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MATH20180: Foundations of Financial Mathematics Vincent Astier email: [email protected] office: room S1.72 (Science South) Lecture 1 Vincent Astier MATH20180 1 / 35

Transcript of MATH20180: Foundations of Financial Mathematicsastier/math20180/chapter1.pdf · MATH20180:...

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MATH20180: Foundations of Financial Mathematics

Vincent Astier

email: [email protected]: room S1.72 (Science South)

Lecture 1

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Our goal: the Black-Scholes Formula

Black-Scholes Formula for pricing a call option:

CT = X0Φ( log(X0/k) + rT

σ√

T+

12σ√

T)−ke−rT Φ

( log(X0/k) + rTσ√

T−1

2σ√

T)

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To reach this goal we need to learn...

Mathematics: some set and measure theory, probability, randomvariables, expected values, Brownian motion.

Finance: interest rates, present value, discounted value, hedging, risk,bonds,shares, call (and put) options, arbitrage.

Text: Sean Dineen "Probability Theory in Finance: A MathematicalGuide to the Black-Scholes Formula" (in small doses).

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Interest Rates

DefinitionThe interest represents the money paid to the lender for the use of theborrowed money. It also represents the money earned when a givenamount of capital is invested.

The principal P is the total amount of money that is either borrowed orinvested.

The rate of interest r (usually given in percentage) is the amountcharged for the use of principal for a given length of time, usually onyearly basis.

The future value S is the principal P plus the accumulated interest.

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Simple interest

DefinitionThe simple interest is the interest computed once on the principal forthe entire period of time when it is borrowed.

Formula: If a principal P is borrowed (invested) at a simple in-terest rate of r per year for a period of t years, then

1 the interest isI = Prt

2 the future value S is given by

S = P + I = P + Prt = P (1 + rt) .

(t does not have to be an integer.)

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Simple interest: Examples

Example 1. John wants to open a special savings account earning6.25% simple interest per annum with an initial deposit of e500. Howmuch will be in the account at the end of 8 months?

Answer: 500 + 500× 0.0625× 8/12.

Example 2. At what annual rate of simple interest will a principal ofe960 accumulate to a future value of e1,000 in 10 months?

Answer: Solve 960 + 960× r × 10/12 = 1000.

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Simple interest

Remark: Simple interest is rarely used by banks.

If e1,000 is deposited for 2 years at a rate of 10% simple interest, thenthe amount accumulated at the end of two years would be e1,200.

If, however, at the end of year one the amount accumulated at thattime e1,100 is withdrawn and immediately deposited for a further yearat the same interest rate, then, after one more year the amountaccumulated would be e1,210, that is, a gain of e10 on the previousamount.

If simple interest would be the norm, people would be in and out ofbanks regularly withdrawing and immediately re-depositing theirsavings.

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Compound interest

DefinitionThe compound interest is obtained by applying simple interest atregular intervals to the amount accumulated at the start of each period.

Details: If a principal P is borrowed (invested) at a compoundedinterest rate r for n periods in a year, then

1 The future value after one period is P + P rn = P(1 + r

n ), i.e.simple interest at the rate of r/n. The new value is obtainedby multiplying the previous one by 1 + r

n . Therefore:2 the future value after k periods is P(1 + r

n )k ,3 the future value S after t years is P(1 + r

n )nt ,4 the interest I after t years is I = S − P.

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Compounded interest: examples

Example 1. A deposit of e400 is placed in an account earning interestat 10% per year compounded quarterly. Find the future value afterthree years. How much compounded interest was earned?

Answer: 400(1 + 0.1/4)12.

Example 2. At what interest rate r per year compounded quarterly wille250 invested today amount to e1,900 in ten years?

Answer: Solve 250(1 + r/4)40 = 1900.

Example 3. Which of the following yields more profit for e50 investedfor a period of one year at the interest rate of 10%:(a) compounded annually;(b) compounded monthly;(c) compounded daily.

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Compounded interest: Solution to Example 3

(a) The compounding period is 1 year. Therefore n = 1 so

S = 50(

1 +0.11

)1

= 55.

(b) The compounding period is 1/12 year. Therefore n = 12 and

S = 50(

1 +0.112

)12

= 55.235653.

(c) The compounding period is 1/365 year.

S = 50(

1 +0.1365

)365

= 55.257789.

Note that the more often the 10% is compounded, the higher thereturn. What if we were to take the 10% per annum and compound itcontinuously? What would the future value S be?

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Continuously compounded interest

LemmaLet a be a real number. Then

limn→∞

(1 +

an

)n= ea.

Remark. In our case limn→∞(1 + r

n

)nt= ert .

DefinitionIf a principal P is invested (borrowed) at an interest rate r per yearcompounded continuously, after t years the future value S is givenby

S = Pert

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Continuously compounded interest

Exercise: Suppose you invest e100 at interest rate of 5% per annumcompounded continuously. How much will you have in 3 years?

Solution: S = Pert = 100e0.05×3 = 116.18.

Remark: We can reverse the process and say that the present value(present worth, principal) of e116.18 in three years’s time is e100. Inthis way we can determine, for a given fixed rate of interest, thepresent value of any amount from any future time.

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Continuously compounded interest

Example 2: If P is the present value of e5,000 in six years’s time at 7%interest rate compounded continuously, then

Pe0.07×6 = 5,000

thereforeP = 5,000× e−0.42 = 3,285.23.

The procedure of finding the principal value from the future amount iscalled discounting back to the present. The present worth of a futureamount is called its discounted value. In the above example,e3,285.23 is the discounted value of e5,000 in six years’s time at 7%interest rate compounded continuously.

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Continuously compounded interest

Formula: The discounted value of an amount S at a future timet assuming a constant continuously compounded interest rate ris given by

Se−rt .

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Other financial Instruments-Bonds

A bond is a contract under which the issuer borrows money from theholder of the bond. The money (called the face value) will be repaid atthe end of the contract (called the maturity date). The bond mayinclude the payment of interest (the coupon) at regular intervals.

Remark 1: Bonds can be traded between investors.

Remark 2: Bond is related to interest rate because of its coupon. Forexample, if we buy a one-year bond with face value e100 and coupone6, then we will get e106 after one year, independent of the interestrate.

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Other financial Instruments-Bonds: examples

Example: A bond has a face value of e1,000 and a coupon worth e50.

The (coupon) interest rate is

501000

× 100% =5

100× 100% = 5%.

If the interest rate increases to 7%, then the actual value of thebond will decrease. An investor would never pay e1,000 for abond with a coupon rate of 5% on the secondary market whennew issues of similar quality are paying 7%.In order to remain competitive with new issues, the bond wouldsell at a discount to its face value. If x is the new value of thebond, then

0.07x = 50

and sox =

500.07

= e714.29.

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Fair Games

DefinitionA game is said to be fair if it is equally favorable to all players.

In Game Theory, by winning we mean net winning, and thus wesubtract off any losses and treat a loss as a win of a negativeamount.

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Fair Games: examples

Example: Consider a simple betting game (e.g. tossing a coin)between two players, John and Mark.

If the game is favorable to John, then his expected winnings, E[WJ ],is at least as large as those of Mark, E[WM ]; that is E[WJ ] ≥ E[WM ].

Similarly, if the game is favorable to Mark, then E[WM ] ≥ E[WJ ].

The game is called fair if

E[WJ ] = E[WM ] (1)

If the bets placed by John and Mark are BJ and BM , respectively, thenthe total input will be BJ + BM .If the total output, that is the sum of all the players’ winnings, is alsoBJ + BM , we call the game a zero-sum game.

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Fair Games

If the game is a zero-sum game we have

E[WJ ] + E[WM ] = Output− Input = 0 (2)

and, combining the above two equalities (1) and (2), we obtain

E[WJ ] = E[WM ] = 0.

The converse is also true.

TheoremA zero-sum game is a fair game if and only if the expected winnings ofeach player is zero.

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Fair Games

Game: John and Mark each bet e5 on the toss of a coin: John winswhen a head (H) comes up, Mark wins when a tail (T) comes up andthe winner gets e10.In tossing an unbiased coin we expect half of the outcomes, no matterhow many games are played, to result in a head, and we interpret thisas the probability that a head appears.

Similarly, the probability of a tail appearing is 1/2.

Thus in 1,000 games John would expect to win 500 games and also tolose 500. His expected winnings are

E[WJ ] = 500 · (10− 5) + 500 · (0− 5) = 0.

Since Mark’s expected winnings are also 0, we have verified the fairgame criterion.

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Fair Games

Question: What happens if we change these parameters in a zero-sumgame?

Consider the following variations:

(A) John bets e3 and Mark e7,

Answer: E[WJ ] = 12(10− 3) + 1

2(0− 3) = 2, so the game is notfair.

(B) on average the coin turns up heads 80% of the time.

Answer: E[WJ ] = 810(10− 5) + 2

10(0− 5) = 3, the game is not fair.

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Fair Games

We conclude:

TheoremProbabilities (or risks) and rewards (or winnings/losses) are both usedin calculating the expected return. In a zero-sum game if one of theseis given, then the other can be chosen to make the game fair.

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Hedging

Motivating example: Consider another game in which John and Markplace a bet with a bookmaker on a two-horse race.

Terminology: A person who accepts bets on horse races (and othersporting events) is called a bookmaker. Members of the public whoplace bets are called punters.

Remark: In contrast to the coin tossing game, John and Mark do notnegotiate the odds on each horse.These are set by the bookmaker, who does not care what odds aregiven as long as he gets his percentage of the total amount wagered.

In fact the bookmaker’s strategy is to eliminate (or minimize) any risk tohis percentage share. This is called hedging.

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Hedging

Simple case: John bets e400 on horse A while Mark bets e100 onhorse B.The bookmaker knows that they are the only ones playing, and wants10% profit from the sum of all bets (e500). So the winner takes e450.

What odds should the bookmaker give on horses A and B?

If horse A wins, John wins e50, so a bet of e400 brings a win of e50,i.e. the odds are 1 to 8 for horse A.

If horse B wins, Mark wins e350, so a bet of e100 brings a win ofe350, i.e. the odds are 3.5 to 1, i.e. 7 to 2 for horse B.

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Hedging

What about the point of view of John and Mark?

Let p be what John thinks is the probability that horse A wins, and qwhat Mark thinks is the probability that horse B wins.

If they both think the game is fair, then

E[WJ ] = p50 + (1− p)(−400) = 0, so p =89.

E[WM ] = q350 + (1− q)(−100) = 0, so q =29.

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Hedging

Motivating example (continued): Consider another game in which Johnand Mark place a bet with a bookmaker on a two-horse race. Johnplaces a bet of e400 on horse A while Mark bets e100 on horse B.The bookmaker wants to make a profit of 10%. Suppose now that afurther bet for e300 is placed on horse B at the odds 7 : 2.

Question: What is the bookmaker’s position in this case?

Let WB be the winnings of the bookmaker. If horse A wins thenWB = 50 + 300 = 350, if horse B wins then WB = 50− 1050 = −1000.To avoid or lower the risk, the bookmaker can: refuse the bet, changethe odds on horse A to increase the bets on it, place a limit on howmuch can be bet on a horse, etc.

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Hedging

Another option is to bet himself on horse B (with another bookmaker)to make up for the losses in case horse B wins. For instance if anotherbookmaker offers odds of 3 to 1 for horse B, and our bookmaker betex on horse B.

Then:If horse A wins, we have WB = 350− xIf horse B wins, we have WB = −1000 + 3x .

Both outcomes will be positive, and thus guarantee a gain for thebookmaker, if 333.3 < x < 350. If he wants to know in advance howmuch he will gain, we must have 350− x = −1000 + 3x , i.e. x = 337.5and the bookmaker will win e12.5, regardless of who wins the race.

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Hedging

Conclusions:

(1) To remove the uncertainty associated with unpredictable futureevents, identify the desired rewards (or penalties) and develop ahedging strategy by working backwards.

(2) To reduce the potential loss due to an unfavorable event occurring,place a bet in favour of the event happening.

Most people follow (2) in their daily lives by taking out insurance, andinsurance companies also lay off bets but call it re-insuring. In thefinancial world this form of playing safe is called hedging.

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Arbitrage

DefinitionArbitrage means any situation, opportunity or price which allows aguaranteed profit without risk.

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Arbitrage: examples

Example 1: Suppose we have two banks A and B, operating side byside. Bank A offers customers a 10% interest per annum on savings,while bank B offers loans to customers at an 8% interest rate perannum.

We can take advantage of this situation: Borrow as much as possiblefrom B, and immediately place it on deposit in Bank A.If, for example, one obtains a loan for e1,000,000 for a year, then atthe end of the year the principal in Bank A amounts to

1,000,000e0.1 = 1,105,171

while the loan repayment to Bank B amounts to

1,000,000e0.08 = 1,083,287.

This gives a risk-free guaranteed profit of e21,884 at the end of theyear.

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Arbitrage

In reality, arbitrage opportunities close down very rapidly. In the aboveexample, the demand on Bank B would increase rapidly, and as aresult, interest rates would quickly be adjusted until equilibrium wasestablished.

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Arbitrage: examples

Example 2: Two bookmakers offer different odds on a race betweentwo horses A and B. The first bookmaker offers the odds 7 : 2 on thehorse B while the second bookmaker offers odds 5 : 2 on the horse A.Mary has e160. How much should she bet on each horse in order tomake a no-risk guaranteed profit?

Solution. Assume Mary bets x on horse A, so she bets 160− x onhorse B. After this she has no money left.

Mary’s position if horse A wins : x +5x2

Mary’s position if horse B wins : (160− x) +7(160− x)

2To run no risk, we must have

x +5x2

= (160− x) +7(160− x)

2which gives x = 90 and so Mary gets 315. Thus her profit is e155regardless of who wins the race.

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Arbitrage

In any game we have:

either

(a) all bets are fair;

or

(b) there exists a betting strategy which produces a positive returnindependent of the outcome of the game.

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More examples: example 3

A bookmaker knows there will be only three bets of e40, e60 ande100 on three different horses in the same race. Determine the oddshe should give on each horse so that he will make a profit of 10%. If,after the odds are fixed, a further bet of e150 is placed on the thirdhorse and the bookmaker responds by placing a bet of ex at odds of 2to 1 on the same horse in order to run no risk, find x and thebookmaker’s profit/loss.

Answer: The odds are 7 to 2 on the first horse, 2 to 1 on the secondhorse, and 4 to 5 on the third horse.After his bet of ex on horse 3 at 2 to 1, we have E[WB] = −100 + 2x ifhorse 3 wins, and E[WB] = 170− x if horse 3 loses. To run no risk, wemust have −100 + 2x = 170− x so x = 90 and E[WB] = 80.

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Exercise

At a horserace a bookmaker has a guaranteed profit of e150. At a lastminute, a new bet of e50 is placed on a certain horse and thebookmaker is able to place a bet on the same horse at odds of a to 1.If he bets e100 it may increase his profits by e50 and if he bets e50 itmay reduce his profits by e50.

(a) Find a and the odds on the horse offered by the bookmaker(b) How much should he bet in order to run no risk? What will his

profit/loss be in this case?

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