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**Project 2- Stock Option Pricing**

Mathematical Tools -Today we will learn Compound Interest

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**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.

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**Compound Interest Discrete Compounding**

-Interest compounded n times per year Continuous Compounding -Interest compounded continuously

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**Compound Interest Discrete Compounding**

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. Compunded n times for one year Compounded once a year 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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**Discrete Compounding Example 1**

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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**Example1 (i) Using Discrete Compounding formula Given P=$74,000**

Goal- To find F

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Example 1 (ii) Using yield formula Given r=0.0525 n=12 Goal- To find y

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**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?

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**Example 2 (i) Using Discrete Compounding formula Given P=$150,000**

Goal- To find F

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Example 2 (ii) Using yield formula Given r=0.062 n=4 Goal- To find y

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**Annual rate for Discrete Compounding**

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**Annual rate for Discrete Compounding**

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

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**Discrete Compounding Example 3**

What rate, r, compounded monthly, will yield 5.25%?

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**Example 3 (i) Using Annual rate formula Given y=0.0525 n=12**

Goal- To find r

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**Compound Interest 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.

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**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?

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**F = Pert Example1 F = 750,000e0.061(40/12) =$ 919,111**

Using Continuous Compounding formula Given P=$750,000 r=0.061 t=(40/12) Goal- To find F F = Pert F = 750,000e0.061(40/12) =$ 919,111

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Example 1 (ii) Using yield formula Given r=0.061 Goal- To find y

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**Logarithms Why do we need logarithms for compound interest ?**

To find r (since r is an exponent) Recall: 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 Logarithms 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 bubv = bu+v and (bu)v = buv,**

The basic properties of exponents, yield properties for the logarithm functions. bubv = bu+v and (bu)v = buv, logb(uv) = logbu + logbv logb(u/v) = logbu logbv logbuv = vlogbu.

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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 Logarithms Example1**

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 formula Given y=6.00% Goal- To find r**

Taking ln on both sides

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**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.

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**Example 2 (ii) Using continuous compounding formula Given y=5.15%**

Goal- To find r Taking ln on both sides

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**Review of Logarithms Example 3**

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

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**Example 3 (ii) Using yield formula Given F=$15,162.65 P=$10,000**

Goal- To find t

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Example 3

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**Value of Money Discrete compounding**

Recall Present value (P) and Future value(F) of money We need to rearrange the formula to find P The present value of money for discrete compounding

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**Value of Money Continuous compounding**

Recall Present value (P) and Future value(F) of money We need to rearrange the formula to find P 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

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Recall- Class Project We suppose that it is Friday, January 11, 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**

The weekly risk-free rate for the Walt Disney The risk-free weekly ratio 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)

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