Project 2- Stock Option Pricing

1

Project 2- Stock Option Pricing

• Mathematical Tools -Today we will learn Compound Interest

2

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.

3

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

4

Compound InterestCompound Interest Discrete Compounding Discrete Compounding

tn

nrPF

1P- dollars invested

r -an annual rate

n- number of times the interest compounded per year

t- number of years

F- dollars after t years.

5

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.

yPnrP

n

11

Compounded once a year for one year

Compunded n times for one year

6

Yield for Discrete Compounding

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

11

n

nry

7

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?

8

Example1tn

nrPF

1

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

8918853

,$).(

12

120.052574,000F 1

9

Example 1

11

n

nry

(ii) Using yield formulaGivenr=0.0525n=12

Goal- To find y

%...y 3785053780112052501

12

10


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

(ii) What is the effective annual yield?

11

Example 2tn

nrPF

1

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

0282045

,$

4

40.062150,000F 1

12

Example 2

11

n

nry

(ii) Using yield formulaGivenr=0.062n=4

Goal- To find y

%...y 34660634601406201

4

13

Annual rate for Discrete Compounding

ryn

nry

nry

nry

n

n

n

n

11

11

11

11

1

1

14

Annual rate for Discrete Compounding

11

1n

ynr

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

15


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

16

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

11

1n

ynr

%...r 1285051280105250112121

17

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

18

Yield for Continuous Compounding

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

1 rey

19

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?

20

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

21

Example 1(ii) Using yield formulaGivenr=0.061

Goal- To find y

1 rey

%..eey .r 2960629011 0610

22

Logarithms

• Why do we need logarithms for compound interest ?

• To find r (since r is an exponent)

1 reyRecall: yield formula for continuous compounding

23

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

24

Review of LogarithmsInverse Functions

-4-3-2-10123456

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

x

func

tions

LogarithmExponential1.The logarithm

logb(x) function is the INVERSE of expb(x)

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

25


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.

26


• 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

27

Review of LogarithmsExample1

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

28

Example 1(ii) Using yield formulaGiveny=6.00%

Goal- To find r

1 rey

%83.50583.0)0600.1ln(

0600.1

1

1

rr

e

ye

ey

r

r

r

Taking ln on both sides

29

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.

30

Example 2(ii) Using continuous

compounding formulaGiveny=5.15%Goal- To find r

1 rey

%022.505022.0)0515.1ln(

0515.1

1

1

rr

e

ye

ey

r

r

r

Taking ln on both sides

31

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?

32

Example 3

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

Goal- To find t

trPF e

33

Example 3

yearsPF

rt

PFrt

PFe

PFe

PeF

rt

rt

rt

65.1610000

65.15162ln025.01ln1

ln

ln)ln(

34

Value of Money Discrete compounding

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

• We need to rearrange the formula to find P

tn

nrPF

1

Recall

t-n

nrFP

1

The present value of money for discrete compounding

35

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 eThe present value of money for continuous compounding

36

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

PFR

37

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)

38

Project Focus I

• Walt Disney-r =4%, compounded continuously

0007695.152/04.0 eRrf

The risk-free weekly ratio for the Walt Disney

0007692.05204.0

rfr

The weekly risk-free rate for the Walt Disney

39

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

Documents

Project 2- Stock Option Pricing