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Consider a stock whose price is S In a short period of time of length Dt, the return on the stock is normally distributed:
DSmDt,s2Dt
S
where m is expected return and s is volatility
x = 1 ln ST
T S0
x
m
s2 2
,
s2 T
Options, Futures, and Other Derivatives, 7th International
Edition, Copyright © John C. Hull 2019
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
The Black-Scholes-Merton Model
Chapter 13
Options, Futures, and Other Derivatives, 7th International Edition,
Copyright © John C. Hull 2019
1
The Stock Price Assumption
This is because ln E (S T [ /S 0 )]an E [d lS T n /S 0 )(]
are not the same
Options, Futures, and Other Derivatives, 7th International Edition, Copyright © John C. Hull
Options, Futures, and Other Derivatives, 7th International
The standard deviation of the return in time Dt is s Dt
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?
Options, Futures, and Other Derivatives, 7th International Edition, Copyright © John C. Hull
2019
2
The Lognormal Property
(Equations 13.2 and 13.3, page 278)
Suppose that returns in successive years are 15%, 20%, 30%, -20% and 25% The arithmetic mean of the returns is 14% The returned that would actually be earned over the five years (the geometric mean) is 12.4%
It follows from this assumption that
ln
ST
ln
S0
m
s2 2
T,
s2T
or
ln ST
ln S0
m
s2 2
T
,
s2T
Since the logarithm of ST is normal, ST is lognormally distributed
Options, Futures, and Other Derivatives, 7th International
Options, Futures, and Other Derivatives, 7th International
Edition, Copyright © John C. Hull 2019
8
The Volatility
The volatility is the standard deviation of the continuously compounded rate of return in 1 year
2019
6
m and m−s2/2
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)
Options, Futures, and Other Derivatives, 7th International
Edition, Copyright © John C. Hull 2019
7
Mutual Fund Returns (See Business
Snapshot 13.1 on page 281)
Edition, Copyright © John C. Hull 2019
3
The LognormaST)S02e2m T(es2T1)
Options, Futures, and Other Derivatives, 7th International
Edition, Copyright © John C. Hull 2019
4
Continuously Compounded Return
Equations 13.6 and 13.7), page 279)
If x is the continuously compounded return
ST S0 exT