1 project 2- stock option pricing mathematical tools -today we will learn compound interest
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1
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
11
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
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