Stochastic
Calculus and
Model of the
Behavior
of Stock Prices
What is calculus?
• To obtain long term estimation from short
term information
Why we need stochastic calculus
• Many important results can not be
obtained by simple averaging.
• Some examples
Investment Return
• Suppose a portfolio either gain 60% with
50% probability or lose 60% with 50%
probability. What is its average rate of
return? What is the most likely value of the
portfolio after 10 years?
Solution
• The average rate of return is
0.6*50% + (-0.6)*50% = 0
• The most likely value of the portfolio after
10 years is therefore 1000. Right?
• Let’s calculate
• 1000*(1+0.6)^5*(1-0.6)^5 = 107.4
• This is much less than 1000.
Examples
• 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?
• Rewards will be given to Olympic medalists
according to the formula 1/x^2 million dollars,
where x is the rank of an athlete in an event.
How much rewards Amy and Betty will get?
Solution
• Amy will get
1
2  2  0.5 million
2
• Betty will get
1
1
 1.11 million
2
3
Discussion
• Although the average ranks of Amy and
Betty are the same, the rewards are not
the same.
Resource and reproductive
success of males and females
• Reproductive successes are closely
related to the amount of resources one
controls. While both male and female
animals benefit from more resources, their
precise relations are different.
• Assume the relation between the amount of
resource under control and the number of
offspring for female is n = 1.52x^0.5 and for
male is n = 0.45x^2, where x is the amount of
resources and n is the number of offspring.
Suppose x can take the discrete values 1, 2 and
3. We further assume the probability of average
numbers of females and males who obtain
resources 1, 2 and 3 are {1/3, 1/3, 1/3}. What
are the expected number of offspring for females
and males? (To be continued)
• Suppose genetic systems can alter the ratios of males
and females who can obtain different amount of
resources. Specifically, under one type of genetic
regulation, the new probability distributions for females
and males to obtain resources of amount 1, 2 and 3 is
{1/3-1/8, 1/3+1/4, 1/3-1/8} and under another type of
genetic regulation, the new probability distributions for
females and males to obtain resources of amount 1, 2
and 3 is {1/3+1/8, 1/3-1/4, 1/3+1/8} . What are the new
average numbers of offspring for females and males?
How genetic systems achieve such regulation?
Solution
• The expected number of offspring for females
is
1/ 31.521  1/ 31.52 2  1/ 31.52 3  2.1
• The expected number of offspring for males is
1 / 3  0.451  1 / 3  0.45 22  1 / 3  0.45 32  2.1
• When female and male distribution is revised to
{1/3-1/8, 1/3+1/4, 1/3-1/8}, the new expected
number of offspring by females and males are
2.12 and 1.99 respectively.
• When female and male distribution is revised to
{1/3+1/8, 1/3-1/4, 1/3+1/8}, the new expected
number of offspring by females and males are
2.08 and 2.21 respectively.
• Female achieve higher reproductive success
with lower variability. Male achieve higher
reproductive success with higher variability.
• The regulation is achieved with the genetic
structure of XX in females and XY in males.
Pairing of XX makes genetic systems stable,
which reduces variance of the systems.
• This is why males are more divergent than
females.
Square root function, concave
Square function, convex
General discussion
• One function is concave while the other
function is convex, which are defined by
the signs of the second order derivatives.
• This means that we need to study second
order derivatives when dealing with
stochastic functions.
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?
Homework 3
• 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 magazine. 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?
Homework 4
• A portfolio has 60% probability to gain
50% per year and 40% probability to lose
45% per year. Please calculate the
arithmetic rate of return and geometric rate
of return of the portfolio. The initial value of
the portfolio is 1 million dollar. After ten
years, what is the most likely value of this
portfolio?