7. Continuously Varying Interest Rates

Size: px
Start display at page:

Download "7. Continuously Varying Interest Rates"

Transcription

1 7. Continuously Varying Interest Rates 7.1 The Continuous Varying Interest Rate Formula. Suppose that interest is continuously compounded with a rate which is changing in time. Let the present time be time, and let r(s denote the interest rate per unit at time s units, s. The quantity r(s is called the spot or the instantaneous interest rate at time s. Let D(t be the amount that you will have in your account at time t if you deposit P at time. In order to determine D(t in terms of the interest rates r(s, s t, note that by the Simple Interest Formula 1.8, for small h, we have D(s + h D(s(1 + r(s h,

2 ( means is approximately equal to D(s + h D(s + D(s r(s h, D(s + h D(s D(s r(s h, D(s + h D(s h D(s r(s. By the definition of the derivative 4.7, taking the limit as h, we have or D (s = D(s r(s D (s D(s = r(s. To solve this differential equation, integrate both sides: Thus D (s D(s ds = [ln D(s] t =.

3 or ln D(t ln D( = r(sds. Since D( = P, we obtain from the preceding equation that ln D(t ln P = exp (ln D(t ln P = exp ( exp (ln D(t exp ( ln P 1 = exp D(t P 1 = exp and D(t = P exp ( ( (.

4 7.2 Example. Suppose that interest is continuously compounded with a rate which is changing in time. Let the present time be time, and let r(s = s be the interest rate per unit at time s units, s. (i Draw a graph of the function r(s. (ii Find the amount D(t that you will have in your account at time t, t, if you deposit 1, at time. (iii Suppose the unit is equal to one year. How much money do you have in your account after 1 years?

5 Solution. (i The graph of the function y 1 (s = s is The graph of the function y 2 (s = s is The graph of the function r(s = s is

6 (ii By the Continuous Varying Interest Rate Formula 7.1, Note that D(t = 1, exp r(sds = ( s ds (by the linearity of the integrals Hence =.1ds s ds =.1 t.5 [ln 1 + s ] t =.1 t.5 (ln 1 + t ln 1 =.1 t.5 ln 1 + t. D(t = 1, exp (.1t.5 ln 1 + t ( D(t = 1, e.1t+ln 1 +t.5 = 1, e.1t eln 1 +t.5.

7 = 1, e.1t 1 + t.5. (iii For t = 1, D(1 = 1, e.1 1 ( = 1, e 11 (Recall that e = 1, = 2, The Continuous Compounded Interest Formula. The Continuous Compounded Interest Formula 4.1 is a special case of the Continuous Varying Interest Rate Formula 7.1, arising when the interest rate r(s, s, is always equal to r 1. Let a principal P be borrowed at an annual rate of interest r% compounded continuously for t years. Then, according to 7.1, for D( = P and r(s = r 1, s, the amount A owed on the principal P at time t is A = D(t = P exp (

8 ( = P exp = P exp r 1 ds ( r 1 t = P e r 1 t. 7.4 The Present Value. The Continuous Varying Interest Rate Formula 7.1 states that a principal P earning a continuously compounded interest r(s per unit at time s units will be worth D(t = P exp at time t. Therefore [ P = D(t exp = D(t exp ( ( ] 1 [ ( ] r(sds.

9 Definition. The present value P (t corresponding to an amount A and to a variable interest rate r(s, s, is given by [ ( P (t = A exp r(sds, t, and is equal to the amount to be invested at time to accumulate A at time t by continuous compounding at the interest rate r(s, s. ] 7.5 Definition. The function [ ( P (t = exp r(sds, t, which is at each point t equal to the present value P (t corresponding to the amount 1 and to a variable interest rate r(s, s, is called the present value function. ] In other words, the present value function at t is equal to the amount to be invested at time to accumulate 1 at time t by continuous compounding at the interest rate r(s, s.

10 7.6 Example. How much money should be invested at time at a continuously compounded interest rate r(s = s per unit at time s units, s, in order to accumulate 5,, at time t? Solution. By 7.4, the present value P (t of 5,, is P (t = 5,, exp [ ( ]. In Example 7.2 we showed that r(sds = s ds =.1 t.5 ln 1 + t. Hence the present value P (t is 5,, exp [ (.1 t.5 ln 1 + t ]

11 = 5,, exp [.1 t +.5 ln 1 + t ] = 5,, exp (.1 t exp ( ln 1 + t.5 = 5,, exp (.1 t 1 + t.5. Suppose t = 1: then P (1 = 5,, exp ( = 5,, 1 e 11.5 = 2, 73, Definition. The average of the spot interest rates up to time t r(t = 1 t is called the yield curve. r(sds, t,

12 7.8 Example. Find the yield curve and the present value function if r(s = r s + r 2 s 1 + s. Solution. Note that we can rewrite r(s as r(s = r 1 + r 2 s 1 + s = r 1 r 2 + r 2 (1 + s 1 + s = r 1 r s + r 2. Hence, for t, r(sds = r 1 r s + r 2ds = r 1 r s ds + r 2ds = (r 1 r s ds + r 2 ds

13 = (r 1 r 2 [ln 1 + s ] t + r 2[s] t = (r 1 r 2 ln(1 + t + r 2 t. Therefore, for t, the yield curve is r(t = 1 t = 1 t [(r 1 r 2 ln(1 + t + r 2 t] = r 1 r 2 t ln(1 + t + r 2. Consequently, for t, the present value function is [ ( ] P (t = exp r(sds = exp [ ((r 1 r 2 ln(1 + t + r 2 t] = exp ( ln[(1 + t (r 2 r 1 ] exp ( r 2 t = (1 + t (r 2 r 1 exp ( r 2 t.

Continuous Compounding and Discounting

Continuous Compounding and Discounting Continuous Compounding and Discounting Philip A. Viton October 5, 2011 Continuous October 5, 2011 1 / 19 Introduction Most real-world project analysis is carried out as we ve been doing it, with the present

More information

Manual for SOA Exam FM/CAS Exam 2.

Manual for SOA Exam FM/CAS Exam 2. Manual for SOA Exam FM/CAS Exam 2. Chapter 2. Cashflows. c 29. Miguel A. Arcones. All rights reserved. Extract from: Arcones Manual for the SOA Exam FM/CAS Exam 2, Financial Mathematics. Fall 29 Edition,

More information

Absolute Value Equations and Inequalities

Absolute Value Equations and Inequalities Key Concepts: Compound Inequalities Absolute Value Equations and Inequalities Intersections and unions Suppose that A and B are two sets of numbers. The intersection of A and B is the set of all numbers

More information

Annuity is defined as a sequence of n periodical payments, where n can be either finite or infinite.

Annuity is defined as a sequence of n periodical payments, where n can be either finite or infinite. 1 Annuity Annuity is defined as a sequence of n periodical payments, where n can be either finite or infinite. In order to evaluate an annuity or compare annuities, we need to calculate their present value

More information

Vilnius University. Faculty of Mathematics and Informatics. Gintautas Bareikis

Vilnius University. Faculty of Mathematics and Informatics. Gintautas Bareikis Vilnius University Faculty of Mathematics and Informatics Gintautas Bareikis CONTENT Chapter 1. SIMPLE AND COMPOUND INTEREST 1.1 Simple interest......................................................................

More information

Essential Topic: Continuous cash flows

Essential Topic: Continuous cash flows Essential Topic: Continuous cash flows Chapters 2 and 3 The Mathematics of Finance: A Deterministic Approach by S. J. Garrett CONTENTS PAGE MATERIAL Continuous payment streams Example Continuously paid

More information

21.1 Arithmetic Growth and Simple Interest

21.1 Arithmetic Growth and Simple Interest 21.1 Arithmetic Growth and Simple Interest When you open a savings account, your primary concerns are the safety and growth of your savings. Suppose you deposit $1000 in an account that pays interest at

More information

Financial Mathematics for Actuaries. Chapter 1 Interest Accumulation and Time Value of Money

Financial Mathematics for Actuaries. Chapter 1 Interest Accumulation and Time Value of Money Financial Mathematics for Actuaries Chapter 1 Interest Accumulation and Time Value of Money 1 Learning Objectives 1. Basic principles in calculation of interest accumulation 2. Simple and compound interest

More information

$496. 80. Example If you can earn 6% interest, what lump sum must be deposited now so that its value will be $3500 after 9 months?

$496. 80. Example If you can earn 6% interest, what lump sum must be deposited now so that its value will be $3500 after 9 months? Simple Interest, Compound Interest, and Effective Yield Simple Interest The formula that gives the amount of simple interest (also known as add-on interest) owed on a Principal P (also known as present

More information

Solutions to Exercises, Section 4.5

Solutions to Exercises, Section 4.5 Instructor s Solutions Manual, Section 4.5 Exercise 1 Solutions to Exercises, Section 4.5 1. How much would an initial amount of $2000, compounded continuously at 6% annual interest, become after 25 years?

More information

Nominal rates of interest and discount

Nominal rates of interest and discount 1 Nominal rates of interest and discount i (m) : The nominal rate of interest payable m times per period, where m is a positive integer > 1. By a nominal rate of interest i (m), we mean a rate payable

More information

5.1 Simple and Compound Interest

5.1 Simple and Compound Interest 5.1 Simple and Compound Interest Question 1: What is simple interest? Question 2: What is compound interest? Question 3: What is an effective interest rate? Question 4: What is continuous compound interest?

More information

Homework 4 Solutions

Homework 4 Solutions Homework 4 Solutions Chapter 4B Does it make sense? Decide whether each of the following statements makes sense or is clearly true) or does not make sense or is clearly false). Explain your reasoning.

More information

Absolute Value Equations and Inequalities

Absolute Value Equations and Inequalities . Absolute Value Equations and Inequalities. OBJECTIVES 1. Solve an absolute value equation in one variable. Solve an absolute value inequality in one variable NOTE Technically we mean the distance between

More information

With compound interest you earn an additional $128.89 ($1628.89 - $1500).

With compound interest you earn an additional $128.89 ($1628.89 - $1500). Compound Interest Interest is the amount you receive for lending money (making an investment) or the fee you pay for borrowing money. Compound interest is interest that is calculated using both the principle

More information

MTH6120 Further Topics in Mathematical Finance Lesson 2

MTH6120 Further Topics in Mathematical Finance Lesson 2 MTH6120 Further Topics in Mathematical Finance Lesson 2 Contents 1.2.3 Non-constant interest rates....................... 15 1.3 Arbitrage and Black-Scholes Theory....................... 16 1.3.1 Informal

More information

Section 4.5 Exponential and Logarithmic Equations

Section 4.5 Exponential and Logarithmic Equations Section 4.5 Exponential and Logarithmic Equations Exponential Equations An exponential equation is one in which the variable occurs in the exponent. EXAMPLE: Solve the equation x = 7. Solution 1: We have

More information

8.7 Exponential Growth and Decay

8.7 Exponential Growth and Decay Section 8.7 Exponential Growth and Decay 847 8.7 Exponential Growth and Decay Exponential Growth Models Recalling the investigations in Section 8.3, we started by developing a formula for discrete compound

More information

Financial Mathematics for Actuaries. Chapter 2 Annuities

Financial Mathematics for Actuaries. Chapter 2 Annuities Financial Mathematics for Actuaries Chapter 2 Annuities Learning Objectives 1. Annuity-immediate and annuity-due 2. Present and future values of annuities 3. Perpetuities and deferred annuities 4. Other

More information

About Compound Interest

About Compound Interest About Compound Interest TABLE OF CONTENTS About Compound Interest... 1 What is COMPOUND INTEREST?... 1 Interest... 1 Simple Interest... 1 Compound Interest... 1 Calculations... 3 Calculating How Much to

More information

Chapter 5 Financial Forwards and Futures

Chapter 5 Financial Forwards and Futures Chapter 5 Financial Forwards and Futures Question 5.1. Four different ways to sell a share of stock that has a price S(0) at time 0. Question 5.2. Description Get Paid at Lose Ownership of Receive Payment

More information

Chapter 22: Borrowings Models

Chapter 22: Borrowings Models October 21, 2013 Last Time The Consumer Price Index Real Growth The Consumer Price index The official measure of inflation is the Consumer Price Index (CPI) which is the determined by the Bureau of Labor

More information

9.2 Summation Notation

9.2 Summation Notation 9. Summation Notation 66 9. Summation Notation In the previous section, we introduced sequences and now we shall present notation and theorems concerning the sum of terms of a sequence. We begin with a

More information

Building a Smooth Yield Curve. University of Chicago. Jeff Greco

Building a Smooth Yield Curve. University of Chicago. Jeff Greco Building a Smooth Yield Curve University of Chicago Jeff Greco email: jgreco@math.uchicago.edu Preliminaries As before, we will use continuously compounding Act/365 rates for both the zero coupon rates

More information

88 CHAPTER 2. VECTOR FUNCTIONS. . First, we need to compute T (s). a By definition, r (s) T (s) = 1 a sin s a. sin s a, cos s a

88 CHAPTER 2. VECTOR FUNCTIONS. . First, we need to compute T (s). a By definition, r (s) T (s) = 1 a sin s a. sin s a, cos s a 88 CHAPTER. VECTOR FUNCTIONS.4 Curvature.4.1 Definitions and Examples The notion of curvature measures how sharply a curve bends. We would expect the curvature to be 0 for a straight line, to be very small

More information

1 Interest rates, and risk-free investments

1 Interest rates, and risk-free investments Interest rates, and risk-free investments Copyright c 2005 by Karl Sigman. Interest and compounded interest Suppose that you place x 0 ($) in an account that offers a fixed (never to change over time)

More information

Lecture 6 Black-Scholes PDE

Lecture 6 Black-Scholes PDE Lecture 6 Black-Scholes PDE Lecture Notes by Andrzej Palczewski Computational Finance p. 1 Pricing function Let the dynamics of underlining S t be given in the risk-neutral measure Q by If the contingent

More information

3. Time value of money. We will review some tools for discounting cash flows.

3. Time value of money. We will review some tools for discounting cash flows. 1 3. Time value of money We will review some tools for discounting cash flows. Simple interest 2 With simple interest, the amount earned each period is always the same: i = rp o where i = interest earned

More information

Future Value of an Annuity Sinking Fund. MATH 1003 Calculus and Linear Algebra (Lecture 3)

Future Value of an Annuity Sinking Fund. MATH 1003 Calculus and Linear Algebra (Lecture 3) MATH 1003 Calculus and Linear Algebra (Lecture 3) Future Value of an Annuity Definition An annuity is a sequence of equal periodic payments. We call it an ordinary annuity if the payments are made at the

More information

Check off these skills when you feel that you have mastered them.

Check off these skills when you feel that you have mastered them. Chapter Objectives Check off these skills when you feel that you have mastered them. Know the basic loan terms principal and interest. Be able to solve the simple interest formula to find the amount of

More information

The Basics of Interest Theory

The Basics of Interest Theory Contents Preface 3 The Basics of Interest Theory 9 1 The Meaning of Interest................................... 10 2 Accumulation and Amount Functions............................ 14 3 Effective Interest

More information

Chapter 21: Savings Models

Chapter 21: Savings Models October 16, 2013 Last time Arithmetic Growth Simple Interest Geometric Growth Compound Interest A limit to Compounding Problems Question: I put $1,000 dollars in a savings account with 2% nominal interest

More information

PowerPoint. to accompany. Chapter 5. Interest Rates

PowerPoint. to accompany. Chapter 5. Interest Rates PowerPoint to accompany Chapter 5 Interest Rates 5.1 Interest Rate Quotes and Adjustments To understand interest rates, it s important to think of interest rates as a price the price of using money. When

More information

MAT116 Project 2 Chapters 8 & 9

MAT116 Project 2 Chapters 8 & 9 MAT116 Project 2 Chapters 8 & 9 1 8-1: The Project In Project 1 we made a loan workout decision based only on data from three banks that had merged into one. We did not consider issues like: What was the

More information

CHAPTER 1. Compound Interest

CHAPTER 1. Compound Interest CHAPTER 1 Compound Interest 1. Compound Interest The simplest example of interest is a loan agreement two children might make: I will lend you a dollar, but every day you keep it, you owe me one more penny.

More information

Chapter 04 - More General Annuities

Chapter 04 - More General Annuities Chapter 04 - More General Annuities 4-1 Section 4.3 - Annuities Payable Less Frequently Than Interest Conversion Payment 0 1 0 1.. k.. 2k... n Time k = interest conversion periods before each payment n

More information

Problems and Solutions

Problems and Solutions Problems and Solutions CHAPTER Problems. Problems on onds Exercise. On /04/0, consider a fixed-coupon bond whose features are the following: face value: $,000 coupon rate: 8% coupon frequency: semiannual

More information

Payment streams and variable interest rates

Payment streams and variable interest rates Chapter 4 Payment streams and variable interest rates In this chapter we consider two extensions of the theory Firstly, we look at payment streams A payment stream is a payment that occurs continuously,

More information

Pre-Session Review. Part 2: Mathematics of Finance

Pre-Session Review. Part 2: Mathematics of Finance Pre-Session Review Part 2: Mathematics of Finance For this section you will need a calculator with logarithmic and exponential function keys (such as log, ln, and x y ) D. Exponential and Logarithmic Functions

More information

Ch. 11.2: Installment Buying

Ch. 11.2: Installment Buying Ch. 11.2: Installment Buying When people take out a loan to make a big purchase, they don t often pay it back all at once in one lump-sum. Instead, they usually pay it back back gradually over time, in

More information

The Black-Scholes Formula

The Black-Scholes Formula FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 The Black-Scholes Formula These notes examine the Black-Scholes formula for European options. The Black-Scholes formula are complex as they are based on the

More information

MAT12X Intermediate Algebra

MAT12X Intermediate Algebra MAT12X Intermediate Algebra Workshop I - Exponential Functions LEARNING CENTER Overview Workshop I Exponential Functions of the form y = ab x Properties of the increasing and decreasing exponential functions

More information

Caput Derivatives: October 30, 2003

Caput Derivatives: October 30, 2003 Caput Derivatives: October 30, 2003 Exam + Answers Total time: 2 hours and 30 minutes. Note 1: You are allowed to use books, course notes, and a calculator. Question 1. [20 points] Consider an investor

More information

, plus the present value of the $1,000 received in 15 years, which is 1, 000(1 + i) 30. Hence the present value of the bond is = 1000 ;

, plus the present value of the $1,000 received in 15 years, which is 1, 000(1 + i) 30. Hence the present value of the bond is = 1000 ; 2 Bond Prices A bond is a security which offers semi-annual* interest payments, at a rate r, for a fixed period of time, followed by a return of capital Suppose you purchase a $,000 utility bond, freshly

More information

Problems and Solutions

Problems and Solutions 1 CHAPTER 1 Problems 1.1 Problems on Bonds Exercise 1.1 On 12/04/01, consider a fixed-coupon bond whose features are the following: face value: $1,000 coupon rate: 8% coupon frequency: semiannual maturity:

More information

ST334 ACTUARIAL METHODS

ST334 ACTUARIAL METHODS ST334 ACTUARIAL METHODS version 214/3 These notes are for ST334 Actuarial Methods. The course covers Actuarial CT1 and some related financial topics. Actuarial CT1 which is called Financial Mathematics

More information

1 Error in Euler s Method

1 Error in Euler s Method 1 Error in Euler s Method Experience with Euler s 1 method raises some interesting questions about numerical approximations for the solutions of differential equations. 1. What determines the amount of

More information

3 More on Accumulation and Discount Functions

3 More on Accumulation and Discount Functions 3 More on Accumulation and Discount Functions 3.1 Introduction In previous section, we used 1.03) # of years as the accumulation factor. This section looks at other accumulation factors, including various

More information

Percent, Sales Tax, & Discounts

Percent, Sales Tax, & Discounts Percent, Sales Tax, & Discounts Many applications involving percent are based on the following formula: Note that of implies multiplication. Suppose that the local sales tax rate is 7.5% and you purchase

More information

Microeconomic Theory: Basic Math Concepts

Microeconomic Theory: Basic Math Concepts Microeconomic Theory: Basic Math Concepts Matt Van Essen University of Alabama Van Essen (U of A) Basic Math Concepts 1 / 66 Basic Math Concepts In this lecture we will review some basic mathematical concepts

More information

L 2 : x = s + 1, y = s, z = 4s + 4. 3. Suppose that C has coordinates (x, y, z). Then from the vector equality AC = BD, one has

L 2 : x = s + 1, y = s, z = 4s + 4. 3. Suppose that C has coordinates (x, y, z). Then from the vector equality AC = BD, one has The line L through the points A and B is parallel to the vector AB = 3, 2, and has parametric equations x = 3t + 2, y = 2t +, z = t Therefore, the intersection point of the line with the plane should satisfy:

More information

ACTS 4302 SOLUTION TO MIDTERM EXAM Derivatives Markets, Chapters 9, 10, 11, 12, 18. October 21, 2010 (Thurs)

ACTS 4302 SOLUTION TO MIDTERM EXAM Derivatives Markets, Chapters 9, 10, 11, 12, 18. October 21, 2010 (Thurs) Problem ACTS 4302 SOLUTION TO MIDTERM EXAM Derivatives Markets, Chapters 9, 0,, 2, 8. October 2, 200 (Thurs) (i) The current exchange rate is 0.0$/. (ii) A four-year dollar-denominated European put option

More information

TIME VALUE OF MONEY PROBLEM #7: MORTGAGE AMORTIZATION

TIME VALUE OF MONEY PROBLEM #7: MORTGAGE AMORTIZATION TIME VALUE OF MONEY PROBLEM #7: MORTGAGE AMORTIZATION Professor Peter Harris Mathematics by Sharon Petrushka Introduction This problem will focus on calculating mortgage payments. Knowledge of Time Value

More information

first complete "prior knowlegde" -- to refresh knowledge of Simple and Compound Interest.

first complete prior knowlegde -- to refresh knowledge of Simple and Compound Interest. ORDINARY SIMPLE ANNUITIES first complete "prior knowlegde" -- to refresh knowledge of Simple and Compound Interest. LESSON OBJECTIVES: students will learn how to determine the Accumulated Value of Regular

More information

2 Applications to Business and Economics

2 Applications to Business and Economics 2 Applications to Business and Economics APPLYING THE DEFINITE INTEGRAL 442 Chapter 6 Further Topics in Integration In Section 6.1, you saw that area can be expressed as the limit of a sum, then evaluated

More information

Logarithmic and Exponential Equations

Logarithmic and Exponential Equations 11.5 Logarithmic and Exponential Equations 11.5 OBJECTIVES 1. Solve a logarithmic equation 2. Solve an exponential equation 3. Solve an application involving an exponential equation Much of the importance

More information

Sample Solutions for Assignment 2.

Sample Solutions for Assignment 2. AMath 383, Autumn 01 Sample Solutions for Assignment. Reading: Chs. -3. 1. Exercise 4 of Chapter. Please note that there is a typo in the formula in part c: An exponent of 1 is missing. It should say 4

More information

The Black-Scholes pricing formulas

The Black-Scholes pricing formulas The Black-Scholes pricing formulas Moty Katzman September 19, 2014 The Black-Scholes differential equation Aim: Find a formula for the price of European options on stock. Lemma 6.1: Assume that a stock

More information

QUADRATIC, EXPONENTIAL AND LOGARITHMIC FUNCTIONS

QUADRATIC, EXPONENTIAL AND LOGARITHMIC FUNCTIONS QUADRATIC, EXPONENTIAL AND LOGARITHMIC FUNCTIONS Content 1. Parabolas... 1 1.1. Top of a parabola... 2 1.2. Orientation of a parabola... 2 1.3. Intercept of a parabola... 3 1.4. Roots (or zeros) of a parabola...

More information

CE 314 Engineering Economy. Interest Formulas

CE 314 Engineering Economy. Interest Formulas METHODS OF COMPUTING INTEREST CE 314 Engineering Economy Interest Formulas 1) SIMPLE INTEREST - Interest is computed using the principal only. Only applicable to bonds and savings accounts. 2) COMPOUND

More information

6.4 Logarithmic Equations and Inequalities

6.4 Logarithmic Equations and Inequalities 6.4 Logarithmic Equations and Inequalities 459 6.4 Logarithmic Equations and Inequalities In Section 6.3 we solved equations and inequalities involving exponential functions using one of two basic strategies.

More information

Lecture 12: The Black-Scholes Model Steven Skiena. http://www.cs.sunysb.edu/ skiena

Lecture 12: The Black-Scholes Model Steven Skiena. http://www.cs.sunysb.edu/ skiena Lecture 12: The Black-Scholes Model Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena The Black-Scholes-Merton Model

More information

Institutional Finance 08: Dynamic Arbitrage to Replicate Non-linear Payoffs. Binomial Option Pricing: Basics (Chapter 10 of McDonald)

Institutional Finance 08: Dynamic Arbitrage to Replicate Non-linear Payoffs. Binomial Option Pricing: Basics (Chapter 10 of McDonald) Copyright 2003 Pearson Education, Inc. Slide 08-1 Institutional Finance 08: Dynamic Arbitrage to Replicate Non-linear Payoffs Binomial Option Pricing: Basics (Chapter 10 of McDonald) Originally prepared

More information

DERIVATIVES AS MATRICES; CHAIN RULE

DERIVATIVES AS MATRICES; CHAIN RULE DERIVATIVES AS MATRICES; CHAIN RULE 1. Derivatives of Real-valued Functions Let s first consider functions f : R 2 R. Recall that if the partial derivatives of f exist at the point (x 0, y 0 ), then we

More information

Recall that two vectors in are perpendicular or orthogonal provided that their dot

Recall that two vectors in are perpendicular or orthogonal provided that their dot Orthogonal Complements and Projections Recall that two vectors in are perpendicular or orthogonal provided that their dot product vanishes That is, if and only if Example 1 The vectors in are orthogonal

More information

Time Value of Money Revisited: Part 1 Terminology. Learning Outcomes. Time Value of Money

Time Value of Money Revisited: Part 1 Terminology. Learning Outcomes. Time Value of Money Time Value of Money Revisited: Part 1 Terminology Intermediate Accounting II Dr. Chula King 1 Learning Outcomes Definition of Time Value of Money Components of Time Value of Money How to Answer the Question

More information

Bond Options, Caps and the Black Model

Bond Options, Caps and the Black Model Bond Options, Caps and the Black Model Black formula Recall the Black formula for pricing options on futures: C(F, K, σ, r, T, r) = Fe rt N(d 1 ) Ke rt N(d 2 ) where d 1 = 1 [ σ ln( F T K ) + 1 ] 2 σ2

More information

About the Gamma Function

About the Gamma Function About the Gamma Function Notes for Honors Calculus II, Originally Prepared in Spring 995 Basic Facts about the Gamma Function The Gamma function is defined by the improper integral Γ) = The integral is

More information

Slope-Intercept Form of a Linear Equation Examples

Slope-Intercept Form of a Linear Equation Examples Slope-Intercept Form of a Linear Equation Examples. In the figure at the right, AB passes through points A(0, b) and B(x, y). Notice that b is the y-intercept of AB. Suppose you want to find an equation

More information

A) 1.8% B) 1.9% C) 2.0% D) 2.1% E) 2.2%

A) 1.8% B) 1.9% C) 2.0% D) 2.1% E) 2.2% 1 Exam FM Questions Practice Exam 1 1. Consider the following yield curve: Year Spot Rate 1 5.5% 2 5.0% 3 5.0% 4 4.5% 5 4.0% Find the four year forward rate. A) 1.8% B) 1.9% C) 2.0% D) 2.1% E) 2.2% 2.

More information

Lecture Notes on the Mathematics of Finance

Lecture Notes on the Mathematics of Finance Lecture Notes on the Mathematics of Finance Jerry Alan Veeh February 20, 2006 Copyright 2006 Jerry Alan Veeh. All rights reserved. 0. Introduction The objective of these notes is to present the basic aspects

More information

Scalar Valued Functions of Several Variables; the Gradient Vector

Scalar Valued Functions of Several Variables; the Gradient Vector Scalar Valued Functions of Several Variables; the Gradient Vector Scalar Valued Functions vector valued function of n variables: Let us consider a scalar (i.e., numerical, rather than y = φ(x = φ(x 1,

More information

Lecture Notes on Actuarial Mathematics

Lecture Notes on Actuarial Mathematics Lecture Notes on Actuarial Mathematics Jerry Alan Veeh May 1, 2006 Copyright 2006 Jerry Alan Veeh. All rights reserved. 0. Introduction The objective of these notes is to present the basic aspects of the

More information

Chapter 27: Taxation. 27.1: Introduction. 27.2: The Two Prices with a Tax. 27.2: The Pre-Tax Position

Chapter 27: Taxation. 27.1: Introduction. 27.2: The Two Prices with a Tax. 27.2: The Pre-Tax Position Chapter 27: Taxation 27.1: Introduction We consider the effect of taxation on some good on the market for that good. We ask the questions: who pays the tax? what effect does it have on the equilibrium

More information

Lab 17: Consumer and Producer Surplus

Lab 17: Consumer and Producer Surplus Lab 17: Consumer and Producer Surplus Who benefits from rent controls? Who loses with price controls? How do taxes and subsidies affect the economy? Some of these questions can be analyzed using the concepts

More information

OPTIMAL CONTROL OF A COMMERCIAL LOAN REPAYMENT PLAN. E.V. Grigorieva. E.N. Khailov

OPTIMAL CONTROL OF A COMMERCIAL LOAN REPAYMENT PLAN. E.V. Grigorieva. E.N. Khailov DISCRETE AND CONTINUOUS Website: http://aimsciences.org DYNAMICAL SYSTEMS Supplement Volume 2005 pp. 345 354 OPTIMAL CONTROL OF A COMMERCIAL LOAN REPAYMENT PLAN E.V. Grigorieva Department of Mathematics

More information

Valuing Stock Options: The Black-Scholes-Merton Model. Chapter 13

Valuing Stock Options: The Black-Scholes-Merton Model. Chapter 13 Valuing Stock Options: The Black-Scholes-Merton Model Chapter 13 Fundamentals of Futures and Options Markets, 8th Ed, Ch 13, Copyright John C. Hull 2013 1 The Black-Scholes-Merton Random Walk Assumption

More information

Engineering Economy. Time Value of Money-3

Engineering Economy. Time Value of Money-3 Engineering Economy Time Value of Money-3 Prof. Kwang-Kyu Seo 1 Chapter 2 Time Value of Money Interest: The Cost of Money Economic Equivalence Interest Formulas Single Cash Flows Equal-Payment Series Dealing

More information

CAPM, Arbitrage, and Linear Factor Models

CAPM, Arbitrage, and Linear Factor Models CAPM, Arbitrage, and Linear Factor Models CAPM, Arbitrage, Linear Factor Models 1/ 41 Introduction We now assume all investors actually choose mean-variance e cient portfolios. By equating these investors

More information

n(n + 1) 2 1 + 2 + + n = 1 r (iii) infinite geometric series: if r < 1 then 1 + 2r + 3r 2 1 e x = 1 + x + x2 3! + for x < 1 ln(1 + x) = x x2 2 + x3 3

n(n + 1) 2 1 + 2 + + n = 1 r (iii) infinite geometric series: if r < 1 then 1 + 2r + 3r 2 1 e x = 1 + x + x2 3! + for x < 1 ln(1 + x) = x x2 2 + x3 3 ACTS 4308 FORMULA SUMMARY Section 1: Calculus review and effective rates of interest and discount 1 Some useful finite and infinite series: (i) sum of the first n positive integers: (ii) finite geometric

More information

3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes

3.3. Solving Polynomial Equations. Introduction. Prerequisites. Learning Outcomes Solving Polynomial Equations 3.3 Introduction Linear and quadratic equations, dealt within Sections 3.1 and 3.2, are members of a class of equations, called polynomial equations. These have the general

More information

Chapter 6. Commodity Forwards and Futures. Question 6.1. Question 6.2

Chapter 6. Commodity Forwards and Futures. Question 6.1. Question 6.2 Chapter 6 Commodity Forwards and Futures Question 6.1 The spot price of a widget is $70.00. With a continuously compounded annual risk-free rate of 5%, we can calculate the annualized lease rates according

More information

CHAPTER 2. Eigenvalue Problems (EVP s) for ODE s

CHAPTER 2. Eigenvalue Problems (EVP s) for ODE s A SERIES OF CLASS NOTES FOR 005-006 TO INTRODUCE LINEAR AND NONLINEAR PROBLEMS TO ENGINEERS, SCIENTISTS, AND APPLIED MATHEMATICIANS DE CLASS NOTES 4 A COLLECTION OF HANDOUTS ON PARTIAL DIFFERENTIAL EQUATIONS

More information

Business 2019. Fundamentals of Finance, Chapter 6 Solution to Selected Problems

Business 2019. Fundamentals of Finance, Chapter 6 Solution to Selected Problems Business 209 Fundamentals of Finance, Chapter 6 Solution to Selected Problems 8. Calculating Annuity Values You want to have $50,000 in your savings account five years from now, and you re prepared to

More information

Average rate of change of y = f(x) with respect to x as x changes from a to a + h:

Average rate of change of y = f(x) with respect to x as x changes from a to a + h: L15-1 Lecture 15: Section 3.4 Definition of the Derivative Recall the following from Lecture 14: For function y = f(x), the average rate of change of y with respect to x as x changes from a to b (on [a,

More information

Comparing Simple and Compound Interest

Comparing Simple and Compound Interest Comparing Simple and Compound Interest GRADE 11 In this lesson, students compare various savings and investment vehicles by calculating simple and compound interest. Prerequisite knowledge: Students should

More information

Homework #2 Solutions

Homework #2 Solutions MAT Spring Problems Section.:, 8,, 4, 8 Section.5:,,, 4,, 6 Extra Problem # Homework # Solutions... Sketch likely solution curves through the given slope field for dy dx = x + y...8. Sketch likely solution

More information

Lecture Notes on MONEY, BANKING, AND FINANCIAL MARKETS. Peter N. Ireland Department of Economics Boston College. irelandp@bc.edu

Lecture Notes on MONEY, BANKING, AND FINANCIAL MARKETS. Peter N. Ireland Department of Economics Boston College. irelandp@bc.edu Lecture Notes on MONEY, BANKING, AND FINANCIAL MARKETS Peter N. Ireland Department of Economics Boston College irelandp@bc.edu http://www2.bc.edu/~irelandp/ec261.html Chapter 16: Determinants of the Money

More information

5 Systems of Equations

5 Systems of Equations Systems of Equations Concepts: Solutions to Systems of Equations-Graphically and Algebraically Solving Systems - Substitution Method Solving Systems - Elimination Method Using -Dimensional Graphs to Approximate

More information

Mathematics. Rosella Castellano. Rome, University of Tor Vergata

Mathematics. Rosella Castellano. Rome, University of Tor Vergata and Loans Mathematics Rome, University of Tor Vergata and Loans Future Value for Simple Interest Present Value for Simple Interest You deposit E. 1,000, called the principal or present value, into a savings

More information

1 The Black-Scholes Formula

1 The Black-Scholes Formula 1 The Black-Scholes Formula In 1973 Fischer Black and Myron Scholes published a formula - the Black-Scholes formula - for computing the theoretical price of a European call option on a stock. Their paper,

More information

Review of Fundamental Mathematics

Review of Fundamental Mathematics Review of Fundamental Mathematics As explained in the Preface and in Chapter 1 of your textbook, managerial economics applies microeconomic theory to business decision making. The decision-making tools

More information

Option Properties. Liuren Wu. Zicklin School of Business, Baruch College. Options Markets. (Hull chapter: 9)

Option Properties. Liuren Wu. Zicklin School of Business, Baruch College. Options Markets. (Hull chapter: 9) Option Properties Liuren Wu Zicklin School of Business, Baruch College Options Markets (Hull chapter: 9) Liuren Wu (Baruch) Option Properties Options Markets 1 / 17 Notation c: European call option price.

More information

5. Time value of money

5. Time value of money 1 Simple interest 2 5. Time value of money With simple interest, the amount earned each period is always the same: i = rp o We will review some tools for discounting cash flows. where i = interest earned

More information

Moreover, under the risk neutral measure, it must be the case that (5) r t = µ t.

Moreover, under the risk neutral measure, it must be the case that (5) r t = µ t. LECTURE 7: BLACK SCHOLES THEORY 1. Introduction: The Black Scholes Model In 1973 Fisher Black and Myron Scholes ushered in the modern era of derivative securities with a seminal paper 1 on the pricing

More information

Derivatives as Rates of Change

Derivatives as Rates of Change Derivatives as Rates of Change One-Dimensional Motion An object moving in a straight line For an object moving in more complicated ways, consider the motion of the object in just one of the three dimensions

More information

10.6 Functions - Compound Interest

10.6 Functions - Compound Interest 10.6 Functions - Compound Interest Objective: Calculate final account balances using the formulas for compound and continuous interest. An application of exponential functions is compound interest. When

More information

APPENDIX. Interest Concepts of Future and Present Value. Concept of Interest TIME VALUE OF MONEY BASIC INTEREST CONCEPTS

APPENDIX. Interest Concepts of Future and Present Value. Concept of Interest TIME VALUE OF MONEY BASIC INTEREST CONCEPTS CHAPTER 8 Current Monetary Balances 395 APPENDIX Interest Concepts of Future and Present Value TIME VALUE OF MONEY In general business terms, interest is defined as the cost of using money over time. Economists

More information

第 9 讲 : 股 票 期 权 定 价 : B-S 模 型 Valuing Stock Options: The Black-Scholes Model

第 9 讲 : 股 票 期 权 定 价 : B-S 模 型 Valuing Stock Options: The Black-Scholes Model 1 第 9 讲 : 股 票 期 权 定 价 : B-S 模 型 Valuing Stock Options: The Black-Scholes Model Outline 有 关 股 价 的 假 设 The B-S Model 隐 性 波 动 性 Implied Volatility 红 利 与 期 权 定 价 Dividends and Option Pricing 美 式 期 权 定 价 American

More information