Project 2- Stock Option Pricing
• Mathematical Tools
-Today we will learn Compound Interest
1

Compounding
• 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 .
2

Compound Interest
• Discrete Compounding
-Interest compounded n times per year
• Continuous Compounding
-Interest compounded continuously
3

Compound Interest
Discrete Compounding
P- dollars invested
r -an annual rate
F

P

1


  r n


 n
 t n- number of times the interest compounded per year t- number of years
F- dollars after t years.
4

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 .
P 1 r n n



1
 y

Compunded n times for one year
Compounded once a year for one year 5

Yield for Discrete Compounding
Interest at an annual rate r, compounded n times per year has yield
y.
y
 
 r n


 n

1
6

Discrete Compounding
Example 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?
7

Example1
(i) Using Discrete
Compounding formula
Given
P =$74,000 r =0.0525
n =12 t =3.5
Goal- To find F
F

P

1
F

74,000
 r n n
 t
0.0525
12
12

( 3 .
5 )

$ 88 , 891
8

Example 1
(ii) Using yield formula
Given r =0.0525
n =12 y

Goal- To find y r n n

1 y
 

1
0 .
0525
12 

 12

1

0 .
05378

5 .
378 %
9

Discrete Compounding
Example 2
(i)What is the value of $150,000 after 5 years at 6.2%, compounded quarterly?
(ii) What is the effective annual yield?
10

Example 2
(i) Using Discrete
Compounding formula
Given
P =$150,000 r =0.062
n =4 t =5
Goal- To find F
F

P

1
F

150,000
 r n n
 t
0.062
4
4

5

$ 204 , 028
11

Example 2
(ii) Using yield formula
Given r =0.062
n =4 y

Goal- To find y r n n

1 y
 

1
0 .
062
4 

 4

1

0 .
06346

6 .
346 %
12

Annual rate for Discrete
Compounding y


 r n 

 n

1 y

1



 y

1

1 n 
1
 n n



  y

1

1 n 
1



 r r r n


 n
13

Annual rate for Discrete
Compounding
Interest compounded n times r .
per year at a yield y , has an annual rate r
 n
 


1
 y
 n
1

1


14

Discrete Compounding
Example 3
(i) What rate, r , compounded monthly, will yield 5.25%?
15

Example 3
(i) Using Annual rate formula
Given y =0.0525
n =12
Goal- To find r r
 n
 





1
 y
 n
1

1




 r

12
 




1

0 .
0525

1 2
1

1 


 
0 .
05128

5 .
128 %
16

Compound Interest
Continuous Compounding
The value of P dollars after t years, when compounded continuously at an annual rate r, is
F = P
 e r
 t
17

Yield for Continuous Compounding
Interest at an annual rate r , compounded continuously has yield y .
y
 e r 
1
18

Continuous Compounding
Example 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?
19

Example1
(i) Using Continuous
Compounding formula
Given
P =$750,000 r =0.061
t =(40/12)
Goal- To find F
F = P
 e r
 t
F = 750,000
 e 0.061

(40/12) =$ 919,111
20

Example 1
(ii) Using yield formula
Given r =0.061
y
 e r 
1
Goal- To find y y
 e r 
1
 e
0 .
061 
1

0 .
0629

6 .
29 %
21

Logarithms
• Why do we need logarithms for compound interest ?
Recall: yield formula for continuous compounding y
 e r 
1
• To find r (since r is an exponent)
22

Review of Logarithms
• For any base b, the logarithm function log b
(x)
•
The equations u = b v and v = log b u are equivalent
•
Eg: 100=10 2 and 2=log
10
100 are equivalent
• Two types
-Common Logarithms (base is 10)
-Natural Logartihms (base is e)- Notation: ln
23

Review of Logarithms
1.The logarithm log b
(x) function is the INVERSE of exp b
(x)
2. log b
(x) is defined for any positive real number x
Inverse Functions
Logarithm
Exponential
2
1
0
4
3
-1 1 2 3 4 5 6
-2
-3
-4
6
5
24 x

Review of Logarithms
The basic properties of exponents, b u
 b v = b u + v and ( b u ) v = b u
 v , yield properties for the logarithm functions.
log b
( u
 v ) = log b u + log b v log b
( u / v ) = log b u
 log b v log b u v = v
 log b u.
25

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
26

Review of Logarithms
Example1
Find the annual rate, r , that produces an effective annual yield of 6.00%, when compounded continuously.
27

Example 1
(ii) Using yield formula
Given y =6.00% y
 e r 
1
Goal- To find r y
 e r 
1 e r 
1
 y e r r

1 .
0600
 ln( 1 .
0600 ) r

0 .
0583

5 .
83 %
Taking ln on both sides
28

Review of Logarithms
Example 2
Find the annual rate, r , that produces an effective annual yield of 5.15%, when compounded continuously. Round your answer to 3 places.
29

Example 2
(ii) Using continuous compounding formula
Given y =5.15%
Goal- To find r y
 e r 
1 e r 
1
 y e r r

1 .
0515
 ln( 1 .
0515 ) r

0 .
05022

5 .
022 % y
 e r
Taking ln on both sides

1
30

Review of Logarithms
Example 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?
31

(ii) Using yield formula
Given
F =$15,162.65
P =$10,000 r =0.025
Example 3
F

P
 e r
 t
Goal- To find t
32

Example 3
F

Pe rt e rt ln( e rt
)


F
P ln


F
P

 rt
 ln


F
P

 t

1 ln r


F
P



1
0 .
025 ln


15162 .
65
10000



16 .
65 years
33

Value of Money
Discrete compounding
Recall
• Present value ( P ) and Future value( F ) of money
• We need to rearrange the formula to find P
F

P

1


  r n


 n
 t
P

F

1


  r n 

 -n
 t
The present value of money for discrete compounding
34

Value of Money
Continuous compounding
Recall
• Present value ( P ) and Future value( F ) of money
• We need to rearrange the formula to find P
F
P

P
 e

F
 e r
 t
-r
 t
The present value of money for continuous compounding
35

Ratio (R)
• Under continuous compounding-The ratio of the future value to the present value
R

F
P

P
 e
P r
 t
 e r
 t
• This allows us to convert the interest rate for a given period to a ratio of future to present value for the same period
36

Recall- Class Project
We 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)
37

Project Focus I
• Walt Disneyr =4%, compounded continuously
The weekly risk-free rate for the
Walt Disney r rf

0 .
04
52

0 .
0007692
R rf
 e
0 .
04 / 52 
1 .
0007695
The risk-free weekly ratio for the
Walt Disney
38

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 ) pv
 fv
 e
 r
 t
 fv
 e

0 .
04

( 20 / 52 )
39