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 ( ) –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 91Gauss Prize

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 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

Taylor Series Expansion A Taylor’s series expansion of G ( x, t ) gives

Ignoring Terms of Higher Order Than  t

Substituting for  x

The  2  t Term

Taking Limits

Differentiation is stochastic and deterministic calculus Ito Lemma can be written in another form In deterministic calculus, the differentiation is

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 where  is the expected return  is the volatility. The discrete time equivalent is

Application of Ito’s Lemma to a Stock Price Process

Examples

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 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 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%

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?