Stochastic Calculus and The Black-Scholes Model

Stochastic Calculus and The Black-Scholes Model

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 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 r1 with probability of p1 or r2 with probability of p2. The arithmetic rate of return: r1*p1+r2*p2 The geometric rate of return: (1+r1)^p1*(1+r2)^p2-1

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.

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 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 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 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 recognize 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 e2Dt Term

Taking Limits

Differentiation in 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 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 – 1/2sigma^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.

The Black-Scholes Model

Randomness matters in nonlinearity
An call option with strike price of 10. Suppose the expected value of a stock at call option’s maturity is 10. If the stock price has 50% chance of ending at 11 and 50% chance of ending at 9, the expected payoff is 0.5. If the stock price has 50% chance of ending at 12 and 50% chance of ending at 8, the expected payoff is 1.

Applying Ito’s Lemma, we can find
Therefore, the geometric rate of return is r-0.5sigma^2. The arithmetic rate of return is r

The history of option pricing models
1900, Bachelier, the purpose, risk management 1950s, the discovery of Bachelier’s work 1960s, Samuelson’s formula, which contains expected return Thorp and Kassouf (1967): Beat the market, long stock and short warrant 1973, Black and Scholes

The influence of Beat the Market
Practical experience is not merely the ultimate test of ideas; it is also the ultimate source. At their beginning, most ideas are dimly perceived. Ideas are most clearly viewed when presented as abstractions, hence the common assumption that academics --- who are proficient at presenting and discussing abstractions --- are the source of most ideas. (p. 6, Treynor, 1973) (quoted in p. 49)

Why Black and Scholes Jack Treynor, developed CAPM theory
CAPM theory: Risk and return is the same thing Black learned CAPM from Treynor. He understood return can be dropped from the formula

Fischer Black (1938 – 1995 ) Start undergraduate in physics
Transfer to computer science Finish PhD in mathematics Looking for something practical Join ADL, meet Jack Treynor, learn finance and economics Developed Black-Scholes Move to academia, in Chicago then to MIT Return to industry at Goldman Sachs for the last 11 years of his life, starting from 1984

Fischer never took a course in either economics or finance, so he never learned the way you were supposed to do things. But that lack of training proved to be an advantage, Treynor suggested, since the traditional methods in those fields were better at producing academic careers than new knowledge. Fischer’s intellectual formation was instead in physics and mathematics, and his success in finance came from applying the methods of astrophysics. Lacking the ability to run controlled experiments on the stars, the astrophysist relies on careful observation and then imagination to find the simplicity underlying apparent complexity. In Fischer’s hands, the same habits of research turned out to be effective for producing new knowledge in finance. (p. 6)

Both CAPM and Black-Scholes are thus much simpler than the world they seek to illuminate, but according to Fischer that’s a good thing, not a bad thing. In a world where nothing is constant, complex models are inherently fragile, and are prone to break down when you lean on them too hard. Simple models are potentially more robust, and easier to adapt as the world changes. Fischer embraced simple models as his anchor in the flux because he thought they were more likely to survive Darwinian selection as the system changes. (p. 14)

John Cox, said it best, ‘Fischer is the only real genius I’ve ever met in finance. Other people, like Robert Merton or Stephen Ross, are just very smart and quick, but they think like me. Fischer came from someplace else entirely.” (p. 17) Why Black is the only genius? No one else can achieve the same level of understanding?

Fischer’s research was about developing clever models ---insightful, elegant models that changed the way we look at the world. They have more in common with the models of physics --- Newton’s laws of motion, or Maxwell’s equations --- than with the econometric “models” --- lists of loosely plausible explanatory variables --- that now dominate the finance journals. (Treynor, 1996, Remembering Fischer Black)

The objective of this course
We will learn Black-Scholes theory. Then we will develop an economic theory of life and social systems from basic physical and economic principles. We will show that the knowledge that helps Black succeed will help everyone succeed. There is really no mystery.

Effect of Variables on Option Pricing
K T  r D c p C P – – + + – – + + + + ? + + + + – – + + – – + +

The Concepts Underlying Black-Scholes
The option price and the stock price depend on the same underlying source of uncertainty We can form a portfolio consisting of the stock and the option which eliminates this source of uncertainty The portfolio is instantaneously riskless and must instantaneously earn the risk-free rate This leads to the Black-Scholes differential equation Thorp and Kassouf (1967): Beat the market, long stock and short warrant. This provided the stimulus for this line of thinking.

The Derivation of the Black-Scholes Differential Equation

The Derivation of the Black-Scholes Differential Equation continued

The Differential Equation
Any security whose price is dependent on the stock price satisfies the differential equation The particular security being valued is determined by the boundary conditions of the differential equation In a forward contract the boundary condition is ƒ = S – K when t =T The solution to the equation is ƒ = S – K e–r (T – t )

The payoff structure When the contract matures, the payoff is
Solving the equation with the end condition, we obtain the Black-Scholes formula

The Black-Scholes Formulas

How they found the solution
The equation had been obtained quite awhile ago. But they could not find a solution for some time. Later they use formulas from others which contains expected rate of return. They set the return to be the risk free rate. That was the formula. It can be solved directly from the equation and the initial condition.

The basic property of Black-Schoels formula

Rearrangement of d1, d2

Properties of B-S formula
When S/Ke-rT increases, the chances of exercising the call option increase, from the formula, d1 and d2 increase and N(d1) and N(d2) becomes closer to 1. That means the uncertainty of not exercising decreases. When σ increase, d1 – d2 increases, which suggests N(d1) and N(d2) diverge. This increase the value of the call option.

Similar properties for put options

Calculating option prices
The stock price is $42. The strike price for a European call and put option on the stock is $40. Both options expire in 6 months. The risk free interest is 6% per annum and the volatility is 25% per annum. What are the call and put prices?

Solution S = 42, K = 40, r = 6%, σ=25%, T=0.5 = =

Solution (continued) =4.7144 =1.5322

The Volatility The volatility of an asset is the standard deviation of the continuously compounded rate of return in 1 year As an approximation it is the standard deviation of the percentage change in the asset price in 1 year

Estimating Volatility from Historical Data
Take observations S0, S1, , Sn at intervals of t years Calculate the continuously compounded return in each interval as: Calculate the standard deviation, s , of the ui´s The historical volatility estimate is:

Implied Volatility The implied volatility of an option is the volatility for which the Black-Scholes price equals the market price The is a one-to-one correspondence between prices and implied volatilities Traders and brokers often quote implied volatilities rather than dollar prices

Causes of Volatility Volatility is usually much greater when the market is open (i.e. the asset is trading) than when it is closed For this reason time is usually measured in “trading days” not calendar days when options are valued

Dividends European options on dividend-paying stocks are valued by substituting the stock price less the present value of dividends into Black-Scholes Only dividends with ex-dividend dates during life of option should be included The “dividend” should be the expected reduction in the stock price expected

Calculating option price with dividends
Consider a European call option on a stock when there are ex-dividend dates in two months and five months. The dividend on each ex-dividend date is expected to be $0.50. The current share price is $30, the exercise price is $30. The stock price volatility is 25% per annum and the risk free interest rate is 7%. The time to maturity is 6 month. What is the value of the call option?

Solution The present value of the dividend is
0.5*exp (-2/12*7%)+0.5*exp(-5/12*7%)=0.9798 S= = , K =30, r=7%, σ=25%, T=0.5 d1=0.0985 d2= c=

Investment strategies and outcomes
With options, we can develop many different investment strategies that could generate high rate of return in different scenarios if we turn out to be right. However, we could lose a lot when market movement differ from our expectation.

Example Four investors. Each with 10,000 dollar initial wealth.
One traditional investor buys stock. One is bullish and buys call option. One is bearish and buy put option. One believes market will be stable and sells call and put options to the second and third investors.

Parameters S 20 K R 0.03 T 0.5 sigma 0.3 d1 0.1768 d2 -0.035 c 1.8299
1.5321

Number of call options the second investor buys
10000/ = Number of put options the second investor buys 10000/ =

Final wealth for four investors with different levels of final stock price.
20 15 30 First investor 10000 7500 15000 Second investor Third investor 32635 Fourth investor 30000 -2635

American Calls An American call on a non-dividend-paying stock should never be exercised early Theoretically, what is the relation between an American call and European call? Which one customers prefer? Why? An American call on a dividend-paying stock should only ever be exercised immediately prior to an ex-dividend date

Put-Call Parity; No Dividends
Consider the following 2 portfolios: Portfolio A: European call on a stock + PV of the strike price in cash Portfolio C: European put on the stock + the stock Both are worth MAX(ST , K ) at the maturity of the options They must therefore be worth the same today This means that c + Ke -rT = p + S0

An alternative way to derive Put-Call Parity
From the Black-Scholes formula

Arbitrage Opportunities
Suppose that c = S0 = 31 T = r = 10% K = D = 0 What are the arbitrage possibilities when p = 2.25 ? p = 1 ?

Application to corporate liabulities
Black, Fischer; Myron Scholes (1973). "The Pricing of Options and Corporate Liabilities

Put-Call parity and capital structure
Assume a company is financed by equity and a zero coupon bond mature in year T and with a face value of K. At the end of year T, the company needs to pay off debt. If the company value is greater than K at that time, the company will payoff debt. If the company value is less than K, the company will default and let the bond holder to take over the company. Hence the equity holders are the call option holders on the company’s asset with strike price of K. The bond holders let equity holders to have a put option on their asset with the strike price of K. Hence the value of bond is

Value of debt = K*exp(-rT) – put
Asset value is equal to the value of financing from equity and debt Asset = call + K*exp(-rT) – put Rearrange the formula in a more familiar manner call + K*exp(-rT) = put + Asset

Example A company has 3 million dollar asset, of which 1 million is financed by equity and 2 million is finance with zero coupon bond that matures in 5 years. Assume the risk free rate is 7% and the volatility of the company asset is 25% per annum. What should the bond investor require for the final repayment of the bond? What is the interest rate on the debt?

equity financing 1 million debt financing 2 total asset 3 debt maturity 5 years risk free rate 7% volatility 25%

S 3 K R 0.07 T 5 sigma 0.25 c 1 p value of debt 2 debt rate

Discussion From the option framework, the equity price, as well as debt price, is determined by the volatility of individual assets. From CAPM framework, the equity price is determined by the part of volatility that co-vary with the market. The inconsistency of two approaches has not been resolved.

Homework1 The stock price is $50. The strike price for a European call and put option on the stock is $50. Both options expire in 9 months. The risk free interest is 6% per annum and the volatility is 25% per annum. If the stock doesn’t distribute dividend, what are the call and put prices?

Homework2 Three investors are bullish about Canadian stock market. Each has ten thousand dollars to invest. Current level of S&P/TSX Composite Index is The first investor is a traditional one. She invests all her money in an index fund. The second investor buys call options with the strike price at The third investor is very aggressive and invests all her money in call options with strike price at Suppose both options will mature in six months. The interest rate is 4% per annum, compounded continuously. The implied volatility of options is 15% per annum. For simplicity we assume the dividend yield of the index is zero. If S&P/TSX index ends up at 12000, and respectively after six months. What is the final wealth of each investor? What conclusion can you draw from the results?

Homework3 The price of a non-dividend paying stock is $19 and the price of a 3 month European call option on the stock with a strike price of $20 is $1. The risk free rate is 5% per annum. What is the price of a 3 month European put option with a strike price of $20?

Homework4 A 6 month European call option on a dividend paying stock is currently selling for $5. The stock price is $64, the strike price is $60. The risk free interest rate is 8% per annum for all maturities. What opportunities are there for an arbitrageur?

Homework5 Use Excel to demonstrate how the change of S, K, T, r and σ affect the price of call and put options. If you don’t know how to use Excel to calculate Black-Scholes option prices, go to COMM423 syllabus page on my teaching website and click on Option calculation Excel sheet

Homework6 A company has 3 million dollar asset, of which 1 million is financed by equity and 2 million is finance with zero coupon bond that matures in 10 years. Assume the risk free rate is 3% and the volatility of the company asset is 25% per annum. What should the bond investor require for the final repayment of the bond? What is the interest rate on the debt? How about the volatility of the company asset is 35%?

Homework 7 The asset values of companies A, B are both at 1000 million dollars. Each companies is purely financed with equity, with100 million shares outstanding. The stock prices of companies A, B are both at 10 dollars per share. Company A provides its CEO 1 million shares. What is the value of these shares?

Homework 7 Company B provides its CEO call options on 5 million shares with strike price at 10 dollars and a maturity of 5 years. Assume the risk free rate is 2% per annum and the volatility is 25% per annum. Please calculate the total value of the call options. Is the value of the option provided to the CEO of company B higher or lower than the value of shares provided to the CEO of company A?