13.14
The Derivation of the Black-Scholes Differential Equation continued
The value of the portfolio is given by
ƒ f S S
The change in its value in time Dt is given by
DS mDt , s Dt S


where m is expected return and s is volatility
13.2
The Lognormal Property
(Equations 13.2 and 13.3, page 282)

It follows from this assumption that
This is because
ln[E(ST / S0 )]
are not the same
and
E[ln(ST / S0 )]
13.6
ln(S0 ) mT E ln(ST ) mT E ln(ST / S0 )
1 ST x= ln T S0
13.7
m and m−s2/2
c0
13.21
13.19
Properties of Black-Scholes Formula



As S0 becomes very large c tends to S0– Ke-rT and p tends to zero As S0 becomes very small c tends to zero and p tends to Ke-rT – S0 What happens when the volatility s approaches zero?
ST S0 e or
xT
ST 1 x = ln T S0 or x s2 m , 2
s2 ln ST ln S0 ~ m T , s T 2
s
T
13.5
The Expected Return

The expected value of the stock price is S0emT The expected return on the stock is m – s2/2 not m
13.13
The Derivation of the Black-Scholes Differential Equation
DS mS Dt sS Dz ƒ ƒ 2 ƒ 2 2 ƒ Dƒ S mS t ½ S 2 s S Dt S sS Dz We set up a portfolio consisting of 1 : derivative ƒ + : shares S
13.8


Mutual Fund Returns (See Business
Snapshot 13.1 on page 285)



Suppose that returns in successive years are 15%, 20%, 30%, -20% and 25% The arithmetic mean of the returns is 14% （15%+20%+30%-20%+25%）/5=14% The returned that would actually be earned over the five years (the geometric mean) is 12.4% 1*1.15*1.2*1.3*0.8*1.25=1*1.1245
S



If a stock price is $50 and its volatility is 25% per year what is the standard deviation of the price change in one day? 1
25% 252
13.10
Estimating Volatility from Historical Data (page 286-88)
ƒ = S – K e–r (T – t ) f f 1 2 2 2f rS s S rf 2 t S 2 S
13.17
The Black-Scholes Formulas
(See pages 295-297)
c S 0 N (d1 ) K e pKe
rT
rT
f f 1 2 2 f rS s S rf 2 t S 2 S
2
13.16
The Differential Equation




Any security whose price is dependent on the stock price satisfies the differential equation The particular security being valued is determined by the boundary conditions of the differential equation In a forward contract the boundary condition is ƒ = S – K when t =T The solution to the equation is
13.12
The Concepts Underlying BlackScholesce and the stock price depend on the same underlying source of uncertainty We can form a portfolio consisting of the stock and the option which eliminates this source of uncertainty The portfolio is instantaneously riskless and must instantaneously earn the risk-free rate This leads to the Black-Scholes differential equation


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
13.3
The Lognormal Distribution
0
E ( ST ) S0 e mT var( ST ) S 0 e
2 2 mT
(e
s 2T
1)
Continuously Compounded Return, x
Equations 13.6 and 13.7), page 283)
13.20

The stock price will grow at rate r, at time T the payoff from a call option is Discounting at rate r, the value of the call today is consider consider
s2 ln ST ln S0 ~ m 2 or s2 ln ST ~ ln S0 m 2 T , s T T , s T

Since the logarithm of ST is normal, ST is lognormally distributed
Suppose we have daily data for a period of several months m is the average of the returns in each day [=E(DS/S)] m−s2/2 is the expected return over the whole period covered by the data measured with continuous compounding (or daily compounding, which is almost the same)
N (d 2 )
N (d 2 ) S 0 N (d1 )
2 ln( S 0 / K ) (r s / 2)T where d1 s T 2 ln( S 0 / K ) (r s / 2)T d2 d1 s T s T
13.18
The N(x) Function
13.9
The Volatility

s2 s x m , 2 T

The volatility is the standard deviation of the continuously compounded rate of return in 1 year The standard deviation of the return in time Dt is s Dt DS mDt , s Dt
3. 4.
t
S0
Calculate the standard deviation, s , of the ui´s The historical volatility estimate is: s
t
ˆ s
t
13.11
S1
S2
Sn
Nature of Volatility