期货期权及其衍生品配套课件Ch13

ƒ = S – K e–r (T – t )
Options, Futures, and Other Derivatives, 7th International
Edition, Copyright © John C. Hull 2019
13
The Derivation of the Black-Scholes Differential Equation continued
Thevalueoftheportfolioisgivenby ƒƒ S S
ThechangienitsvalueintimeDt isgivenby DDƒ ƒ DS S
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
Байду номын сангаас
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?
DS mS Dt sS Dz
Dƒ
ƒ S
mS
ƒ t
½
2ƒ S2
s2S2
Dt
ƒ S
sS
Dz
We setupa portfoli oconsi sti nogf
1: derivative
+ ƒ : shares S
Options, Futures, and Other Derivatives, 7th International
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)
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
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The Expected Return
The expected value of the stock price is S0emT The expected return on the stock is m – s2/2 not m
1. Take observations S0, S1, . . . , Sn at intervals of t years
2. Calculate the continuously compounded return in each interval as:
ui
ln
Si
Si1
3. Calculate the standard deviation, s , of the
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4
Continuously Compounded Return
Equations 13.6 and 13.7), page 279)
If x is the continuously compounded return
ST S0 exT
Options, Futures, and Other Derivatives, 7th International
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Estimating Volatility from Historical Data (page 282-84)
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The Lognormal Distribution
E(ST)S0em T var(ST)S02e2m T(es2T1)
Options, Futures, and Other Derivatives, 7th International
Thereturnontheportfoliomustbetherisk- free
rate.Hence
Dr Dt
WesubstituteforăandDS intheseequations
togettheB lack- S choleds i fferenatilequati o:n
ƒ rS ƒ ? t S
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
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
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%
Options, Futures, and Other Derivatives, 7th International
Edition, Copyright © John C. Hull 2019
12
The Derivation of the Black-Scholes Differential Equation
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
Volatility is usually much greater when the market is open (i.e. the asset is trading) than when it is closed For this reason time is usually measured in “trading days” not calendar days when options are valued
σ2S2
2ƒ S2
r?
Options, Futures, and Other Derivatives, 7th International
Edition, Copyright © John C. Hull 2019
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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
Options, Futures, and Other Derivatives, 7th International
Edition, Copyright © John C. Hull 2019
11
The Concepts Underlying BlackScholes
The option price 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
4.
ui´s The historical volatility estimate is:
sˆ s
t
Options, Futures, and Other Derivatives, 7th International
Edition, Copyright © John C. Hull 2019
10
Nature of Volatility
Options, Futures, and Other Derivatives, 7th International
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14
The Derivation of the Black-Scholes Differential Equation continued
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
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