Lecture Basic Finance

Basic Finance

Peter Ouwehand

Department of Mathematical SciencesUniversity of Stellenbosch

November 2010

P. Ouwehand (Stellenbosch Univ.) Basic Finance November 2010 1 / 30

What is Finance?

Finance: ”The study of how people allocate scarce resources overtime.”

Individuals and corporations useI Savings accounts;I Mortgages;I Pension funds;I Annuities;I Stock market;

The outcomes — the costs and benefits — of financial decisions areusually:

I spread over time;I uncertain, i.e. subject to risk;

To make intelligent investment and consumption decisions, individualsmust be able to value and compare different risky cashflows overtime.


What is Finance?







What is Finance?


Individuals and corporations use

I Savings accounts;I Mortgages;I Pension funds;I Annuities;I Stock market;





What is Finance?


Individuals and corporations useI Savings accounts;

I Mortgages;I Pension funds;I Annuities;I Stock market;





What is Finance?


Individuals and corporations useI Savings accounts;I Mortgages;

I Pension funds;I Annuities;I Stock market;





What is Finance?


Individuals and corporations useI Savings accounts;I Mortgages;I Pension funds;

I Annuities;I Stock market;





What is Finance?


Individuals and corporations useI Savings accounts;I Mortgages;I Pension funds;I Annuities;

I Stock market;





What is Finance?







What is Finance?







What is Finance?




I spread over time;

I uncertain, i.e. subject to risk;



What is Finance?







What is Finance?







The Three Pillars of Finance

To make investment decisions, individuals must consider thefollowing:

I. Time value of money: Individuals must compare the value ofdifferent payments at different times — R100 today is worth more thanR100 next year.

II. Risk management: Individuals must be able to assess and managethe riskiness of investments.

III. Asset valuation: Individuals must be able to determine and compareasset prices.


























The Time Value of Money 1

Which do you prefer? R1000 in hand today, or the promise of R1000in one year’s time.Why?

I Opportunity cost: You can invest the R1000 now, with theexpectation of receiving a greater sum in the future.

I Inflation: R1000 in one year’s time may buy fewer goods than R1000today.

I Risk/Uncertainty: You can’t be sure that you will actually receive theR1000 in one year’s time.

So borrowing isn’t free: The borrower must pay a premium to inducethe lender to part temporarily with his/her money — the interest.

The interest rate depends on many factors, e.g. inflation, moneysupply, credit rating, etc.


















































Time Value of Money 2

Interest rate modelling is a complex part of mathematical finance, butwe are going to keep things simple:

Definition

If an amount A0 is deposited in a bank account at a simple rate r forone time–period, it will grow to A1 = A0(1 + r).

If the interest is compounded twice per year, then an amount A0 willbe worth:

I A0(1 + r2 ) after 0.5 yr.. . .

I . . . and thus to A0(1 + r2 )(1 + r

2 ) after 1 yr.

If the interest is compounded n times per year, an amount A0 willgrow to A0(1 + r

n )n after one yr.




Definition



I A0(1 + r2 ) after 0.5 yr.. . .

I . . . and thus to A0(1 + r2 )(1 + r

2 ) after 1 yr.


n )n after one yr.




Definition



I A0(1 + r2 ) after 0.5 yr.. . .

I . . . and thus to A0(1 + r2 )(1 + r

2 ) after 1 yr.


n )n after one yr.




Definition



I A0(1 + r2 ) after 0.5 yr.. . .

I . . . and thus to A0(1 + r2 )(1 + r

2 ) after 1 yr.


n )n after one yr.




Definition



I A0(1 + r2 ) after 0.5 yr.. . .

I . . . and thus to A0(1 + r2 )(1 + r

2 ) after 1 yr.


n )n after one yr.




Definition



I A0(1 + r2 ) after 0.5 yr.. . .

I . . . and thus to A0(1 + r2 )(1 + r

2 ) after 1 yr.


n )n after one yr.



If the interest is compounded continuously, an amount A0 will growto limn→∞ A0(1 + r

n )n = A0er after one yr.. . . ,

. . . and thus to A0erT after T yr.

We are now able to compare different payments at different times: Toobtain an amount A in n years time, you must deposit Ae−rT in thebank today, i.e. the present value of the amount A in T years’ time is

A = Ae−rT . . . continuous rate

OR

A =A

(1 + r)T. . . simple rate

etc.








OR

A =A


etc.








OR

A =A


etc.








OR

A =A


etc.


Returns I

Returns are similar to interest rates. The main difference is that aninterest rate is a promised return on a deposit, whereas the returns onother assets are generally uncertain (i.e. risky)

Example: You bought one share of Xcor one year ago for R123.45.Today the share pays a dividend of R12.00 and the share price is nowR135.40. The net income provided by the share is

135.40 + 12.00− 123.45 = 23.95

The investment cost R123.45, so the rate of return is23.95

123.45= 19.4%.

Example: You deposited R123.45 in a bank one year ago. Today, youwithdraw R12.00, and R135.40 remains in your bank account. Whatwas the simple rate of interest?

I Before withdrawal, total was R147.40.I Thus 123.45(1 + r) = 147.40.I And so r = 19.4%


Returns I



135.40 + 12.00− 123.45 = 23.95


123.45= 19.4%.




Returns I



135.40 + 12.00− 123.45 = 23.95


123.45= 19.4%.




Returns I



135.40 + 12.00− 123.45 = 23.95


123.45= 19.4%.




Returns I



135.40 + 12.00− 123.45 = 23.95


123.45= 19.4%.


I Before withdrawal, total was R147.40.

I Thus 123.45(1 + r) = 147.40.I And so r = 19.4%


Returns I



135.40 + 12.00− 123.45 = 23.95


123.45= 19.4%.


I Before withdrawal, total was R147.40.I Thus 123.45(1 + r) = 147.40.

I And so r = 19.4%


Returns I



135.40 + 12.00− 123.45 = 23.95


123.45= 19.4%.




Returns II

Fundamental relationship in finance:

E[Return] = f (Risk)

where f is an increasing function.

Shares are riskier investments than deposits. Thus the expectedreturn on a share should be greater than the interest offered by abank account.

Note that returns can be negative, whereas interest rates must bepositive.

The return on an investment is roughly the percentage by which itsvalue has increased in one year, i.e.

Return =Final Price + Interim Cashflows – Initial Price

Initial Price


Returns II








Initial Price


Returns II








Initial Price


Returns II








Initial Price


Returns II








Initial Price


Returns III

Shares with a higher expected return are therefore riskier than shareswith a low expected return.

I If two shares had the same risk, but different expected returns,everyone would buy the share with the higher return (and short theshare with the lower return).

I This would drive the price of the “high return” share up, thus loweringits return.

The riskiness of a share is measured by a quantity called volatility: Itis the standard deviation of annualized returns.

Returns may also be measured as discretely– or continuouslycompounded.

Returns on bonds are called yields.


Returns III








Returns III








Returns III








Returns III








Returns III








Returns III








Markets and Instruments I

Traders in a financial market exchange securities for money.

Securities are contracts for future delivery of goods or money, e.g.I sharesI bondsI derivatives

One distinguishes between underlying (primary) and derivative(secondary) instruments.

Examples of underlying instruments are shares, bonds, currencies,interest rates, and indexes.

A derivative is a financial instruments whose value is derived from anunderlying asset.

Examples of derivatives are forward contracts, futures, options, swapsand bonds.












Securities are contracts for future delivery of goods or money, e.g.

I sharesI bondsI derivatives








Securities are contracts for future delivery of goods or money, e.g.I shares

I bondsI derivatives








Securities are contracts for future delivery of goods or money, e.g.I sharesI bonds

I derivatives














































Markets and Instruments II

One also distinguishes between primary and secondary markets.Securities are issued for the first time on the primary market, andthen traded on the secondary market. The secondary market providesimportant liquidity.

Borrowing and lending is done in fixed–income markets. The moneymarket is for very short–term debt (maturities ≤ 1 yr.)

Finally, we distinguish between the spot market and the forwardmarket.

I Most transactions are spot transactions: Pay now, and receive goodsnow.

I To hedge/speculate on future market movements, it is possible to sellgoods for delivery in the future. Forward and futures contracts arederivatives which make this possible.





































Markets and Instruments III

Equity: Stocks, shares. Ownership of a small piece of a company.I Shareholders own a corporation. Directors act in the shareholders’

best interest.I Public limited companies are listed on a stock exchange. Ownership is

easily transferred. The shareholders share the profits of the company,but have limited liability: At most, they can lose their investment.

Most shares pay regular dividends, whose amount varies according toprofitability and opportunities for growth.

I A share may be bought cum– or ex–dividend.I On the ex-dividend date, the share price decreases by the amount of

the dividend.

Occasionally a company announces a stock split: Suppose, forexample, that you own a single stock whose current price is R600.00.After a 3–for–1 stock split you will own 3 shares each valued atR200.00.



Equity: Stocks, shares. Ownership of a small piece of a company.

I Shareholders own a corporation. Directors act in the shareholders’best interest.

I Public limited companies are listed on a stock exchange. Ownership iseasily transferred. The shareholders share the profits of the company,but have limited liability: At most, they can lose their investment.



the dividend.





best interest.

I Public limited companies are listed on a stock exchange. Ownership iseasily transferred. The shareholders share the profits of the company,but have limited liability: At most, they can lose their investment.



the dividend.









the dividend.









the dividend.








I A share may be bought cum– or ex–dividend.

I On the ex-dividend date, the share price decreases by the amount ofthe dividend.









the dividend.









the dividend.



Markets and Instruments IV

Short selling: Selling a share you don’t own, hoping to pick them upmore cheaply later on.

I Your broker borrows the share from a client.I You may now sell these shares, even though you don’t own them.I Later, you buy the shares in the market and return them to your

broker, who returns them to the other client. You also pay anydividends that were issued in the interim.

Commodities: Raw materials such as metals, oil, agriculturalproducts, etc. These are often traded by people who have no need forthe material, but are speculating on the direction of the commodity.Most of this trading is done in the futures market, and contracts areclosed out before the delivery date.

Currencies: FOREX.







Currencies: FOREX.




I Your broker borrows the share from a client.

I You may now sell these shares, even though you don’t own them.I Later, you buy the shares in the market and return them to your



Currencies: FOREX.




I Your broker borrows the share from a client.I You may now sell these shares, even though you don’t own them.

I Later, you buy the shares in the market and return them to yourbroker, who returns them to the other client. You also pay anydividends that were issued in the interim.


Currencies: FOREX.







Currencies: FOREX.







Currencies: FOREX.







Currencies: FOREX.


Markets and Instruments V

Indices: An index tracks the changes in a hypothetical portfolio ofinstruments (S&P500, DIJA, FTSE100, DAX–30, NIKKEI225,NASDAQ100, ALSI40, INDI25, EMBI+, GSCI). A typical indexconsists of a weighted sum of a basket of representative stocks. Theserepresentatives and their weights may change from time to time.

Fixed income securities:I Bonds, notes, bills. These are debt instruments, and promise to pay a

certain rate of interest, which may be fixed or floating.I Example: A 10–year, 5% semi–annual coupon bond with a face

value of $1m promises to pay $25 000 every six months for 10 years,and a balloon of $1m at maturity.










Fixed income securities:

I Bonds, notes, bills. These are debt instruments, and promise to pay acertain rate of interest, which may be fixed or floating.

I Example: A 10–year, 5% semi–annual coupon bond with a facevalue of $1m promises to pay $25 000 every six months for 10 years,and a balloon of $1m at maturity.





certain rate of interest, which may be fixed or floating.

I Example: A 10–year, 5% semi–annual coupon bond with a facevalue of $1m promises to pay $25 000 every six months for 10 years,and a balloon of $1m at maturity.








Derivative Securities

Individuals and corporations face risks, and many of these risks entailfinancial gain or loss. Often, gain or loss is a simple result of a changein the value of a market variable, such as a price or rate:

I Commodity prices: Oil, maize, wheat, wool, etc.I Interest rates.I The prices of stocks that make up a pension portfolio.I Foreign currency exchange rates.I . . .

A derivative security is a financial instrument whose value is derivedfrom another, underlying or primary, variable, such as a stock price,an interest rate, a commodity price, a forex rate, etc.

Derivatives are used to transfer risk:I They can be used to hedge — i.e. as insurance against adverse risk.I They can be used to speculate — to take on extra risk in the hope of

greater returns.







greater returns.




I Commodity prices: Oil, maize, wheat, wool, etc.

I Interest rates.I The prices of stocks that make up a pension portfolio.I Foreign currency exchange rates.I . . .



greater returns.




I Commodity prices: Oil, maize, wheat, wool, etc.I Interest rates.

I The prices of stocks that make up a pension portfolio.I Foreign currency exchange rates.I . . .



greater returns.




I Commodity prices: Oil, maize, wheat, wool, etc.I Interest rates.I The prices of stocks that make up a pension portfolio.

I Foreign currency exchange rates.I . . .



greater returns.




I Commodity prices: Oil, maize, wheat, wool, etc.I Interest rates.I The prices of stocks that make up a pension portfolio.I Foreign currency exchange rates.

I . . .



greater returns.







greater returns.







greater returns.






Derivatives are used to transfer risk:

I They can be used to hedge — i.e. as insurance against adverse risk.I They can be used to speculate — to take on extra risk in the hope of

greater returns.






Derivatives are used to transfer risk:I They can be used to hedge — i.e. as insurance against adverse risk.

I They can be used to speculate — to take on extra risk in the hope ofgreater returns.







greater returns.


Call Options

An option gives the holder the right, but not the obligation to buy or sellan asset.

A European call option gives the holder the right to buy an asset S (theunderlying) for an agreed amount K (the strike price or exercise price) ona specified future date T (maturity or expiry).

The party who undertakes to deliver the asset is called the writer ofthe option.

The buyer of a European call would exercise at time T only ifK < S(T ), for a profit of S(T )− K .

If the spot price is less than the strike, the holder would discard theoption: Why pay K if you can pay S(T ) < K ?

Thus the payoff to the holder is max{S(T )− K , 0} ≥ 0.

Unlike forward contracts, options cost money. You have to pay thewriter of an option a premium upfront to enter into the contract.


Call Options









Call Options









Call Options









Call Options









Call Options









More Options

A European put option confers the right to sell an asset S for anagreed amount K at a specified future date T .

Similarly, an American call (put) option confers the right to buy (sell)an asset S for an agreed amount K , but at any time at or beforematurity T .

An Asian option has a payoff that depends on the average stock priceover a certain time period.

A knock–out barrier call will pay the same as a European call, butonly if the underlying asset price hasn’t crossed a predeterminedbarrier level.

The list of examples of derivatives is endless: Interest rate swaps,interest rate caps and floors, forward rate agreements, credit defaultswaps. . .


More Options







More Options







More Options







More Options







More Options







Hedging with Options

Example

An investor owns 1 000 shares of Anglo, with value R60.00 pershare.If the share price drops to R50.00, this will lead to a loss of R10000.To hedge against possible loss, the investor buys a put option tosell 1 000 shares in 3 months time at a price of R55.00 per share.This limits the losses to R5000 + option premium.If the stock price rises to R63.00, the investor will not exercise theoption.In that case the investor’s profit will be R3000 - option premium.The investor thus has put a cap on possible losses withoutrestraining the possible gains.



Example

An investor owns 1 000 shares of Anglo, with value R60.00 pershare.

If the share price drops to R50.00, this will lead to a loss of R10000.To hedge against possible loss, the investor buys a put option tosell 1 000 shares in 3 months time at a price of R55.00 per share.This limits the losses to R5000 + option premium.If the stock price rises to R63.00, the investor will not exercise theoption.In that case the investor’s profit will be R3000 - option premium.The investor thus has put a cap on possible losses withoutrestraining the possible gains.



Example

An investor owns 1 000 shares of Anglo, with value R60.00 pershare.If the share price drops to R50.00, this will lead to a loss of R10000.

To hedge against possible loss, the investor buys a put option tosell 1 000 shares in 3 months time at a price of R55.00 per share.This limits the losses to R5000 + option premium.If the stock price rises to R63.00, the investor will not exercise theoption.In that case the investor’s profit will be R3000 - option premium.The investor thus has put a cap on possible losses withoutrestraining the possible gains.



Example

An investor owns 1 000 shares of Anglo, with value R60.00 pershare.If the share price drops to R50.00, this will lead to a loss of R10000.To hedge against possible loss, the investor buys a put option tosell 1 000 shares in 3 months time at a price of R55.00 per share.

This limits the losses to R5000 + option premium.If the stock price rises to R63.00, the investor will not exercise theoption.In that case the investor’s profit will be R3000 - option premium.The investor thus has put a cap on possible losses withoutrestraining the possible gains.



Example

An investor owns 1 000 shares of Anglo, with value R60.00 pershare.If the share price drops to R50.00, this will lead to a loss of R10000.To hedge against possible loss, the investor buys a put option tosell 1 000 shares in 3 months time at a price of R55.00 per share.This limits the losses to R5000 + option premium.

If the stock price rises to R63.00, the investor will not exercise theoption.In that case the investor’s profit will be R3000 - option premium.The investor thus has put a cap on possible losses withoutrestraining the possible gains.



Example

An investor owns 1 000 shares of Anglo, with value R60.00 pershare.If the share price drops to R50.00, this will lead to a loss of R10000.To hedge against possible loss, the investor buys a put option tosell 1 000 shares in 3 months time at a price of R55.00 per share.This limits the losses to R5000 + option premium.If the stock price rises to R63.00, the investor will not exercise theoption.

In that case the investor’s profit will be R3000 - option premium.The investor thus has put a cap on possible losses withoutrestraining the possible gains.



Example

An investor owns 1 000 shares of Anglo, with value R60.00 pershare.If the share price drops to R50.00, this will lead to a loss of R10000.To hedge against possible loss, the investor buys a put option tosell 1 000 shares in 3 months time at a price of R55.00 per share.This limits the losses to R5000 + option premium.If the stock price rises to R63.00, the investor will not exercise theoption.In that case the investor’s profit will be R3000 - option premium.

The investor thus has put a cap on possible losses withoutrestraining the possible gains.



Example

An investor owns 1 000 shares of Anglo, with value R60.00 pershare.If the share price drops to R50.00, this will lead to a loss of R10000.To hedge against possible loss, the investor buys a put option tosell 1 000 shares in 3 months time at a price of R55.00 per share.This limits the losses to R5000 + option premium.If the stock price rises to R63.00, the investor will not exercise theoption.In that case the investor’s profit will be R3000 - option premium.The investor thus has put a cap on possible losses withoutrestraining the possible gains.


Speculating with Options

Example

Investor X believes that the shares of pharmaceuticals will rise sharply.

She is willing to speculate with a capital of R10 000.

Today, the shares of PharmCor trade at R50.00.I If Investor X buys 200 shares and the share price rises to R60.00, she

will make a profit of $2 000.I If the price drops to R40.00, her loss will be $2 000.

A call option to buy 100 PharmCor shares at strike R53.00 costsR200.

I If Investor X buys 50 call options and the share price rises to 60.00, shewill exercise the options and buy 5 000 shares at R53.00 per share.

I She will immediately sell these at R60.00 per share.I Her profit is therefore

5 000× 60− 5 000× 53− 50× 200 = 25 000

i.e. a profit of R25 000, instead of just R2 000.I BUT: Should the share price remain below R53.00, she will lose all.



Example








5 000× 60− 5 000× 53− 50× 200 = 25 000




Example








5 000× 60− 5 000× 53− 50× 200 = 25 000




Example



Today, the shares of PharmCor trade at R50.00.

I If Investor X buys 200 shares and the share price rises to R60.00, shewill make a profit of $2 000.

I If the price drops to R40.00, her loss will be $2 000.




5 000× 60− 5 000× 53− 50× 200 = 25 000




Example




will make a profit of $2 000.

I If the price drops to R40.00, her loss will be $2 000.




5 000× 60− 5 000× 53− 50× 200 = 25 000




Example








5 000× 60− 5 000× 53− 50× 200 = 25 000




Example








5 000× 60− 5 000× 53− 50× 200 = 25 000




Example








5 000× 60− 5 000× 53− 50× 200 = 25 000




Example







I She will immediately sell these at R60.00 per share.

I Her profit is therefore

5 000× 60− 5 000× 53− 50× 200 = 25 000




Example








5 000× 60− 5 000× 53− 50× 200 = 25 000

i.e. a profit of R25 000, instead of just R2 000.

I BUT: Should the share price remain below R53.00, she will lose all.



Example








5 000× 60− 5 000× 53− 50× 200 = 25 000



World Derivatives Markets

Value in $ trillion

OTC Derivatives Notional 516

OTC Derivatives Value 11

World GDP 54

USA GDP 14

RSA GDP 0.283

Derivatives figures: BIS 2007GDP figures: IMF 2007


Pricing Derivative Securities I

Because the value of a derivative is derived from another asset ormarket variable, it is sometimes possible to find a mathematicalformula for the price.

Example

Tomorrow, Allegra and Darcy will face each other in the finals atWimbledon.Tickets are available to gamble on the outcome of the game:

I If Allegra wins, the holder of a ticket gets R10 000.I If Darcy wins, the holder gets nothing.

Because the payoff is non–negative, such a ticket cannot be free.What would you be willing to pay for such a ticket?Mathematics cannot be used to determine the price of this ticket.It is determined by punters’ combined views on who is likely towin, as well as their risk preferences.




Example







Example

Tomorrow, Allegra and Darcy will face each other in the finals atWimbledon.

Tickets are available to gamble on the outcome of the game:I If Allegra wins, the holder of a ticket gets R10 000.I If Darcy wins, the holder gets nothing.





Example







Example


I If Allegra wins, the holder of a ticket gets R10 000.

I If Darcy wins, the holder gets nothing.





Example







Example



Because the payoff is non–negative, such a ticket cannot be free.What would you be willing to pay for such a ticket?

Mathematics cannot be used to determine the price of this ticket.It is determined by punters’ combined views on who is likely towin, as well as their risk preferences.




Example





Pricing Derivative Securities II

Example

(Continued)

Suppose the market price of the ticket is P.Suppose also that there is a second type of ticket available: Thisticket pays R10 000 if Darcy wins, and R0 if Allegra wins.We can determine the price of the second ticket mathematically:If you own one of each kind, you will definitely get R10 000. Sothe price of both tickets must be R10 000, and hence the price ofthe second ticket is 10 000− P.The second ticket is a derivative of the first ticket — once themarket decides the price of the first ticket, the price of the secondticket is determined, independent of views and risk preferences ofpunters.



Example

(Continued)Suppose the market price of the ticket is P.

Suppose also that there is a second type of ticket available: Thisticket pays R10 000 if Darcy wins, and R0 if Allegra wins.We can determine the price of the second ticket mathematically:If you own one of each kind, you will definitely get R10 000. Sothe price of both tickets must be R10 000, and hence the price ofthe second ticket is 10 000− P.The second ticket is a derivative of the first ticket — once themarket decides the price of the first ticket, the price of the secondticket is determined, independent of views and risk preferences ofpunters.



Example

(Continued)Suppose the market price of the ticket is P.Suppose also that there is a second type of ticket available: Thisticket pays R10 000 if Darcy wins, and R0 if Allegra wins.

We can determine the price of the second ticket mathematically:If you own one of each kind, you will definitely get R10 000. Sothe price of both tickets must be R10 000, and hence the price ofthe second ticket is 10 000− P.The second ticket is a derivative of the first ticket — once themarket decides the price of the first ticket, the price of the secondticket is determined, independent of views and risk preferences ofpunters.



Example

(Continued)Suppose the market price of the ticket is P.Suppose also that there is a second type of ticket available: Thisticket pays R10 000 if Darcy wins, and R0 if Allegra wins.We can determine the price of the second ticket mathematically:

If you own one of each kind, you will definitely get R10 000. Sothe price of both tickets must be R10 000, and hence the price ofthe second ticket is 10 000− P.The second ticket is a derivative of the first ticket — once themarket decides the price of the first ticket, the price of the secondticket is determined, independent of views and risk preferences ofpunters.



Example

(Continued)Suppose the market price of the ticket is P.Suppose also that there is a second type of ticket available: Thisticket pays R10 000 if Darcy wins, and R0 if Allegra wins.We can determine the price of the second ticket mathematically:If you own one of each kind, you will definitely get R10 000. Sothe price of both tickets must be R10 000, and hence the price ofthe second ticket is 10 000− P.

The second ticket is a derivative of the first ticket — once themarket decides the price of the first ticket, the price of the secondticket is determined, independent of views and risk preferences ofpunters.



Example

(Continued)Suppose the market price of the ticket is P.Suppose also that there is a second type of ticket available: Thisticket pays R10 000 if Darcy wins, and R0 if Allegra wins.We can determine the price of the second ticket mathematically:If you own one of each kind, you will definitely get R10 000. Sothe price of both tickets must be R10 000, and hence the price ofthe second ticket is 10 000− P.The second ticket is a derivative of the first ticket — once themarket decides the price of the first ticket, the price of the secondticket is determined, independent of views and risk preferences ofpunters.


Law of One Price

In order to be able to use mathematics to find prices, we assume onlythat you can’t make money from nothing:

Law of One Price:Two securities that are guaranteed to have thesame value at time t = T must have the samevalue at time t = 0.

For suppose that X ,Y are securities, and that XT = YT in all statesof the world.

I If X0 < Y0, you can buy X and sell Y at time t = 0 — for animmediate profit of Y0 − X0.

I At time T , you have XT and you owe YT — and these cancel!I So if X0 < Y0, you can make money from nothing!!!!!I If X0 > Y0 do the opposite.


Law of One Price







Law of One Price







Law of One Price







Law of One Price





I At time T , you have XT and you owe YT — and these cancel!

I So if X0 < Y0, you can make money from nothing!!!!!I If X0 > Y0 do the opposite.


Law of One Price





I At time T , you have XT and you owe YT — and these cancel!I So if X0 < Y0, you can make money from nothing!!!!!

I If X0 > Y0 do the opposite.


Law of One Price







Option Pricing in a Single–Period Model 1CAN WE PRICE THIS CALL OPTION?

r = 10% K = 11

p 22 11

10 C0 = ?

1p 5.5 0

STOCK CALL


Option Pricing in a Single–Period Model 2

Pricing by Expectation: (Huygens–Bernoulli) The “fair” price is theexpected (discounted) payoff:

C0 = E[

CT

1 + r

]= p × 11

1.1+ (1− p)× 0

1.1

I So the price depends on p.I If p = 1

2 , then C0 = 5.

Supply and demand: The “correct” price is the one at which thesupply is equal to the demand.

I If demand goes up(down), so must the price: Higher prices will make itmore attractive to sell(buy).

I The higher the probability p of an ↑–move, the more attractive theoption, and thus the higher its price.

Both the above methods are WRONG!!




C0 = E[

CT

1 + r

]= p × 11

1.1+ (1− p)× 0

1.1


2 , then C0 = 5.








C0 = E[

CT

1 + r

]= p × 11

1.1+ (1− p)× 0

1.1

I So the price depends on p.

I If p = 12 , then C0 = 5.








C0 = E[

CT

1 + r

]= p × 11

1.1+ (1− p)× 0

1.1


2 , then C0 = 5.








C0 = E[

CT

1 + r

]= p × 11

1.1+ (1− p)× 0

1.1


2 , then C0 = 5.








C0 = E[

CT

1 + r

]= p × 11

1.1+ (1− p)× 0

1.1


2 , then C0 = 5.








C0 = E[

CT

1 + r

]= p × 11

1.1+ (1− p)× 0

1.1


2 , then C0 = 5.








C0 = E[

CT

1 + r

]= p × 11

1.1+ (1− p)× 0

1.1


2 , then C0 = 5.







Consider a portfolio θ := (θ0, θ1) consisting of an θ0–many rands in abank account and θ1–many shares. At t = 0 the portfolio’s value is

V0(θ) = θ0 + 10θ1

at t = T the portfolio’s value is

VT (θ) =

{1.1θ0 + 22θ1 if S ↑ 22

1.1θ0 + 5.5θ1 if S ↓ 5.5

We choose θ so that VT (θ) = CT , whether the stock price goes ↑ or↓:

↑: 1.1θ0 + 22θ1 = 11

↓: 1.1θ0 + 5.5θ1 = 0⇒ θ0 = −10

3 , θ1 = 23




V0(θ) = θ0 + 10θ1


VT (θ) =

{1.1θ0 + 22θ1 if S ↑ 22

1.1θ0 + 5.5θ1 if S ↓ 5.5


↑: 1.1θ0 + 22θ1 = 11

↓: 1.1θ0 + 5.5θ1 = 0⇒ θ0 = −10

3 , θ1 = 23




V0(θ) = θ0 + 10θ1


VT (θ) =

{1.1θ0 + 22θ1 if S ↑ 22

1.1θ0 + 5.5θ1 if S ↓ 5.5


↑: 1.1θ0 + 22θ1 = 11

↓: 1.1θ0 + 5.5θ1 = 0⇒ θ0 = −10

3 , θ1 = 23




V0(θ) = θ0 + 10θ1


VT (θ) =

{1.1θ0 + 22θ1 if S ↑ 22

1.1θ0 + 5.5θ1 if S ↓ 5.5


↑: 1.1θ0 + 22θ1 = 11

↓: 1.1θ0 + 5.5θ1 = 0⇒ θ0 = −10

3 , θ1 = 23



As VT (θ) = CT no matter what, the Law of One Price dictates that

C0 = V0(θ) = −103 + 2

3 × 10 = 103

The probability p of an ↑–move is completely IRRELEVANT!!

The call option has been priced using an arbitrage argument.

An arbitrage is a portfolio θ with the following properties:I V0(θ) = 0I VT (θ) ≥ 0 in all states of the world.I P(VT (θ) > 0) > 0

Thus an arbitrage is like a free lottery ticket.

The only assumption we make is:

There are no arbitrage opportunities in the market




C0 = V0(θ) = −103 + 2

3 × 10 = 103










C0 = V0(θ) = −103 + 2

3 × 10 = 103










C0 = V0(θ) = −103 + 2

3 × 10 = 103










C0 = V0(θ) = −103 + 2

3 × 10 = 103



An arbitrage is a portfolio θ with the following properties:

I V0(θ) = 0I VT (θ) ≥ 0 in all states of the world.I P(VT (θ) > 0) > 0







C0 = V0(θ) = −103 + 2

3 × 10 = 103



An arbitrage is a portfolio θ with the following properties:I V0(θ) = 0

I VT (θ) ≥ 0 in all states of the world.I P(VT (θ) > 0) > 0







C0 = V0(θ) = −103 + 2

3 × 10 = 103



An arbitrage is a portfolio θ with the following properties:I V0(θ) = 0I VT (θ) ≥ 0 in all states of the world.

I P(VT (θ) > 0) > 0







C0 = V0(θ) = −103 + 2

3 × 10 = 103










C0 = V0(θ) = −103 + 2

3 × 10 = 103










C0 = V0(θ) = −103 + 2

3 × 10 = 103








Pricing by Expectation — Reprise! 1

Consider our first method for pricing the call option as (discounted)expected payoff (Huygens–Bernoulli):

C0 = E[

CT

1 + r

]This seems like a good idea, but it went horribly wrong:

I If p = 12 , H-B give a price of C0 = 5,

I but we know that C0 = 3..3 (or else there will be arbitrage).

However, the stock itself is not priced correctly via H-B.I We ought to have

S0 = p × 22

1.1+ (1− p)× 5.5

1.1= 12.5 when p = 1

2

I Instead, we have S0 = 10.

We now find a probability p∗ for which H-B does price the stockcorrectly.




C0 = E[

CT

1 + r





S0 = p × 22

1.1+ (1− p)× 5.5

1.1= 12.5 when p = 1

2






C0 = E[

CT

1 + r





S0 = p × 22

1.1+ (1− p)× 5.5

1.1= 12.5 when p = 1

2






C0 = E[

CT

1 + r





S0 = p × 22

1.1+ (1− p)× 5.5

1.1= 12.5 when p = 1

2






C0 = E[

CT

1 + r




However, the stock itself is not priced correctly via H-B.

I We ought to have

S0 = p × 22

1.1+ (1− p)× 5.5

1.1= 12.5 when p = 1

2






C0 = E[

CT

1 + r





S0 = p × 22

1.1+ (1− p)× 5.5

1.1= 12.5 when p = 1

2






C0 = E[

CT

1 + r





S0 = p × 22

1.1+ (1− p)× 5.5

1.1= 12.5 when p = 1

2






C0 = E[

CT

1 + r





S0 = p × 22

1.1+ (1− p)× 5.5

1.1= 12.5 when p = 1

2





We want

S0 = p∗ × 22

1.1+ (1− p∗)× 5.5

1.1⇒ p∗ =

1

3

If we use this new risk–neutral probability p∗ to price the option viaH-B, we obtain:

C0 = E∗[

CT

1 + r

]=

1

3× 11

1.1+

2

3× 0

1.1=

10

3

which is correct!!

Thus H-B yields the correct price, provided we use risk–neutralprobabilities.



We want

S0 = p∗ × 22

1.1+ (1− p∗)× 5.5

1.1⇒ p∗ =

1

3


C0 = E∗[

CT

1 + r

]=

1

3× 11

1.1+

2

3× 0

1.1=

10

3

which is correct!!




We want

S0 = p∗ × 22

1.1+ (1− p∗)× 5.5

1.1⇒ p∗ =

1

3


C0 = E∗[

CT

1 + r

]=

1

3× 11

1.1+

2

3× 0

1.1=

10

3

which is correct!!




We want

S0 = p∗ × 22

1.1+ (1− p∗)× 5.5

1.1⇒ p∗ =

1

3


C0 = E∗[

CT

1 + r

]=

1

3× 11

1.1+

2

3× 0

1.1=

10

3

which is correct!!



The Fundamental Theorem of Mathematical Finance

Theorem

A market–model is arbitrage–free if and only if there exists arisk–neutral probability measure.

Prices of derivative securities must be obtained via H-B, but usingrisk–neutral probabilities.

This theorem is easy to prove for this simple unrealistic model,

But it holds in general, for all models,

And makes it possible to numerically price options in very complicatedand realistic models, using Monte Carlo Simulation.



Theorem








Theorem








Theorem








Theorem








Theorem







Lecture Basic Finance

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