Transcript Model of the Behavior of Stock Prices Chapter 12

Stochastic
Calculus and
Model of the
Behavior
of Stock Prices
Why we need stochastic calculus
• Many important issues can not be
averaged out.
• Some examples
Examples
• Adam and Benny took two course. Adam got two
Cs. Benny got one B and one D. What are their
average grades? Who will have trouble
graduating?
• Amy and Betty are Olympic athletes. Amy got
two silver medals. Betty got one gold and one
bronze. What are their average ranks in two
sports? Who will get more attention from media,
audience and advertisers?
Examples
• Fancy and Mundane each manage two new mutual
funds. Last year, fancy’s funds got returns of 30% and 10%, while Mundane’s funds got 11% and 9%. One of
Fancy’s fund was selected as “One of the Best New
Mutual Funds” by a finance journal. As a result, the size
of his fund increased by ten folds. The other fund
managed by Fancy was quietly closed down. Mundane’s
funds didn’t get any media coverage. The fund sizes
stayed more or less the same. What are the average
returns of funds managed by Fancy and Mundane? Who
have better management skill according to CAPM?
Which fund manager is better off?
Scientific and social background
• Dominant thinkings of the time are like the
air around us. Our minds absorb them all
the time, conscious or not.
• Modern astronomy
– Copernicus: Sun centered universe
– Kepler: Three laws, introducing physics into
astronomy
– Newton: Newton’s laws, calculus on
deterministic curves
Scientific and social background
(Continued)
• Modern astronomy, with its success in explaining
the planetary movement, conquered the mind of
people to today.
• 1870’s
• The birth of neo-classical economics
– Gossen: 1854 published his book, died 1858
– Jevons: 1871 (1835-1882)
– Walras: 1873
• The birth of statistical physics
– Boltzmann in 1870s: Random movement can be understood
analytically
Mathematical derivatives and
financial derivatives
• Calculus is the most important intellectual
invention. Derivatives on deterministic variables
• Mathematically, financial derivatives are
derivatives on stochastic variables.
• In this course we will show the theory of financial
derivatives, developed by Black-Scholes, will
lead to fundamental changes in the
understanding social and life sciences.
The history of stochastic calculus
and derivative theory
• 1900, Bachelier: A student of Poincare
– His Ph.D. dissertation: The Mathematics of Speculation
– Stock movement as normal processes
– Work never recognized in his life time
• No arbitrage theory
– Harold Hotelling
• Ito Lemma
– Ito developed stochastic calculus in 1940s near the end of WWII,
when Japan was in extreme difficult time
– Ito was awarded the inaugural Gauss Prize in 2006 at
age of 91
The history of stochastic calculus
and derivative theory (continued)
• Feynman (1948)-Kac (1951) formula,
• 1960s, the revival of stochastic theory in
economics
• 1973, Black-Scholes
– Fischer Black died in 1995, Scholes and Merton were
awarded Nobel Prize in economics in 1997.
• Recently, real option theory and an analytical
theory of project investment inspired by the
option theory
• It often took many years for people to recognize
the importance of a new heory
Ito’s Lemma
• If we know the stochastic process
followed by x, Ito’s lemma tells us the
stochastic process followed by some
function G (x, t )
• Since a derivative security is a function of
the price of the underlying and time, Ito’s
lemma plays an important part in the
analysis of derivative securities
• Why it is called a lemma?
The Question
Suppose
dx  a( x, t )dt  b( x, t )dz
How G(x, t) changes with the change of x and t?
Taylor Series Expansion
• A Taylor’s series expansion of G(x, t)
gives
G
G
 2G 2
G 
x 
t  ½ 2 x
x
t
x
 2G
 2G 2

x t  ½ 2 t  
xt
t
Ignoring Terms of Higher Order
Than t
In ordinary calculus w ehave
G
G
G 
x 
t
x
t
In stochastic calculus this becomes
G
G
 2G 2
G 
x 
t  ½
x
2
x
t
x
because x has a component w hichis
of order t
Substituting for x
Suppose
dx  a( x, t )dt  b( x, t )dz
so that
x = a t + b  t
Then ignoring terms of higher order than t
G
G
G 2 2
G 
x 
t  ½ 2 b  t
x
t
x
2
The 2Dt Term
Since   (0,1) E ()  0
E ( 2 )  [ E ()]2  1
E ( )  1
2
It follow sthat E ( 2 t )  t
The variance of t is proportional to t and can
2
be ignored. Hence
G
G
1 G 2
G 
x 
t 
b t
2
x
t
2 x
2
Taking Limits
Taking limits
G
G
2G 2
dG 
dx 
dt  ½ 2 b dt
x
t
x
Substituting
dx  a dt  b dz
We obtain
 G
G
 2G 2 
G
dG  
a
 ½ 2 b  dt 
b dz
t
x
x
 x

This is Ito's Lemma
Differentiation is stochastic and
deterministic calculus
• Ito Lemma can be written in another form
G
G
1  2G 2
dG 
dx 
dt 
b dt
2
x
t
2 x
• In deterministic calculus, the differentiation
is
G
G
dG 
dx 
dt
x
t
The simplest possible model of
stock prices
• Over long term, there is a trend
• Over short term, randomness dominates.
It is very difficult to know what the stock
price tomorrow.
A Process for Stock Prices
dS  mSdt  sSdz
where m is the expected return s is
the volatility.
The discrete time equivalent is
S  mSt  sS t
Application of Ito’s Lemma
to a Stock Price Process
The stock price process is
d S  mS dt  sS d z
For a f unction G of S and t
 G
G
 2G 2 2 
G


dG  
mS 
 ½ 2 s S dt 
sS dz
t
S
S
 S

Examples
1. T heforwardpriceof a stock for a contract
maturingat timeT
G  S e r (T t )
dG  ( m  r )G dt  sG dz
2. G  ln S

s2 
dt  s dz
dG   m 
2 

Expected return and variance
• A stock’s return over the past six years are
• 19%, 25%, 37%, -40%, 20%, 15%.
• Question:
–
–
–
–
What is the arithmetic return
What is the geometric return
What is the variance
What is mu – 1/2sigma^2? Compare it with the
geometric return.
– Which number: arithmetic return or geometric return
is more relevant to investors?
Answer
•
•
•
•
•
Arithmetic mean: 12.67%
Geometric mean: 9.11%
Variance: 7.23%
Arithmetic mean -1/2*variance: 9.05%
Geometric mean is more relevant because
long term wealth growth is determined by
geometric mean.
Arithmetic mean and geometric
mean
• The annual return of a mutual fund is
• 0.15 0.2 0.3 -0.2 0.25
• Which has an arithmetic mean of 0.14 and
geometric mean of 0.124, which is the true
rate of return.
• Calculating r- 0.5*sigma^2 yields 0.12,
which is close to the geometric mean.
Homework 1
• The returns of a mutual fund in the last five years are
30%
25%
35%
-30%
25%
• What is the arithmetic mean of the return? What is
the geometric mean of the return? What is

s2 
 m 


2 
• where mu is arithmetic mean and sigma is standard
deviation of the return series. What conclusion you
will get from the results?
Homework 2
• Rewards will be given to Olympic medalists
according to the formula 1/x^2, where x is the
rank of an athlete in an event. Suppose Amy and
betty are expected to reach number 2 in their
competitions. But Amy’s performance is more
volatile than Betty’s. Specifically, Amy has (0.3,
0.4, 03) chance to get gold, silver and bronze
while Betty has (0.1, 0.8,0.1) respectively. How
much rewards Amy and Betty are expected to
get? Can we calculate them from Ito’s lemma?