Stochastic Calculus and Model of the Behavior of Stock Prices

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Stochastic Calculus and Model of the Behaviorof
Stock Prices
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Why we need stochastic calculus

• Many important issues can not be averaged out.
• Some examples

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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?

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Examples

• 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?

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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 Newtons laws, calculus on deterministic
  curves

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Scientific 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

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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.

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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

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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

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Itos 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?

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The Question
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Taylor Series Expansion

• A Taylors series expansion of G(x, t) gives

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Ignoring Terms of Higher Order Than dt
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Substituting for dx
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The e2Dt Term
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Taking Limits
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Differentiation is stochastic and deterministic
calculus

• Ito Lemma can be written in another form
• In deterministic calculus, the differentiation is
  

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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.

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A Process for Stock Prices

• where m is the expected return s is the
  volatility.
• The discrete time equivalent is

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Application of Itos Lemmato a Stock Price
Process
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Examples
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Expected 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?

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Answer

• 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.

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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- 0.5sigma2 yields 0.12, which is
  close to the geometric mean.

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Homework 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
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Homework 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?