Stochastic Calculus and Model of the Behavior of

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 • The initial value of a portfolio is 1000 dollars. Suppose the 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.

The value distribution for the first four years

Discussion • The arithmetic mean of value is always 1000 dollars each year. But high end, as well as low end, values are extremely rare. • The most likely value decline over time.

Arithmetic mean and geometric mean • Suppose a portfolio either change r 1 with probability of p 1 or r 2 with probability of p 2. • The arithmetic rate of return: r 1*p 1+r 2*p 2 • The geometric rate of return: (1+r 1)^p 1*(1+r 2)^p 2 -1

• The geometric rate of return of the last portfolio • (1+0. 6)^0. 5*(1 -0. 6)^0. 5 -1= -0. 2 • The value of portfolio after 10 years, calculated from geometric rate of return • 1000*(1 -0. 2)^10 = 107. 4 • The same as earlier calculation • The value of portfolio after 4 years, calculated from geometric rate of return • 1000*(1 -0. 2)^4 = 409. 6 • The same as the middle number in year 4.

Another Investment Problem • The initial value of a portfolio is 1000 dollars. Suppose the portfolio either gain 100% with 50% probability or lose 50% with 50% probability. What is its average rate of return? What is the probability that value of the portfolio will be at or higher than the expected value, i. e. , initial value compounded with average rate of return for 4 years, after 4 years?

Solution • The average rate of return is 1. 0*50% + (-0. 5)*50% = 0. 25 • The expected value after 4 years • 1000*(1+0. 25)^4 = 2441. 4

The value distribution for the first four years 16000 8000 4000 2000 1000 500 250 125 62. 5

• What is the probability that the value will be higher than the expected value? • 1/16+4*1/16 = 5/16 • This is much less than half

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 • Betty will get

Discussion • Although the average ranks of Amy and Betty are the same, the expected rewards are not the same.

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 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 1940 s 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, • 1960 s, the revival of stochastic theory in economics • Thorp, E. O. , & Kassouf, S. T. (1967). Beat the market: a scientific stock market system. • 1973, Black-Scholes – Black, F. and Scholes, M. (1973). The Pricing of Options and Corporate Liabilities – Fischer Black died in 1995, Scholes and Merton were awarded Nobel Prize in economics in 1997.

The history of stochastic calculus and derivative theory (continued) • Recently, real option theory and an analytical theory of project investment inspired by the option theory • It often took many years for people to acknowledge and accept the importance of a new theory

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 dt

Substituting for dx

The e 2 Dt Term

Taking Limits

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

Comments • If the second order derivatives with respect to x is positive, we the function G is convex, if negative, the function is concave.

Sports and Education • In sports, if you are excellent in one item, you will be very happy. If you are an NHL player, you don’t worry about your basketball skill. • In education, if you failed one course, you cannot graduate, even if you do great on other subjects.

• We say the reward system in sports in convex, or risk taking, reward system in education is concave, or risk avoiding. Which one is better? • They fit different purposes. The real purpose of education is to train competent workers, although the advertised purpose is much loftier. • We often claim to encourage risk taking. But the whole educational system itself is based on risk avoiding.

• What we can do about it when we go through the educational system? • Try to enjoy what we learn. While the way we are educated is not very adventurous, most educational contents were created in very adventurous ways. The life and experience of many scientific pioneers were far from boring. If we understand more about what we learn, we often find the contents are fascinating. • If we really don’t like what we learn, don’t try to get A in every subject. Just spend enough effort so we won’t fail. We can concentrate more on what we like.

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

Application of Ito’s Lemma to a Stock Price Process

Examples

Arithmetic mean and geometric mean • The initial price of a stock is 1000 dollars. The stock’s return over the past six years are • 19%, 25%, 37%, -40%, 20%, 15%.

Questions – – What is the arithmetic return What is the geometric return What is the variance What is mu – 1/2 sigma^2? Compare it with the geometric return. – What is the final price of the stock? – What are the final prices of the stock calculated from arithmetic and geometric rate of returns? – 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.

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 • 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?

Homework 5 • A portfolio has 60% probability to gain 40% per year and 40% probability to lose 60% per year. Please list all possible values of the portfolio in four years. Calculate the arithmetic rate of return and geometric rate of return of the portfolio. The initial value of the portfolio is 1000 dollar. After four years, what is the most likely value of this portfolio?

Homework 6 • There are two stocks. The first one has expected return at 0. 12 and standard deviation of 0. 4. The second one has expected return at 0. 1 and standard deviation of 0. 2. Suppose the initial investment is 1000 dollars for each stock. After 10 years, what are the most likely outcome for each investment?

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. 52 x^0. 5 and for male is n = 0. 45 x^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}. 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 • The expected number of offspring for males is

• 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.