 21 Option Valuation.

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Generalizing the Two-State Approach

100

110

121

9590.25

104.50

Page 13: McGraw-Hill/Irwin Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter 21 Option Valuation.

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Assume that we can break the year into three intervals.For each interval the stock could increase by 5% or decrease by 3%.Assume the stock is initially selling at 100.

Expanding to Consider Three Intervals

Page 14: McGraw-Hill/Irwin Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter 21 Option Valuation.

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S

S +

S + +

S -S - -

S + -

S + + +

S + + -

S + - -

S - - -

Expanding to Consider Three Intervals

Page 15: McGraw-Hill/Irwin Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter 21 Option Valuation.

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Possible Outcomes with Three Intervals

Event Probability Stock Price

3 up 1/8 100 (1.05)3 =115.76

2 up 1 down 3/8 100 (1.05)2 (.97) =106.94

1 up 2 down 3/8 100 (1.05) (.97)2 = 98.79

3 down 1/8 100 (.97)3 = 91.27

Page 16: McGraw-Hill/Irwin Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter 21 Option Valuation.

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Co = SoN(d1) - Xe-rTN(d2)d1 = [ln(So/X) + (r + 2/2)T] / (T1/2)d2 = d1 + (T1/2)whereCo = Current call option value.So = Current stock priceN(d) = probability that a random draw from a

normal dist. will be less than d.

Black-Scholes Option Valuation

Page 17: McGraw-Hill/Irwin Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter 21 Option Valuation.

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X = Exercise pricee = 2.71828, the base of the natural logr = Risk-free interest rate (annualizes

continuously compounded with the same maturity as the option)

T = time to maturity of the option in yearsln = Natural log functionStandard deviation of annualized cont.

compounded rate of return on the stock

Black-Scholes Option Valuation

Page 18: McGraw-Hill/Irwin Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter 21 Option Valuation.

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So = 100 X = 95r = .10 T = .25 (quarter)= .50d1 = [ln(100/95) + (.10+(5 2/2))] / (5.251/2)

= .43 d2 = .43 + ((5.251/2)

= .18

Call Option Example

Page 19: McGraw-Hill/Irwin Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter 21 Option Valuation.

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N (.43) = .6664Table 17.2

d N(d) .42 .6628 .43 .6664 Interpolation .44 .6700

Probabilities from Normal Dist

Page 20: McGraw-Hill/Irwin Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter 21 Option Valuation.

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N (.18) = .5714Table 17.2

d N(d) .16 .5636 .18 .5714 .20 .5793

Probabilities from Normal Dist.

Page 21: McGraw-Hill/Irwin Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter 21 Option Valuation.

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Co = SoN(d1) - Xe-rTN(d2)Co = 100 X .6664 - 95 e- .10 X .25 X .5714 Co = 13.70Implied VolatilityUsing Black-Scholes and the actual price

of the option, solve for volatility.Is the implied volatility consistent with the

stock?

Call Option Value

Page 22: McGraw-Hill/Irwin Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter 21 Option Valuation.

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Put Value Using Black-Scholes

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

Using the sample call dataS = 100 r = .10 X = 95 g = .5 T = .2595e-10x.25(1-.5714)-100(1-.6664) = 6.35

Page 23: McGraw-Hill/Irwin Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter 21 Option Valuation.

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P = C + PV (X) - So = C + Xe-rT - So

Using the example dataC = 13.70 X = 95 S = 100r = .10 T = .25P = 13.70 + 95 e -.10 X .25 - 100P = 6.35

Put Option Valuation: Using Put-Call Parity

Page 24: McGraw-Hill/Irwin Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter 21 Option Valuation.

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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)

Page 25: McGraw-Hill/Irwin Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter 21 Option Valuation.

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

Using the Black-Scholes Formula

Page 26: McGraw-Hill/Irwin Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter 21 Option Valuation.

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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

Page 27: McGraw-Hill/Irwin Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter 21 Option Valuation.

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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.

Page 28: McGraw-Hill/Irwin Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter 21 Option Valuation.

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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.

Delta = Change in the value of the option

Change of the value of the stock

Page 29: McGraw-Hill/Irwin Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter 21 Option Valuation.

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Mispriced Option: Text Example

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

Page 30: McGraw-Hill/Irwin Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter 21 Option Valuation.

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Hedged Put Portfolio

Cost to establish the hedged position

1000 put options at $4.495 / option $ 4,495

453 shares at $90 / share 40,770

Total outlay 45,265

Page 31: McGraw-Hill/Irwin Copyright © 2005 by The McGraw-Hill Companies, Inc. All rights reserved. Chapter 21 Option Valuation.

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Profit Position on Hedged Put PortfolioValue of put option: implied vol. = 35%

Stock Price 89 90 91

Put Price $5.254 $4.785 $4.347

Profit (loss) for each put .759 .290 (.148)

Value of and profit on hedged portfolio

Stock Price 89 90 91

Value of 1,000 puts $ 5,254 $ 4,785 $ 4,347

Value of 453 shares 40,317 40,770 41,223

Total 45,571 45,555 5,570

Profit 306 290 305