Post on 23-Dec-2015
VIII: Options
26: Options Pricing
Chapter 26: Options Pricing © Oltheten & Waspi 2012
Options Pricing Models
Binomial Model Black Scholes Options Pricing Model
© Oltheten & Waspi 2012
Chapter 26: Options Pricing © Oltheten & Waspi 2012
Binomial Model
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.
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
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
Chapter 26: Options Pricing © Oltheten & Waspi 2012
Option
C
$75 Call Option$25
$0
P=½
P=½
$100 - $75
expire
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
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
Chapter 26: Options Pricing © Oltheten & Waspi 2012
Black Scholes Option Pricing Model
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
Chapter 26: Options Pricing © Oltheten & Waspi 2012
Discovery Café
$0
$2,000
$4,000
30 60 90 120 150
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
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
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
Chapter 26: Options Pricing © Oltheten & Waspi 2012
The second factor: d2
.250.40-0.3643d
tdd
2
12
σ
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
Chapter 26: Options Pricing © Oltheten & Waspi 2012
Discovery Café
$0
$2,000
$4,000
36 54 72 90 108 126 144
Chapter 26: Options Pricing © Oltheten & Waspi 2012
Black-Scholes Options Pricing Model
rt2Exercise1StockCall e dNPdNPP
Chapter 26: Options Pricing © Oltheten & Waspi 2012
Put Call Parity
Engineer two portfolios
rtExercisestockCallPut ePPPP
Put Portfolio
CallPortfolio
Same payout →same price
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
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
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
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
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)
Chapter 26: Options Pricing © Oltheten & Waspi 2012
Put Call Parity
P = $3.93 - $90 + $100 e-0.05*0.25
P =
rtExercisestockCallPut ePPPP
Chapter 26: Options Pricing © Oltheten & Waspi 2012
Excel Spreadsheet
Built with Spreadsheet Exercise 26-1
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
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
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
Chapter 26: Options Pricing © Oltheten & Waspi 2012
Exercise
26-5
Options IV