Presentation on theme: "Project 2- Stock Option Pricing"— Presentation transcript:
1Project 2- Stock Option Pricing Mathematical Tools-Today we will learn Compound Interest
2Compounding 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.
3Compound Interest Discrete Compounding -Interest compounded n times per yearContinuous Compounding-Interest compounded continuously
4Compound Interest Discrete Compounding P- dollars investedr -an annual raten- number of times the interest compounded per yeart- number of yearsF- dollars after t years.
5Yield 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 yearCompounded once a yearfor one year
6Yield for Discrete Compounding Interest at an annual rate r, compounded n times per year has yield y.
7Discrete Compounding Example 1 What is the value of $74,000 after3-1/2 years at 5.25%,compounded monthly?(ii) What is the effective annual yield?
8Example1 (i) Using Discrete Compounding formula Given P=$74,000 Goal- To find F
9Example 1(ii) Using yield formulaGivenr=0.0525n=12Goal- To find y
10Discrete Compounding Example 2 (i)What is the value of $150,000 after 5 yearsat 6.2%, compounded quarterly?(ii) What is the effective annual yield?
11Example 2 (i) Using Discrete Compounding formula Given P=$150,000 Goal- To find F
12Example 2(ii) Using yield formulaGivenr=0.062n=4Goal- To find y
14Annual rate for Discrete Compounding Interest compounded n timesper year at a yield y, has an annual rate r.
15Discrete Compounding Example 3 What rate, r, compounded monthly,will yield 5.25%?
16Example 3 (i) Using Annual rate formula Given y=0.0525 n=12 Goal- To find r
17Compound Interest Continuous Compounding The value of P dollars after t years, when compounded continuously at an annual rate r, isF = Pert
18Yield for Continuous Compounding Interest at an annual rate r, compounded continuously has yield y.
19Continuous Compounding Example 1 (i)Find the value, rounded to whole dollars,of $750,000 after 3 years and 4 months,if it is invested at a rate of 6.1%compounded continuously.(ii) What is the yield, rounded to 3 places,on this investment?
20F = Pert Example1 F = 750,000e0.061(40/12) =$ 919,111 Using ContinuousCompounding formulaGivenP=$750,000r=0.061t=(40/12)Goal- To find FF = PertF = 750,000e0.061(40/12) =$ 919,111
21Example 1(ii) Using yield formulaGivenr=0.061Goal- To find y
22Logarithms Why do we need logarithms for compound interest ? To find r (since r is an exponent)Recall: yield formula for continuous compounding
23Review of Logarithms For any base b, the logarithm function logb (x) The equations u = bv and v = logbu are equivalentEg: 100=102 and 2=log10100 are equivalentTwo types-Common Logarithms (base is 10)-Natural Logartihms (base is e)- Notation: ln
24Review of Logarithms1.The logarithm logb(x) function is the INVERSE of expb(x)2. logb(x) is defined for any positive real number x
25Review 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 + logbvlogb(u/v) = logbu logbvlogbuv = vlogbu.
26Review of Logarithms ln u = ln v if and only if u=v Most commonly used to obtain solution of equationsWe can transform an equation into an equivalent form by taking ln of both sides
27Review of Logarithms Example1 Find the annual rate, r, that produces an effective annual yield of 6.00%, when compounded continuously.
28Example 1 (ii) Using yield formula Given y=6.00% Goal- To find r Taking ln on both sides
29Review 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.
30Example 2 (ii) Using continuous compounding formula Given y=5.15% Goal- To find rTaking ln on both sides
31Review 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?
32Example 3 (ii) Using yield formula Given F=$15,162.65 P=$10,000 Goal- To find t
34Value of Money Discrete compounding RecallPresent value (P) and Future value(F) of moneyWe need to rearrange the formula to find PThe present value of money for discrete compounding
35Value of Money Continuous compounding RecallPresent value (P) and Future value(F) of moneyWe need to rearrange the formula to find PThe present value of money for continuous compounding
36Ratio (R)Under continuous compounding-The ratio of the future value to the present valueThis allows us to convert the interest rate for a given period to a ratio of future to present value for the same period
37Recall- Class ProjectWe 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 laterstrike 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)
38Project Focus I Walt Disney- r =4%, compounded continuously The weekly risk-free rate for the Walt DisneyThe risk-free weekly ratio for the Walt Disney
39Project Focus IISuppose we know the future value (fv) for our 20 week option at the end of 20 weeksrisk-free rate annual interest 4%Can find the Present value (pv)