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Project 2- Stock Option Pricing

• Mathematical Tools

-Today we will learn Compound Interest

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CompoundingCompounding

• Suppose that money left on deposit earns interest.

• Interest is normally paid at regular intervals, while the money is on deposit.

• This is called compounding.

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Compound Interest

• Discrete CompoundingDiscrete Compounding

-Interest compounded n times per year-Interest compounded n times per year

• Continuous CompoundingContinuous Compounding

-Interest compounded continuously-Interest compounded continuously

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Compound InterestCompound Interest Discrete Compounding Discrete Compounding

tn

nr

PF

1

P- dollars invested

r -an annual rate

n- number of times the interest compounded per year

t- number of years

F- dollars after t years.

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Yield for Discrete Compounding

• The annual rate that would produce the same amount as in discrete compounding for one year.

• Such a rate is called an effective annual yield, annual percentage yield, or just the yield.

yPn

rP

n

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Compounded once a year for one year

Compunded n times for one year

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Yield for Discrete Compounding

Interest at an annual rate r, compounded n times per year has yield y.

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n

nr

y

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Discrete CompoundingDiscrete CompoundingExample 1Example 1

(i) What is the value of $74,000 after 3-1/2 years at 5.25%,compounded monthly?

(ii) What is the effective annual yield?

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Example1tn

nr

PF

1

(i) Using Discrete Compounding formulaGivenP=$74,000r=0.0525n=12t=3.5Goal- To find F

8918853

,$).(

12

12

0.052574,000F 1

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Example 1

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n

nr

y

(ii) Using yield formulaGivenr=0.0525n=12

Goal- To find y

%...

y 3785053780112

052501

12

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Discrete CompoundingDiscrete CompoundingExample 2Example 2

(i)What is the value of $150,000 after 5 years at 6.2%, compounded quarterly?

(ii) What is the effective annual yield?

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Example 2tn

nr

PF

1

(i) Using Discrete Compounding formulaGivenP=$150,000r=0.062n=4t=5Goal- To find F

0282045

,$

4

4

0.062150,000F 1

12

Example 2

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n

nr

y

(ii) Using yield formulaGivenr=0.062n=4

Goal- To find y

%...

y 346606346014

06201

4

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Annual rate for Discrete Compounding

ryn

nr

y

nr

y

nr

y

n

n

n

n

11

11

11

11

1

1

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Annual rate for Discrete Compounding

11

1

nynr

Interest compounded n times per year at a yield y, has an annual rate r.

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Discrete CompoundingDiscrete CompoundingExample 3Example 3

(i) What rate, r, compounded monthly, will yield 5.25%?

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Example 3(i) Using Annual rate formulaGiveny=0.0525n=12Goal- To find r

11

1

nynr

%...r 128505128010525011212

1

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Compound InterestCompound Interest Continuous Compounding Continuous Compounding

The value of P dollars after t years, when compounded continuously at an annual rate r, is

F = Pert

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Yield for Continuous Compounding

Interest at an annual rate r, compounded continuously has yield y.

1 rey

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ContinuousContinuous CompoundingCompoundingExample 1Example 1

(i)Find the value, rounded to whole dollars, of $750,000 after 3 years and 4 months, if it is invested at a rate of 6.1% compounded continuously.

(ii) What is the yield, rounded to 3 places, on this investment?

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Example1(i) Using Continuous

Compounding formulaGivenP=$750,000r=0.061t=(40/12)Goal- To find F

F = Pert

F = 750,000e0.061(40/12) =$ 919,111

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Example 1(ii) Using yield formulaGivenr=0.061

Goal- To find y

1 rey

%..eey .r 2960629011 0610

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Logarithms

• Why do we need logarithms for compound interest ?

• To find r (since r is an exponent)

1 reyRecall: yield formula for continuous compounding

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Review of Logarithms

• For any base b, the logarithm function

logb (x)• The equations u = bv and v = logbu are equivalent• Eg: 100=102 and 2=log10100 are equivalent• Two types -Common Logarithms (base is 10)

-Natural Logartihms (base is e)- Notation: ln

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Review of LogarithmsInverse Functions

-4

-3

-2

-1

0

1

2

3

4

5

6

-4 -3 -2 -1 0 1 2 3 4 5 6

x

func

tion

s

LogarithmExponential

1.The logarithm logb(x) function is the INVERSE of expb(x)

2. logb(x) is defined for any positive real number x

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Review of Logarithms

logb(uv) = logbu + logbv

logb(u/v) = logbu logbv

logbuv = vlogbu.

bubv = bu+v and (bu)v = buv,

The basic properties of exponents,

yield properties for the logarithm functions.

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Review of Logarithms

• ln u = ln v if and only if u=v

• Most commonly used to obtain solution of equations

• We can transform an equation into an equivalent form by taking ln of both sides

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Review of LogarithmsExample1

Find the annual rate, r, that produces an effective annual yield of 6.00%, when compounded continuously.

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Example 1(ii) Using yield formulaGiveny=6.00%

Goal- To find r

1 rey

%83.50583.0

)0600.1ln(

0600.1

1

1

r

r

e

ye

ey

r

r

r

Taking ln on both sides

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Review of LogarithmsExample 2

Find the annual rate, r, that produces an effective annual yield of 5.15%, when compounded continuously. Round your answer to 3 places.

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Example 2(ii) Using continuous

compounding formulaGiveny=5.15%Goal- To find r

1 rey

%022.505022.0

)0515.1ln(

0515.1

1

1

r

r

e

ye

ey

r

r

r

Taking ln on both sides

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Review of LogarithmsExample 3

How long will it take $10,000 to grow to $15,162.65 if interest is paid at an annual rate of 2.5% compounded continuously?

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Example 3

(ii) Using yield formulaGivenF=$15,162.65P=$10,000r=0.025

Goal- To find t

trPF e

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Example 3

yearsP

F

rt

P

Frt

P

Fe

P

Fe

PeF

rt

rt

rt

65.1610000

65.15162ln

025.0

1ln

1

ln

ln)ln(

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Value of Money Discrete compounding

• Present value (P) and Future value(F) of money

• We need to rearrange the formula to find P

tn

nr

PF

1

Recall

t-n

n

rFP

1

The present value of money for discrete compounding

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Value of Money Continuous compounding

• Present value (P) and Future value(F) of money

• We need to rearrange the formula to find P

trPF e

Recall

t-rFP e

The present value of money for continuous compounding

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Ratio (R)

• Under continuous compounding-The ratio of the future value to the present value

• This allows us to convert the interest rate for a given period to a ratio of future to present value for the same period

trtr

ePeP

PF

R

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Recall- Class ProjectWe suppose that it is Friday, January 11, 2002. Our

goal is to find the present value, per share, of a European call on Walt Disney Company stock.

• The call is to expire 20 weeks later• strike price of $23. • stock’s price record of weekly closes for the past 8

years(work basis).• risk free rate 4% (this means that on Jan 11,2002

the annual interest rate for a 20 week Treasury Bill was 4% compounded continuously)

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Project Focus I

• Walt Disney-

r =4%, compounded continuously

0007695.152/04.0 eRrf

The risk-free weekly ratio for the Walt Disney

0007692.052

04.0rfr

The weekly risk-free rate for the Walt Disney

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Project Focus II

• Suppose we know the future value (fv) for our 20 week option at the end of 20 weeks

• risk-free rate annual interest 4%

• Can find the Present value (pv)

)52/20(04.0

efv

efvpv tr