Title: Stochastic Calculus and Model of the Behavior of Stock Prices
1Stochastic Calculus and Model of the Behaviorof
Stock Prices
2Why we need stochastic calculus
- Many important issues can not be averaged out.
- Some examples
3Examples
- 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?
4Examples
- Fancy and Mundane each manage two new mutual
funds. Last year, fancys funds got returns of
30 and -10, while Mundanes funds got 11 and
9. One of Fancys 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. Mundanes funds didnt 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?
5Scientific 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 Newtons laws, calculus on deterministic
curves
6Scientific and social background (Continued)
- Modern astronomy, with its success in explaining
the planetary movement, conquered the mind of
people to today. - 1870s
- 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
7Mathematical 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.
8The 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
9The 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
10Itos Lemma
- If we know the stochastic process followed by x,
Itos 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, Itos lemma
plays an important part in the analysis of
derivative securities - Why it is called a lemma?
-
11The Question
12Taylor Series Expansion
- A Taylors series expansion of G(x, t) gives
13Ignoring Terms of Higher Order Than dt
14Substituting for dx
15The e2Dt Term
16Taking Limits
17Differentiation is stochastic and deterministic
calculus
- Ito Lemma can be written in another form
- In deterministic calculus, the differentiation is
18The 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.
19A Process for Stock Prices
- where m is the expected return s is the
volatility. - The discrete time equivalent is
20Application of Itos Lemmato a Stock Price
Process
21Examples
22Expected return and variance
- A stocks 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 1/2sigma2? Compare it with the
geometric return. - Which number arithmetic return or geometric
return is more relevant to investors?
23Answer
- Arithmetic mean 12.67
- Geometric mean 9.11
- Variance 7.23
- Arithmetic mean -1/2variance 9.05
- Geometric mean is more relevant because long term
wealth growth is determined by geometric mean.
24Arithmetic 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- 0.5sigma2 yields 0.12, which is
close to the geometric mean.
25Homework 1
- The returns of a mutual fund in the last five
years are - What is the arithmetic mean of the return? What
is the geometric mean of the return? What is - where mu is arithmetic mean and sigma is standard
deviation of the return series. What conclusion
you will get from the results?
30 25 35 -30 25
26Homework 2
- Rewards will be given to Olympic medalists
according to the formula 1/x2, 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 Amys performance is more
volatile than Bettys. 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 Itos
lemma?