Post on 27-Dec-2015
transcript
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Chapter 12Chapter 12The Black-Scholes FormulaThe Black-Scholes Formula
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Black-Scholes Formula
Call Options:
Put Options:
where
anddln S / K r T
T1
212
( ) ( )
d d T2 1
C S,K, ,r,T, Se N d Ke N d- T -rT( ) = ( ) ( ) 1 2
P S,K, ,r,T, Ke N d Se N d-rT - T ( ) = ( ) ( ) 2 1
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Black-Scholes (BS) assumptions
Assumptions about stock return distribution Continuously compounded returns on the stock are
normally distributed and independent over time (no “jumps”)
The volatility of continuously compounded returns is known and constant
Future dividends are known, either as dollar amount or as a fixed dividend yield
Assumptions about the economic environment The risk-free rate is known and constant There are no transaction costs or taxes It is possible to short-sell costlessly and to borrow at the
risk-free rate
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Applying BS to other assets
Call Options:
where
,
and
d
ln F S / F K T
T
TP
TP
1
0 021
2, ,( ) ( )
d d T2 1
C F S ,F K , ,T F S N d F K N dTP
TP
TP
TP( ) = ( ) ( )0 0 0 1 0 2, , , ,( ) ( ) ( ) ( )
F K KeTP -rT
0, ( ) =
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Applying BS to other assets (cont.)
Options on stocks with discrete dividends:
The prepaid forward price for stock with discrete dividends is:
Examples 12.3 and 12.1: S = $41, K = $40, = 0.3, r = 8%, t = 0.25, Div = $3 in one
month PV (Div) = $3e-0.08/12 = $2.98 Use $41– $2.98 = $38.02 as the stock price in BS formula The BS European call price is $1.763 Compare this to European call on stock without dividends:
$3.399
F S S PV DivTP
T0 0 0, ,( ) = ( )
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Applying BS to other assets (cont.)
Options on currencies: The prepaid forward price for the currency is:
where x is domestic spot rate and rf is foreign interest rate
Example 12.4: x = $0.92/ , K = $0.9, = 0.10, r = 6%, T = 1, and
= 3.2% The dollar-denominated euro call price is $0.0606 The dollar-denominated euro put price is $0.0172
F x x eTP -r Tf
0 0, ( ) =
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Applying BS to other assets
Options on futures: The prepaid forward price for a futures contract is the PV of the
futures price. Therefore:
where
and
Example 12.5: Suppose 1-yr. futures price for natural gas is $2.10/MMBtu, r =
5.5% Therefore, F=$2.10, K=$2.10, and = 5.5% If = 0.25, T= 1, call price = put price = $0.197721
C F,K, ,r,t Fe N d Ke N d-rt -rt( ) = ( ) ( ) 1 2
d
ln F / K T
T1
212
d d T2 1
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Option Greeks
What happens to option price when one input changes? Delta (): change in option price when stock price increases by
$1 Gamma (): change in delta when option price increases by $1 Vega: change in option price when volatility increases by 1% Theta (): change in option price when time to maturity
decreases by 1 day Rho (): change in option price when interest rate increases by
1% Greek measures for portfolios
The Greek measure of a portfolio is weighted average of Greeks
of individual portfolio components:
portfolio i i
i
n
1
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Option Greeks (cont.)
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Option Greeks (cont.)
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Option Greeks (cont.)
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Option Greeks (cont.)
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Option Greeks (cont.)
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Option Greeks (cont.)
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Option Greeks (cont.)
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Option Greeks (cont.)
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Option Greeks (cont.)
Option elasticity (: describes the risk of the option relative to the risk of the stock in percentage terms: If stock
price (S) changes by 1%, what is the percent change in the value of the option (C)?
Example 12.8: S = $41, K = $40, = 0.30, r = 0.08, T = 1, = 0 Elasticity for call: = S /C = $41 x 0.6911 / $6.961 = 4.071 Elasticity for put: = S /C = $41 x – 0.3089 / $2.886 = – 4.389
%
%
C
S
CC
SS
S C
S S
S
C
change in
change in
change in
change in
change in
change in
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Option Greeks (cont.)
Option elasticity (: (cont.)
The volatility of an option:
The risk premium of an option:
The Sharpe ratio of an option:
where | . | is the absolute value, : required return on option, : expected return on stock, and r: risk-free rate
option stock
r r( )
Sharpe ratio for call Sharpe ratio for stock
r
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Profit diagrams before maturity
For purchased call option:
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Implied volatility
The volatility of the returns consistent with observed option prices and the pricing model (typically Black-Scholes) One can use the implied volatility from an option with an
observable price to calculate the price of another option on the same underlying asset
Checking the uniformity of implied volatilities across various options on the same underlying assets allows one to verify the validity of the pricing model in pricing those options
In practice implied volatilities of in, at, and out-of-the money options are generally different resulting in the volatility skew
Implied volatilities of puts and calls with same strike and time to expiration must be the same if options are European because of put-call parity
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Implied volatility (cont.)
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Perpetual American options
Perpetual American options (options that never expire) are optimally exercised when the underlying asset ever reaches the optimal exercise barrier H* (if = 0, H* = infinity) For a perpetual call option: and
For a perpetual put option: and
where and
H Kh
h*
1
1 1Pr ice
K
h
h
h
S
K
h
1
1
11
1 1
hr r r
1 2 2
2
2
1
2
1
2
2
H Kh
h*
2
2 1Pr ice
K
h
h
h
S
K
h
1
1
2
2
2
2
hr r r
2 2 2
2
2
1
2
1
2
2