Black Scholes Option Pricing Model
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Transcript of Black Scholes Option Pricing Model
Black Scholes Black Scholes Option Pricing Option Pricing
ModelModel Finance (Derivative Securities) 312
Tuesday, 10 October 2006
Readings: Chapter 12
Random Walk Random Walk AssumptionAssumption
Consider a stock worth SIn a short period of time, t, change in
stock price is assumed to be normal with mean St and standard deviation
where is expected return, is volatility
tS
Lognormal PropertyLognormal Property
Implies that ln ST is normally distributed with mean:
and standard deviation:
Since logarithm of ST is normal, ST is lognormally distributed
TS )2/(ln 20
T
Lognormal PropertyLognormal Property
where Φ[m,s] is a normal distribution with mean m and standard deviation s
TTS
S
TTSS
T
T
,)2(ln
or
,)2(lnln
2
0
20
Lognormal DistributionLognormal Distribution
E S S e
S S e e
TT
TT T
( )
( ) ( )
0
02 2 2
1
var
Expected ReturnExpected Return
Expected value of stock price is S0eT
Expected return on stock with continuous compounding is – 2/2
Arithmetic mean of returns over short periods of length t is
Geometric mean of returns is – 2/2
VolatilityVolatility
Volatility is standard deviation of the continuously compounded rate of return in one year
Standard deviation of return in time t:
If stock price is $50 and volatility is 30% per year what is the standard deviation of the price change in one week?• 30 x √(1/52) = 4.16% => 50 x 0.0416 = $2.08
t
Estimating VolatilityEstimating Volatility
Using historical data:• Take observations S0, S1, . . . , Sn at intervals of years
• Define the continuously compounded return as:
• Calculate the standard deviation, s , of the ui values
• Historical volatility estimate is:
uS
Sii
i
ln1
sˆ
Concepts Underlying Black Concepts Underlying Black ScholesScholes
Option price and stock price depend on the same underlying source of uncertainty
Can form a portfolio consisting of the stock and option which eliminates this source of uncertainty• Portfolio is instantaneously riskless and must
instantaneously earn the risk-free rate
Black Scholes FormulaeBlack Scholes Formulae
TdT
TrKSd
T
TrKSd
dNSdNeKp
dNeKdNScrT
rT
10
2
01
102
210
)2/2()/ln(
)2/2()/ln(
)()(
)()(
where
NN((xx) ) TablesTables
N(x) is the probability that a normally distributed variable with a mean of zero and a standard deviation of 1 is less than x
See tables at the end of the book
Black Scholes PropertiesBlack Scholes Properties
P
A
As S0 becomes very large c tends to
S – Ke–rT and p tends to zero
As S0 becomes very small c tends to zero and p tends to Ke–rT – S
Black Scholes ExampleBlack Scholes Example
Suppose that:• Stock price in six months from expiration of
option is $42• Exercise price of option is $40• Risk-free rate is 10%, volatility is 20% p.a.
What is the price of the option?
Black Scholes ExampleBlack Scholes Example
Using Black Scholes:• d1 = 0.7693, d2 = 0.6278
• Ke–rT = 40e–0.1(0.5) = 38.049 • If European call, c = 4.76• If European put, p = 0.81
Thus, stock price must:• rise by $2.76 for call holder to breakeven• fall by $2.81 for put holder to breakeven
Risk-Neutral ValuationRisk-Neutral Valuation
does not appear in the Black-Scholes equation• equation is independent of all variables affected by
risk preference, consistent with the risk-neutral valuation principle
Assume the expected return from an asset is the risk-free rate
Calculate expected payoff from the derivative Discount at the risk-free rate
Application to ForwardsApplication to Forwards
Payoff is ST – K
Expected payoff in a risk-neutral world is SerT – K
Present value of expected payoff is
e–rT(SerT – K) = S – Ke–rT
Implied VolatilityImplied Volatility
Implied volatility of an option is the volatility for which the Black-Scholes price equals the market price
One-to-one correspondence between prices and implied volatilities
Traders and brokers often quote implied volatilities rather than dollar prices
Effect of DividendsEffect of Dividends
European options on dividend-paying stocks valued by substituting stock price less PV of dividends into Black-Scholes formula
Only dividends with ex-dividend dates during life of option should be included
“Dividend” should be expected reduction in the stock price expected
Effect of DividendsEffect of Dividends
Suppose that:• Share price is $40, ex-div dates in 2 and 5
months, with dividend value of $0.50• Exercise price of a European call is $40,
maturing in 6 months• Risk-free rate is 9%, volatility is 30% p.a.
What is the price of the call?
Effect of DividendsEffect of Dividends
PV of dividends• 0.5e–0.09(2/12) + 0.5e–0.09(5/12) = 0.9741
• S0 is 39.0259
d1 = 0.2017, N(d1) = 0.5800
d2 = –0.0104, N(d2) = 0.4959
c = 39.0259 x 0.5800 – 40e–0.09(0.5) x 0.4959
= $3.67
American CallsAmerican Calls
American call on non-dividend-paying stock should never be exercised early
American call on dividend-paying stock should only ever be exercised immediately prior to an ex-dividend date
Set American price equal to maximum of two European prices:• The 1st European price is for an option maturing at
the same time as the American option• The 2nd European price is for an option maturing just
before the final ex-dividend date