The Black-ScholesModel

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 average rate of return is r-0.5sigma^2. (But there could be problem because of the last term.)

dzrdts

ds

)( )2

1(

0

2tztr

eeSS

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

• 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

c p C PVariable

S0

KTrD

+ + –+

? ? + ++ + + ++ – + –

–– – +

– + – +

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

The Derivation of the Black-Scholes Differential Equation

shares :ƒ

+

derivative :

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ƒƒ

½ƒƒ

ƒ

S

zSS

tSSt

SS

zStSS

1

222

2

ƒ

ƒ

by given is time in value its in change The

ƒ

ƒ

by given is portfolio the of value The

SS

t

SS

The Derivation of the Black-Scholes Differential Equation continued

The Derivation of the Black-Scholes Differential Equation continued

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:equation aldifferenti Scholes-Black the get to

equations these in and ƒ for substitute We

Hence rate.

free-risk the be must portfolio the on return The

rS

SS

rSt

S

tr

2

222

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 isƒ = 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

)0,max()0,( KSSC

The Black-Scholes Formulas

TdT

TrKSd

T

TrKSd

dNSdNeKp

dNeKdNScrT

rT

10

2

01

102

210

)2/2()/ln(

)2/2()/ln( where

)( )(

)( )(

The basic property of Black-Schoels formula

rTKeSC

Rearrangement of d1, d2

TT

Ke

S

d

TT

Ke

S

d

rT

rT

2

1)ln(

2

1)ln(

2

1

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

TTS

Ke

d

TTS

Ke

d

rT

rT

2

1)ln(

2

1)ln(

1

2

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

• = 0.5341

• = 0.3573

T

TrKSd

)2/2()/ln( 0

1

T

TrKSd

)2/2()/ln( 0

2

Solution (continued)

• =4.7144

• =1.5322

)( )( 210 dNeKdNSc rT

)( )( 102 dNSdNeKp rT

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

1. Take observations S0, S1, . . . , Sn at intervals of years

2. Calculate the continuously compounded return in each interval as:

3. Calculate the standard deviation, s , of the ui

´s

4. The historical volatility estimate is:

uS

Sii

i

ln1

sˆ

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=30-0.9798=29.0202, K =30, r=7%, σ=25%, T=0.5

• d1=0.0985

• d2=-0.0782

• c= 2.0682

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.

Parameters

S 20

K 20

R 0.03

T 0.5

sigma 0.3

d1 0.1768

d2 -0.035

c 1.8299

p 1.5321

• Number of call options the second investor buys10000/ 1.8299 = 5464.84

• Number of put options the second investor buys10000/ 1.5321 = 6526.91

Final wealth for four investors with different levels of final stock price.

Final stock price 20 15 30

First investor 10000 7500 15000

Second investor 0 0 54648.4

Third investor 0 32635 0

Fourth investor 30000 -2635 -24648.4

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?– What are the market prices? 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 (Equation 8.3, page 174)

• 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

rTKeS

dSNdNrTKedNrTKedSNPC

)}1

()2

({)2

()1

(

Arbitrage Opportunities• Suppose that

c = 3 S0 = 31

T = 0.25 r = 10%

K =30 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 tructure

• 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 there 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?

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? If the stock is expected to distribute $1.5 dividend after 5 months, 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 12000. 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 12000. The third investor is very aggressive and invests all her money in call options with strike price at 13000. 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, 13500 and 15000 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 and a dividend of $0.80 is expected in 1 month. 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 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? How about the volatility of the company asset is 35%?