1 Interest rates and bonds
|
|
- Aubrey Reeves
- 7 years ago
- Views:
Transcription
1 1 Interest rates and bonds 1.1 Copounding There are different ways of easuring interest rates Exaple 1 The interest rate on a one-year deposit is 10% per annu. This stateent eans different things depending on the copounding frequency. With annual copounding it eans that $100 grows to $100 With seiannual copounding it eans that $100 grows to $ ) ) = $ With copounding ties per year it eans that $100 grows to $ ) With continuous copounding let ) it eans that $100 grows to $100 e For ost practical purposes continuous copounding can be thought of as being equivalent to daily copounding, which here eans that $100 grows to $ ) Let R c be a rate of interest with continuous copounding and R the equivalent rate with copounding with copounding ties per year. Then e Rc 1 = ), so and R c = ln 1+ R ) ) R = e Rc/ 1. 1
2 1.2 Zero coupon bonds and zero rates As stated before the ost basic interest rate derivatives are zero coupon bonds. Definition 1 A zero coupon bond with $1 principal and aturity T is a T-clai paying $1 at tie T. The price at tie t of the bond is denoted pt,t) and we will assue that pt,t) = 1. Spot rates are defined fro zero coupon bond prices. They are also known as zero coupon bond rates or zero rates. Definition 2 The continuously copounded) spot rate rt, T) at tie t for the tie interval [t,t] is defined by pt,t) = e rt,t)t t) or rt,t) = lnpt,t). T t The price at tie t of the bond is denoted pt,t) and we will assue that pt,t) = 1. For each aturity T there exists a separate spot rate. The collection of all tie t spot rates rt,t) : T = t,..., is referred to as the ter structure or zero coupon yield curve, or zero curve or yield curve) at tie t. Exaple 2 Given prices of zero coupon bonds T p0,t) 0.3 $ $ $ the ter structure is T r0,t) 0.3 5% 0.6 8% % 1.3 Money arket account Before the oney arket account or risk less asset has been given by B T = e rt if B 0 = 1) where r has been constant, but now we will have in discrete tie) B t+ t = B t e rt,t+ t) t so if we divide the tie interval [t,t] into n intervals of size t = T t)/n n 1 B T = B t e i=0 rt+i t,t+i+1) t) t Here rt,t+ t) is the over night rate, and it is deterined at tie t. 2
3 Exaple 3 Consider a three onth investent of $10 in a oney arket account with onthly capitalization eans that t= 1 onth = 1/12 year). Assue that the one onth rate is 5% per annu the first onth, 5.5% the second, and 6% the third. Then 3 1 B 3/12 = B 0 e i=0 r0+i t,0+i+1) t) t { = 10exp r0, 1 12 ) r 1 12, 2 12 ) r 2 12, 3 } 12 ) 1 12 = 10e ) 1 12 = Note that r1/12,2/12) and r2/12,3/12) are not really known at tie t = 0! 1.4 Fixed coupon bonds Most bonds pay coupons to the holder periodically. The siplest coupon bond is the fixed coupon bond. The foral description is as follows. Fix ties T 0,T 1,...,T n. Here T 0 is thought of as the eission date of the bond and T 1,...,T n as coupon dates. At tie T i, i = 1,...,n the owner of the bond receives the deterinistic coupon c i. At tie T n the owner of the bond receives the principal face value) K. The coupon bond is siply a portfolio of zero coupon bonds. We have that the price of the bond for t < T 1 is p fixed t) = c i pt,t i )+Kpt,T n ). 1) Often the coupons are deterined in ters of return rather than onetary ters. The return forthei:thcouponistypically quotedasasiplerateactingontheprincipalk over[t i 1,T i ]. Here coes a sall digression on siple rates. We have seen that the continuously copounded spot rate rt,t) for the interval [t,t] solves or pt,t) = e rt,t)t t) e rt,t)t t) = 1 pt,t). This is because an investent of $1 at tie t can create a payoff of 1/pt,T) at tie T. Just buy 1/pt,T) T-bonds at tie t, this will cost you exactly $1, and at tie T you will get the payoff 1/pt,T). The siple spot rate for [t,t], henceforth referred to as LIBOR spot rate solves and is defined as 1+Lt,T)T t) = 1 pt,t), Lt,T) = pt,t) 1 T t)pt,t). The zero rates and LIBOR spot rates are thus used to express the sae return, they are just quoted in different ways. 3
4 Back to the coupon bonds. A siple rate r i acting on K over [T i 1,T i ] results in K[1+r i T i T i 1 )] and by definition if the i:th coupon has a return r i and the principal is K then c i = Kr i T i T i 1 ). Reark 1 Recall that the rate R copounded ties per year akes $1 grow to 1+ R ) Over a period of 1/ years $1 grows to 1+R 1 ) which is the siple rate for the period 1/! Deterining zero rates by bootstrapping Given arket bond prices, how do we deterine the zero coupon curve? Recall that ost traded bonds are coupon bonds, particularly for the longer aturities. Since the price of a fixed coupon bond is ade up of prices of zero coupon bonds, see Equation 1), which are in turn deterined by the zero rates, there is a one-to-one correspondence, between bond prices and zero rates. Starting fro the short end of the ter structure that is short tie to aturity) zero rates are obtained fro zero coupon bonds. When oving to longer aturities bonds are coupon bonds, but now we can use that we have the short zero rates to back out the longer rates. Exaple 4 Assue that in addition to the zero coupon bonds given in Exaple 2 there is also a coupon bond traded. This bond pays a $5 annual coupon, has a principal of $100, and atures in 1.6 years. The price of the coupon bond is $ We can use this bond to find the 1.6-year zero rate. The price is given by p fixed t) = c i pt,t i )+Kpt,T n ) = 5e rt;t+0.6) )e rt;t+1.6)1.6. Now using that we know that p fixed = and that rt;t + 0.6) = 8%, we can solve for rt;t+1.6) and obtain rt;t+1.6) = e ln We can thus extend the table of zero rates to T t rt,t) 0.3 5% 0.6 8% % %
5 1.4.2 Yield and duration Definition 3 The yield to aturity of a fixed coupon bond is the interest rate y that when used to discount all coupons and the principal results in the arket price. The yield thus solves p arket t) = c i e yti t) +Ke ytn t). Reark 2 The yield to aturity of a zero coupon bond is siply the zero rate. This represents the bonds internal rate of interest and the above definition is an attept to extend the concept to fixed coupon bonds. Exaple 5 The yield to aturity of the coupon bond considered in Exaple 4 solves = 5e y )e y 1.6. It should lie between 8% and 11% the zero rates for aturities 0.6 and 1.6, and it should be closer to 11%, since ost weight is given to the large final payent. Trial and error gives y 10.94%. To siplify notation includethe principal K in the last coupon so c new n that t = 0. Also let p = p fixed, then p = c i e yti t). Definition 4 The duration D of a fixed coupon bond is defined as n T i c i e yt i c i e yt i D = = T i. p p = K+c old n and assue The duration is thus a weighted average of the coupon dates of the bond where the weight for a certain coupon date is the present value of the coupon payent divided by the value of the bond which is the present value of all the payents). Duration is a easure of how long on average the holder of the bond has to wait before receiving cash payents or ean tie to coupon payent. Note that for a zero coupon bond with aturity T the duration is T. Duration also acts as a easure of sensitivity of the bond price to changes in the yield. Proposition 1 With notation as above we have dp dy = d { n } c i e yt i t) dy = Dp So duration is essentially for bonds with respect to yield) what delta is for derivatives with respect to the underlying price). Approxiately it holds that p Dp y or p p D y For concrete coputations, see Exaple 6.5 in Hull. 5
6 Reark 3 Note that the change in yield is supposed to be the sae for all aturities. This eans that we are looking at what happens when there is a parallel shift in the yield curve zero curve). Corresponding to gaa for derivatives there is convexity for bonds. It is defined as 1.5 Forward rates C = d2 p dy 2 We have seen that the continuously copounded spot rate rt,t) solves e rt,t)t t) = 1 pt,t). 2) This is because an investent of $1 at tie t can create a risk less payoff of $1/pt,T) at tie T. Now fix t < S < T ans suppose t is today and we want to offer an interest rate, deterined today, over the future interval [S,T]. We will extend the arguent that led to 2). At tie t sell an S-bond. This will earn you pt,s). Invest the oney in T-bonds. This will give you pt,s)/pt,t) T-bonds. The net investent at tie t is zero. At tie S you will have to pay $1 to the owner of the S-bond. At tie T you will receive $pt,s)/pt,t) 1. Thus an investent of $1 at tie S results in a payoff of pt,s)/pt,t) at tie T. We would therefore offer the rate e ft;s,t)t S) = pt,s) pt,t). 3) Definition 5 The continuously copounded forward rate for [S, T] contracted at t is defined as ft;s,t) = lnpt,t) lnpt,s) 4) T S Reark 4 Note that as S t the forward rate tends to the spot rate lift;s,t) = rt,t). S t There exists a ter structure of forward rates as well; it is in two diensions since for each future tie S there exists one forward rate for each T S. Forally we have that the ter structure of forward rates is given by {ft;s,t) : S [t,t], T [S, )}. If we use that pt,t) = e rt,t)t t) in 4) we get the following relationship between forward rates and spot rates rt,t)t t) = rt,s)s t)+ft;s,t)t S). The ter structure of forward rates is therefore deterined by the ter structure of spot rates. For a concrete exaple of coputations using the relation, see Table 4.5 in Hull. 6
OpenGamma Documentation Bond Pricing
OpenGaa Docuentation Bond Pricing Marc Henrard arc@opengaa.co OpenGaa Docuentation n. 5 Version 2.0 - May 2013 Abstract The details of the ipleentation of pricing for fixed coupon bonds and floating rate
More informationHow To Get A Loan From A Bank For Free
Finance 111 Finance We have to work with oney every day. While balancing your checkbook or calculating your onthly expenditures on espresso requires only arithetic, when we start saving, planning for retireent,
More informationConstruction Economics & Finance. Module 3 Lecture-1
Depreciation:- Construction Econoics & Finance Module 3 Lecture- It represents the reduction in arket value of an asset due to age, wear and tear and obsolescence. The physical deterioration of the asset
More informationCalculating the Return on Investment (ROI) for DMSMS Management. The Problem with Cost Avoidance
Calculating the Return on nvestent () for DMSMS Manageent Peter Sandborn CALCE, Departent of Mechanical Engineering (31) 45-3167 sandborn@calce.ud.edu www.ene.ud.edu/escml/obsolescence.ht October 28, 21
More information550.444 Introduction to Financial Derivatives
550.444 Introduction to Financial Derivatives Week of October 7, 2013 Interest Rate Futures Where we are Last week: Forward & Futures Prices/Value (Chapter 5, OFOD) This week: Interest Rate Futures (Chapter
More informationPricing Forwards and Swaps
Chapter 7 Pricing Forwards and Swaps 7. Forwards Throughout this chapter, we will repeatedly use the following property of no-arbitrage: P 0 (αx T +βy T ) = αp 0 (x T )+βp 0 (y T ). Here, P 0 (w T ) is
More informationTIME VALUE OF MONEY PROBLEMS CHAPTERS THREE TO TEN
TIME VLUE OF MONEY PROBLEMS CHPTERS THREE TO TEN Probles In how any years $ will becoe $265 if = %? 265 ln n 933844 9 34 years ln( 2 In how any years will an aount double if = 76%? ln 2 n 9 46 years ln76
More informationA quantum secret ballot. Abstract
A quantu secret ballot Shahar Dolev and Itaar Pitowsky The Edelstein Center, Levi Building, The Hebrerw University, Givat Ra, Jerusale, Israel Boaz Tair arxiv:quant-ph/060087v 8 Mar 006 Departent of Philosophy
More informationPhysics 211: Lab Oscillations. Simple Harmonic Motion.
Physics 11: Lab Oscillations. Siple Haronic Motion. Reading Assignent: Chapter 15 Introduction: As we learned in class, physical systes will undergo an oscillatory otion, when displaced fro a stable equilibriu.
More informationAnalysis of Deterministic Cash Flows and the Term Structure of Interest Rates
Analysis of Deterministic Cash Flows and the Term Structure of Interest Rates Cash Flow Financial transactions and investment opportunities are described by cash flows they generate. Cash flow: payment
More informationCHAPTER 14. Z(t; i) + Z(t; 5),
CHAPER 14 FIXED-INCOME PRODUCS AND ANALYSIS: YIELD, DURAION AND CONVEXIY 1. A coupon bond pays out 3% every year, with a principal of $1 and a maturity of five years. Decompose the coupon bond into a set
More informationChapter 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 informationFactor Model. Arbitrage Pricing Theory. Systematic Versus Non-Systematic Risk. Intuitive Argument
Ross [1],[]) presents the aritrage pricing theory. The idea is that the structure of asset returns leads naturally to a odel of risk preia, for otherwise there would exist an opportunity for aritrage profit.
More information1 The Black-Scholes model: extensions and hedging
1 The Black-Scholes model: extensions and hedging 1.1 Dividends Since we are now in a continuous time framework the dividend paid out at time t (or t ) is given by dd t = D t D t, where as before D denotes
More informationInvesting in corporate bonds?
Investing in corporate bonds? This independent guide fro the Australian Securities and Investents Coission (ASIC) can help you look past the return and assess the risks of corporate bonds. If you re thinking
More informationInvesting in corporate bonds?
Investing in corporate bonds? This independent guide fro the Australian Securities and Investents Coission (ASIC) can help you look past the return and assess the risks of corporate bonds. If you re thinking
More informationInterest Rate and Currency Swaps
Interest Rate and Currency Swaps Eiteman et al., Chapter 14 Winter 2004 Bond Basics Consider the following: Zero-Coupon Zero-Coupon One-Year Implied Maturity Bond Yield Bond Price Forward Rate t r 0 (0,t)
More informationPricing Interest Rate Derivatives Using Black-Derman-Toy Short Rate Binomial Tree
Pricing Interest Rate Derivatives Using Black-Derman-Toy Short Rate Binomial Tree Shing Hing Man http://lombok.demon.co.uk/financialtap/ 3th August, 0 Abstract This note describes how to construct a short
More informationFIN 472 Fixed-Income Securities Forward Rates
FIN 472 Fixed-Income Securities Forward Rates Professor Robert B.H. Hauswald Kogod School of Business, AU Interest-Rate Forwards Review of yield curve analysis Forwards yet another use of yield curve forward
More informationProblems 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 informationA Gas Law And Absolute Zero Lab 11
HB 04-06-05 A Gas Law And Absolute Zero Lab 11 1 A Gas Law And Absolute Zero Lab 11 Equipent safety goggles, SWS, gas bulb with pressure gauge, 10 C to +110 C theroeter, 100 C to +50 C theroeter. Caution
More informationThe Velocities of Gas Molecules
he Velocities of Gas Molecules by Flick Colean Departent of Cheistry Wellesley College Wellesley MA 8 Copyright Flick Colean 996 All rights reserved You are welcoe to use this docuent in your own classes
More informationManual for SOA Exam FM/CAS Exam 2.
Manual for SOA Exa FM/CAS Exa 2. Chapter 1. Basic Interest Theory. c 2008. Miguel A. Arcones. All rights reserved. Extract fro: Arcones Manual for the SOA Exa FM/CAS Exa 2, Financial Matheatics. Spring
More informationCaput 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 informationBond 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 informationInterest Rate Futures. Chapter 6
Interest Rate Futures Chapter 6 1 Day Count Convention The day count convention defines: The period of time to which the interest rate applies. The period of time used to calculate accrued interest (relevant
More informationA Gas Law And Absolute Zero
A Gas Law And Absolute Zero Equipent safety goggles, DataStudio, gas bulb with pressure gauge, 10 C to +110 C theroeter, 100 C to +50 C theroeter. Caution This experient deals with aterials that are very
More informationLecture L9 - Linear Impulse and Momentum. Collisions
J. Peraire, S. Widnall 16.07 Dynaics Fall 009 Version.0 Lecture L9 - Linear Ipulse and Moentu. Collisions In this lecture, we will consider the equations that result fro integrating Newton s second law,
More informationLesson 44: Acceleration, Velocity, and Period in SHM
Lesson 44: Acceleration, Velocity, and Period in SHM Since there is a restoring force acting on objects in SHM it akes sense that the object will accelerate. In Physics 20 you are only required to explain
More informationHOW CLOSE ARE THE OPTION PRICING FORMULAS OF BACHELIER AND BLACK-MERTON-SCHOLES?
HOW CLOSE ARE THE OPTION PRICING FORMULAS OF BACHELIER AND BLACK-MERTON-SCHOLES? WALTER SCHACHERMAYER AND JOSEF TEICHMANN Abstract. We copare the option pricing forulas of Louis Bachelier and Black-Merton-Scholes
More informationIntroduction to Unit Conversion: the SI
The Matheatics 11 Copetency Test Introduction to Unit Conversion: the SI In this the next docuent in this series is presented illustrated an effective reliable approach to carryin out unit conversions
More informationInterest Rate and Credit Risk Derivatives
Interest Rate and Credit Risk Derivatives Interest Rate and Credit Risk Derivatives Peter Ritchken Kenneth Walter Haber Professor of Finance Weatherhead School of Management Case Western Reserve University
More information10. Fixed-Income Securities. Basic Concepts
0. Fixed-Income Securities Fixed-income securities (FIS) are bonds that have no default risk and their payments are fully determined in advance. Sometimes corporate bonds that do not necessarily have certain
More informationPractice Set #7: Binomial option pricing & Delta hedging. What to do with this practice set?
Derivatives (3 credits) Professor Michel Robe Practice Set #7: Binomial option pricing & Delta hedging. What to do with this practice set? To help students with the material, eight practice sets with solutions
More informationHalloween Costume Ideas for the Wii Game
Algorithica 2001) 30: 101 139 DOI: 101007/s00453-001-0003-0 Algorithica 2001 Springer-Verlag New York Inc Optial Search and One-Way Trading Online Algoriths R El-Yaniv, 1 A Fiat, 2 R M Karp, 3 and G Turpin
More informationCFA Level -2 Derivatives - I
CFA Level -2 Derivatives - I EduPristine www.edupristine.com Agenda Forwards Markets and Contracts Future Markets and Contracts Option Markets and Contracts 1 Forwards Markets and Contracts 2 Pricing and
More informationThe Mathematics of Pumping Water
The Matheatics of Puping Water AECOM Design Build Civil, Mechanical Engineering INTRODUCTION Please observe the conversion of units in calculations throughout this exeplar. In any puping syste, the role
More informationInterest rate Derivatives
Interest rate Derivatives There is a wide variety of interest rate options available. The most widely offered are interest rate caps and floors. Increasingly we also see swaptions offered. This note will
More informationESTIMATING LIQUIDITY PREMIA IN THE SPANISH GOVERNMENT SECURITIES MARKET
ESTIMATING LIQUIDITY PREMIA IN THE SPANISH GOVERNMENT SECURITIES MARKET Francisco Alonso, Roberto Blanco, Ana del Río and Alicia Sanchis Banco de España Banco de España Servicio de Estudios Docuento de
More information5.7 Chebyshev Multi-section Matching Transformer
/9/ 5_7 Chebyshev Multisection Matching Transforers / 5.7 Chebyshev Multi-section Matching Transforer Reading Assignent: pp. 5-55 We can also build a ultisection atching network such that Γ f is a Chebyshev
More informationDerivatives Interest Rate Futures. Professor André Farber Solvay Brussels School of Economics and Management Université Libre de Bruxelles
Derivatives Interest Rate Futures Professor André Farber Solvay Brussels School of Economics and Management Université Libre de Bruxelles Interest Rate Derivatives Forward rate agreement (FRA): OTC contract
More informationASSET LIABILITY MANAGEMENT Significance and Basic Methods. Dr Philip Symes. Philip Symes, 2006
1 ASSET LIABILITY MANAGEMENT Significance and Basic Methods Dr Philip Symes Introduction 2 Asset liability management (ALM) is the management of financial assets by a company to make returns. ALM is necessary
More informationFixed Income Portfolio Management. Interest rate sensitivity, duration, and convexity
Fixed Income ortfolio Management Interest rate sensitivity, duration, and convexity assive bond portfolio management Active bond portfolio management Interest rate swaps 1 Interest rate sensitivity, duration,
More informationVALUATION OF FIXED INCOME SECURITIES. Presented By Sade Odunaiya Partner, Risk Management Alliance Consulting
VALUATION OF FIXED INCOME SECURITIES Presented By Sade Odunaiya Partner, Risk Management Alliance Consulting OUTLINE Introduction Valuation Principles Day Count Conventions Duration Covexity Exercises
More informationProblem Set 2: Solutions ECON 301: Intermediate Microeconomics Prof. Marek Weretka. Problem 1 (Marginal Rate of Substitution)
Proble Set 2: Solutions ECON 30: Interediate Microeconoics Prof. Marek Weretka Proble (Marginal Rate of Substitution) (a) For the third colun, recall that by definition MRS(x, x 2 ) = ( ) U x ( U ). x
More informationYield to Maturity Outline and Suggested Reading
Yield to Maturity Outline Outline and Suggested Reading Yield to maturity on bonds Coupon effects Par rates Buzzwords Internal rate of return, Yield curve Term structure of interest rates Suggested reading
More informationProblems 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 informationForward Contracts and Forward Rates
Forward Contracts and Forward Rates Outline and Readings Outline Forward Contracts Forward Prices Forward Rates Information in Forward Rates Reading Veronesi, Chapters 5 and 7 Tuckman, Chapters 2 and 16
More informationExample 1. Consider the following two portfolios: 2. Buy one c(s(t), 20, τ, r) and sell one c(s(t), 10, τ, r).
Chapter 4 Put-Call Parity 1 Bull and Bear Financial analysts use words such as bull and bear to describe the trend in stock markets. Generally speaking, a bull market is characterized by rising prices.
More informationContinuous 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 informationIn terms of expected returns, MSFT should invest in the U.K.
Rauli Susmel Dept. of Finance Univ. of Houston FINA 4360 International Financial Management 12/4/02 Chapter 21 Short-term Investing MNCs have many choices for investing Home return(usd) = deposit interest
More information3. 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 informationFINANCE 1. DESS Gestion des Risques/Ingéniérie Mathématique Université d EVRY VAL D ESSONNE EXERCICES CORRIGES. Philippe PRIAULET
FINANCE 1 DESS Gestion des Risques/Ingéniérie Mathématique Université d EVRY VAL D ESSONNE EXERCICES CORRIGES Philippe PRIAULET 1 Exercise 1 On 12/04/01 consider a fixed coupon bond whose features are
More information2. Determine the appropriate discount rate based on the risk of the security
Fixed Income Instruments III Intro to the Valuation of Debt Securities LOS 64.a Explain the steps in the bond valuation process 1. Estimate the cash flows coupons and return of principal 2. Determine the
More informationLecture L26-3D Rigid Body Dynamics: The Inertia Tensor
J. Peraire, S. Widnall 16.07 Dynaics Fall 008 Lecture L6-3D Rigid Body Dynaics: The Inertia Tensor Version.1 In this lecture, we will derive an expression for the angular oentu of a 3D rigid body. We shall
More informationOption 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 informationEurodollar Futures, and Forwards
5 Eurodollar Futures, and Forwards In this chapter we will learn about Eurodollar Deposits Eurodollar Futures Contracts, Hedging strategies using ED Futures, Forward Rate Agreements, Pricing FRAs. Hedging
More informationOnline Appendix I: A Model of Household Bargaining with Violence. In this appendix I develop a simple model of household bargaining that
Online Appendix I: A Model of Household Bargaining ith Violence In this appendix I develop a siple odel of household bargaining that incorporates violence and shos under hat assuptions an increase in oen
More informationChapter Nine Selected Solutions
Chapter Nine Selected Solutions 1. What is the difference between book value accounting and market value accounting? How do interest rate changes affect the value of bank assets and liabilities under the
More informationExamination II. Fixed income valuation and analysis. Economics
Examination II Fixed income valuation and analysis Economics Questions Foundation examination March 2008 FIRST PART: Multiple Choice Questions (48 points) Hereafter you must answer all 12 multiple choice
More informationCompound Interest. Invest 500 that earns 10% interest each year for 3 years, where each interest payment is reinvested at the same rate:
Compound Interest Invest 500 that earns 10% interest each year for 3 years, where each interest payment is reinvested at the same rate: Table 1 Development of Nominal Payments and the Terminal Value, S.
More informationLecture 7: Bounds on Options Prices Steven Skiena. http://www.cs.sunysb.edu/ skiena
Lecture 7: Bounds on Options Prices Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena Option Price Quotes Reading the
More informationChapter 4: Common Stocks. Chapter 5: Forwards and Futures
15.401 Part B Valuation Chapter 3: Fixed Income Securities Chapter 4: Common Stocks Chapter 5: Forwards and Futures Chapter 6: Options Lecture Notes Introduction 15.401 Part B Valuation We have learned
More informationFixed Income: Practice Problems with Solutions
Fixed Income: Practice Problems with Solutions Directions: Unless otherwise stated, assume semi-annual payment on bonds.. A 6.0 percent bond matures in exactly 8 years and has a par value of 000 dollars.
More informationOptions. Moty Katzman. September 19, 2014
Options Moty Katzman September 19, 2014 What are options? Options are contracts conferring certain rights regarding the buying or selling of assets. A European call option gives the owner the right to
More informationA short note on American option prices
A short note on American option Filip Lindskog April 27, 2012 1 The set-up An American call option with strike price K written on some stock gives the holder the right to buy a share of the stock (exercise
More informationSolutions to Practice Questions (Bonds)
Fuqua Business School Duke University FIN 350 Global Financial Management Solutions to Practice Questions (Bonds). These practice questions are a suplement to the problem sets, and are intended for those
More informationCHAPTER 11 CURRENCY AND INTEREST RATE FUTURES
Answers to end-of-chapter exercises ARBITRAGE IN THE CURRENCY FUTURES MARKET 1. Consider the following: Spot Rate: $ 0.65/DM German 1-yr interest rate: 9% US 1-yr interest rate: 5% CHAPTER 11 CURRENCY
More informationIs Pay-as-You-Drive Insurance a Better Way to Reduce Gasoline than Gasoline Taxes?
Is Pay-as-You-Drive Insurance a Better Way to Reduce Gasoline than Gasoline Taxes? By Ian W.H. Parry Despite concerns about US dependence on a volatile world oil arket, greenhouse gases fro fuel cobustion,
More informationReliability Constrained Packet-sizing for Linear Multi-hop Wireless Networks
Reliability Constrained acket-sizing for inear Multi-hop Wireless Networks Ning Wen, and Randall A. Berry Departent of Electrical Engineering and Coputer Science Northwestern University, Evanston, Illinois
More informationInterest Rate Swaps. Key Concepts and Buzzwords. Readings Tuckman, Chapter 18. Swaps Swap Spreads Credit Risk of Swaps Uses of Swaps
Interest Rate Swaps Key Concepts and Buzzwords Swaps Swap Spreads Credit Risk of Swaps Uses of Swaps Readings Tuckman, Chapter 18. Counterparty, Notional amount, Plain vanilla swap, Swap rate Interest
More informationOne Period Binomial Model
FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 One Period Binomial Model These notes consider the one period binomial model to exactly price an option. We will consider three different methods of pricing
More informationVALUATION OF DEBT CONTRACTS AND THEIR PRICE VOLATILITY CHARACTERISTICS QUESTIONS See answers below
VALUATION OF DEBT CONTRACTS AND THEIR PRICE VOLATILITY CHARACTERISTICS QUESTIONS See answers below 1. Determine the value of the following risk-free debt instrument, which promises to make the respective
More informationDuration and convexity
Duration and convexity Prepared by Pamela Peterson Drake, Ph.D., CFA Contents 1. Overview... 1 A. Calculating the yield on a bond... 4 B. The yield curve... 6 C. Option-like features... 8 D. Bond ratings...
More informationAssumptions: No transaction cost, same rate for borrowing/lending, no default/counterparty risk
Derivatives Why? Allow easier methods to short sell a stock without a broker lending it. Facilitates hedging easily Allows the ability to take long/short position on less available commodities (Rice, Cotton,
More informationPERFORMANCE METRICS FOR THE IT SERVICES PORTFOLIO
Bulletin of the Transilvania University of Braşov Series I: Engineering Sciences Vol. 4 (53) No. - 0 PERFORMANCE METRICS FOR THE IT SERVICES PORTFOLIO V. CAZACU I. SZÉKELY F. SANDU 3 T. BĂLAN Abstract:
More informationLecture 09: Multi-period Model Fixed Income, Futures, Swaps
Lecture 09: Multi-period Model Fixed Income, Futures, Swaps Prof. Markus K. Brunnermeier Slide 09-1 Overview 1. Bond basics 2. Duration 3. Term structure of the real interest rate 4. Forwards and futures
More informationCHAPTER 20: OPTIONS MARKETS: INTRODUCTION
CHAPTER 20: OPTIONS MARKETS: INTRODUCTION 1. Cost Profit Call option, X = 95 12.20 10 2.20 Put option, X = 95 1.65 0 1.65 Call option, X = 105 4.70 0 4.70 Put option, X = 105 4.40 0 4.40 Call option, X
More informationValuing 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 informationWork, Energy, Conservation of Energy
This test covers Work, echanical energy, kinetic energy, potential energy (gravitational and elastic), Hooke s Law, Conservation of Energy, heat energy, conservative and non-conservative forces, with soe
More informationDetermination of Forward and Futures Prices. Chapter 5
Determination of Forward and Futures Prices Chapter 5 Fundamentals of Futures and Options Markets, 8th Ed, Ch 5, Copyright John C. Hull 2013 1 Consumption vs Investment Assets Investment assets are assets
More informationReview for Exam 1. Instructions: Please read carefully
Review for Exam 1 Instructions: Please read carefully The exam will have 20 multiple choice questions and 5 work problems. Questions in the multiple choice section will be either concept or calculation
More informationChapter 5 - Determination of Forward and Futures Prices
Chapter 5 - Determination of Forward and Futures Prices Investment assets vs. consumption assets Short selling Assumptions and notations Forward price for an investment asset that provides no income Forward
More informationMoney Market and Debt Instruments
Prof. Alex Shapiro Lecture Notes 3 Money Market and Debt Instruments I. Readings and Suggested Practice Problems II. Bid and Ask III. Money Market IV. Long Term Credit Markets V. Additional Readings Buzz
More informationFNCE 301, Financial Management H Guy Williams, 2006
REVIEW We ve used the DCF method to find present value. We also know shortcut methods to solve these problems such as perpetuity present value = C/r. These tools allow us to value any cash flow including
More information24. Pricing Fixed Income Derivatives. through Black s Formula. MA6622, Ernesto Mordecki, CityU, HK, 2006. References for this Lecture:
24. Pricing Fixed Income Derivatives through Black s Formula MA6622, Ernesto Mordecki, CityU, HK, 2006. References for this Lecture: John C. Hull, Options, Futures & other Derivatives (Fourth Edition),
More informationFixed-Income Securities. Assignment
FIN 472 Professor Robert B.H. Hauswald Fixed-Income Securities Kogod School of Business, AU Assignment Please be reminded that you are expected to use contemporary computer software to solve the following
More informationSingle Name Credit Derivatives:
Single ame Credit Derivatives: Products & Valuation Stephen M Schaefer London Business School Credit Risk Elective Summer 2012 Objectives To understand What single-name credit derivatives are How single
More informationSAMPLING METHODS LEARNING OBJECTIVES
6 SAMPLING METHODS 6 Using Statistics 6-6 2 Nonprobability Sapling and Bias 6-6 Stratified Rando Sapling 6-2 6 4 Cluster Sapling 6-4 6 5 Systeatic Sapling 6-9 6 6 Nonresponse 6-2 6 7 Suary and Review of
More informationCORPORATE FINANCE # 2: INTERNAL RATE OF RETURN
CORPORATE FINANCE # 2: INTERNAL RATE OF RETURN Professor Ethel Silverstein Mathematics by Dr. Sharon Petrushka Introduction How do you compare investments with different initial costs ( such as $50,000
More informationDetermination of Forward and Futures Prices
Determination of Forward and Futures Prices Options, Futures, and Other Derivatives, 8th Edition, Copyright John C. Hull 2012 Short selling A popular trading (arbitrage) strategy is the shortselling or
More informationOn Black-Scholes Equation, Black- Scholes Formula and Binary Option Price
On Black-Scholes Equation, Black- Scholes Formula and Binary Option Price Abstract: Chi Gao 12/15/2013 I. Black-Scholes Equation is derived using two methods: (1) risk-neutral measure; (2) - hedge. II.
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?
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 informationChapter 1 - Introduction
Chapter 1 - Introduction Derivative securities Futures contracts Forward contracts Futures and forward markets Comparison of futures and forward contracts Options contracts Options markets Comparison of
More informationBinomial lattice model for stock prices
Copyright c 2007 by Karl Sigman Binomial lattice model for stock prices Here we model the price of a stock in discrete time by a Markov chain of the recursive form S n+ S n Y n+, n 0, where the {Y i }
More informationTreasury Bond Futures
Treasury Bond Futures Concepts and Buzzwords Basic Futures Contract Futures vs. Forward Delivery Options Reading Veronesi, Chapters 6 and 11 Tuckman, Chapter 14 Underlying asset, marking-to-market, convergence
More informationModel-Free Boundaries of Option Time Value and Early Exercise Premium
Model-Free Boundaries of Option Time Value and Early Exercise Premium Tie Su* Department of Finance University of Miami P.O. Box 248094 Coral Gables, FL 33124-6552 Phone: 305-284-1885 Fax: 305-284-4800
More information2 Stock Price. Figure S1.1 Profit from long position in Problem 1.13
Problem 1.11. A cattle farmer expects to have 12, pounds of live cattle to sell in three months. The livecattle futures contract on the Chicago Mercantile Exchange is for the delivery of 4, pounds of cattle.
More informationOptions. + Concepts and Buzzwords. Readings. Put-Call Parity Volatility Effects
+ Options + Concepts and Buzzwords Put-Call Parity Volatility Effects Call, put, European, American, underlying asset, strike price, expiration date Readings Tuckman, Chapter 19 Veronesi, Chapter 6 Options
More informationOption Values. Option Valuation. Call Option Value before Expiration. Determinants of Call Option Values
Option Values Option Valuation Intrinsic value profit that could be made if the option was immediately exercised Call: stock price exercise price : S T X i i k i X S Put: exercise price stock price : X
More information