VIII: Options 26: Options Pricing. Chapter 26: Options Pricing © Oltheten & Waspi 2012 Options...

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VIII: Options 26: Options Pricing

Transcript of VIII: Options 26: Options Pricing. Chapter 26: Options Pricing © Oltheten & Waspi 2012 Options...

Page 1: VIII: Options 26: Options Pricing. Chapter 26: Options Pricing © Oltheten & Waspi 2012 Options Pricing Models  Binomial Model  Black Scholes Options.

VIII: Options

26: Options Pricing

Page 2: VIII: Options 26: Options Pricing. Chapter 26: Options Pricing © Oltheten & Waspi 2012 Options Pricing Models  Binomial Model  Black Scholes Options.

Chapter 26: Options Pricing © Oltheten & Waspi 2012

Options Pricing Models

Binomial Model Black Scholes Options Pricing Model

© Oltheten & Waspi 2012

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Chapter 26: Options Pricing © Oltheten & Waspi 2012

Binomial Model

Page 4: VIII: Options 26: Options Pricing. Chapter 26: Options Pricing © Oltheten & Waspi 2012 Options Pricing Models  Binomial Model  Black Scholes Options.

Chapter 26: Options Pricing © Oltheten & Waspi 2012

Law of One Price

Two assets with the same risk and return characteristics will have the same price.

Arbitrage Taking advantage of different prices in different

markets of two assets of the same risk and return characteristics.

Page 5: VIII: Options 26: Options Pricing. Chapter 26: Options Pricing © Oltheten & Waspi 2012 Options Pricing Models  Binomial Model  Black Scholes Options.

Chapter 26: Options Pricing © Oltheten & Waspi 2012

Two Portfolios Leveraged Stock

Shares of the underlying asset and debt

Option Call option on

the underlying asset

Constructed to have the same risk and return characteristics

Leveraged Stock Call

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Chapter 26: Options Pricing © Oltheten & Waspi 2012

Leveraged Stock

$50

Discovery Café$100

$25

P=½

P=½R=3%

Borrow $24.27

$25.73

Repay $25

$75

$0

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Chapter 26: Options Pricing © Oltheten & Waspi 2012

Option

C

$75 Call Option$25

$0

P=½

P=½

$100 - $75

expire

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Chapter 26: Options Pricing © Oltheten & Waspi 2012

Binomial Model

$25.73

$75

$0

P=½

P=½

C

$25

$0

P=½

P=½

Leveraged Stock

$75 Call Option

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Chapter 26: Options Pricing © Oltheten & Waspi 2012

Multi-period Binomial Model

$50

$59.46

$42.04

$70.71

$84.09

$100.

$35.36

$29.73

$25

1

3

2

5

6

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Chapter 26: Options Pricing © Oltheten & Waspi 2012

Black Scholes Option Pricing Model

Page 11: VIII: Options 26: Options Pricing. Chapter 26: Options Pricing © Oltheten & Waspi 2012 Options Pricing Models  Binomial Model  Black Scholes Options.

Chapter 26: Options Pricing © Oltheten & Waspi 2012

tdd

t

t2

rPP

ln

d

e dNPdNPP

12

2

Ex

S

1

rt2Exercise1StockCall

σ

σ

Black-Scholes Options Pricing Model

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Chapter 26: Options Pricing © Oltheten & Waspi 2012

Discovery Café

$0

$2,000

$4,000

30 60 90 120 150

Page 13: VIII: Options 26: Options Pricing. Chapter 26: Options Pricing © Oltheten & Waspi 2012 Options Pricing Models  Binomial Model  Black Scholes Options.

Chapter 26: Options Pricing © Oltheten & Waspi 2012

Example:

Discovery Café Call P Exercise= $100 91 days to maturity

P stock = $90 σ=40%

r = 5%

© Oltheten & Waspi 2012

Page 14: VIII: Options 26: Options Pricing. Chapter 26: Options Pricing © Oltheten & Waspi 2012 Options Pricing Models  Binomial Model  Black Scholes Options.

Chapter 26: Options Pricing © Oltheten & Waspi 2012

The first factor: d1

.250.40

0.25 2

0.400.05

$100$90

ln

d

t

t2

rPP

ln

d

2

1

2

Ex

S

1

σ

σ

Page 15: VIII: Options 26: Options Pricing. Chapter 26: Options Pricing © Oltheten & Waspi 2012 Options Pricing Models  Binomial Model  Black Scholes Options.

Chapter 26: Options Pricing © Oltheten & Waspi 2012

Delta Hedge Ratio

N[d1] = 0.35942 Delta Hedge Ratio = 36

If P Stock $1 then P Call $0.36 36 shares hedges 1 short call Example:

36 shares *(+$1) = +$361 written call: -100 shares * (+$0.36) = -$36

Net effect = $0

Page 16: VIII: Options 26: Options Pricing. Chapter 26: Options Pricing © Oltheten & Waspi 2012 Options Pricing Models  Binomial Model  Black Scholes Options.

Chapter 26: Options Pricing © Oltheten & Waspi 2012

The second factor: d2

.250.40-0.3643d

tdd

2

12

σ

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Chapter 26: Options Pricing © Oltheten & Waspi 2012

In-the-Money

N[d2] = 0.28774 Probability (In-the-Money) = 28.8%

σ = 0.40 annual volatility → σ√t = 0.40(√.25) = 0.20 over 91 days 20% of $90 = $18 volatility

© Oltheten & Waspi 2012

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Chapter 26: Options Pricing © Oltheten & Waspi 2012

Discovery Café

$0

$2,000

$4,000

36 54 72 90 108 126 144

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Chapter 26: Options Pricing © Oltheten & Waspi 2012

Black-Scholes Options Pricing Model

rt2Exercise1StockCall e dNPdNPP

Page 20: VIII: Options 26: Options Pricing. Chapter 26: Options Pricing © Oltheten & Waspi 2012 Options Pricing Models  Binomial Model  Black Scholes Options.

Chapter 26: Options Pricing © Oltheten & Waspi 2012

Put Call Parity

Engineer two portfolios

rtExercisestockCallPut ePPPP

Put Portfolio

CallPortfolio

Same payout →same price

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Chapter 26: Options Pricing © Oltheten & Waspi 2012

Put Call Parity

Put Portfolio $100 Put Option 100 Shares DVC

Call Portfolio $100 Call Option NPV $10,000

$0

$5,000

$10,000

$15,000

36 54 72 90 108 126 144

$0

$5,000

$10,000

$15,000

36 54 72 90 108 126 144

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Chapter 26: Options Pricing © Oltheten & Waspi 2012

Put Portfolio

$0

$5,000

$10,000

$15,000

36 54 72 90 108 126 144

Put ExercisedPut Expires

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Chapter 26: Options Pricing © Oltheten & Waspi 2012

Call Portfolio

$0

$5,000

$10,000

$15,000

36 54 72 90 108 126 144

Call ExpiresCall Exercised

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Chapter 26: Options Pricing © Oltheten & Waspi 2012

Put Call Parity

Put Portfolio $100 Put Option 100 Shares DVC

Call Portfolio $100 Call Option NPV $10,000

$0

$5,000

$10,000

$15,000

36 54 72 90 108 126 144

$0

$5,000

$10,000

$15,000

36 54 72 90 108 126 144

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Chapter 26: Options Pricing © Oltheten & Waspi 2012

Put Call Parity

Put Portfolio $100 Put Option 100 Shares DVC

Call Portfolio $100 Call Option NPV 100 shares

P (Put) = pP (Shares) = $90

P (Call) = $3.93NPV (Ex) = $100e- (0.05) (0.25)

Page 26: VIII: Options 26: Options Pricing. Chapter 26: Options Pricing © Oltheten & Waspi 2012 Options Pricing Models  Binomial Model  Black Scholes Options.

Chapter 26: Options Pricing © Oltheten & Waspi 2012

Put Call Parity

P = $3.93 - $90 + $100 e-0.05*0.25

P =

rtExercisestockCallPut ePPPP

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Chapter 26: Options Pricing © Oltheten & Waspi 2012

Excel Spreadsheet

Built with Spreadsheet Exercise 26-1

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Chapter 26: Options Pricing © Oltheten & Waspi 2012

Implied Volatility

Black Scholes Options Pricing

Model

Volatility: σ Call Option PricePut Option Price

Call Option PricePut Option Price

Volatility: σ

Page 29: VIII: Options 26: Options Pricing. Chapter 26: Options Pricing © Oltheten & Waspi 2012 Options Pricing Models  Binomial Model  Black Scholes Options.

Chapter 26: Options Pricing © Oltheten & Waspi 2012

Volatility

$3.93

$12.69

$-

$5

$10

$15

$20

$25

0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%

Put Price

Call Price

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Chapter 26: Options Pricing © Oltheten & Waspi 2012

Volatility Smile on GE Options

Jan

March

June

20%

25%

30%

35%

40%

45%

50%

55%

25 30 35 40 45 50 55

Strike Price

Impli

ed V

olatili

ty

Page 31: VIII: Options 26: Options Pricing. Chapter 26: Options Pricing © Oltheten & Waspi 2012 Options Pricing Models  Binomial Model  Black Scholes Options.

Chapter 26: Options Pricing © Oltheten & Waspi 2012

Exercise

26-5

Page 32: VIII: Options 26: Options Pricing. Chapter 26: Options Pricing © Oltheten & Waspi 2012 Options Pricing Models  Binomial Model  Black Scholes Options.

Options IV