P2.T5. Tuckman Chapter 7 The Science of Term Structure Models. Bionic Turtle FRM Video Tutorials. By: David Harper CFA, FRM, CIPM
|
|
- Chrystal Parker
- 6 years ago
- Views:
Transcription
1 P2.T5. Tuckman Chapter 7 The Science of Term Structure Models Bionic Turtle FRM Video Tutorials By: David Harper CFA, FRM, CIPM Note: This tutorial is for paid members only. You know who you are. Anybody else is using an illegal copy and also violates GARP s ethical standards.
2 P2.T5. Tuckman Chapter 7 P1.T5. Tuckman Chapter 7 Workbook Exam Relevance (XLS not topic) Worksheet T Term Structure Low Exp Disc Value (Zero) Low Low Low Low Risk Neutral Pricing Replicate Port (Option) Three Steps CMS Note: If you are unable to view the content within this document we recommend the following: MAC Users: The built-in pdf reader will not display our non-standard fonts. Please use adobe s pdf reader ( PC Users: We recommend you use the foxit pdf reader ( or adobe s pdf reader ( Mobile and Tablet users: We recommend you use the foxit pdf reader app or the adobe pdf reader app. All of these products are free. We apologize for any inconvenience. If you have any additional problems, please Suzanne at suzanne@bionicturtle.com. 2
3 Chapters 7: The Science of Term Structure Models
4 Calculate the expected discounted value of a zero coupon bond using a binomial tree. Six-month interest rate tree T Exp Disc Value (Zero) Spot rates 0 s(0.5) s(1.0) Spot (0,T) 5.00% 5.15% PV of $1,000 Par Interest Rate Binomial Tree T= 0.0 T= 0.5 T= % p % 1-p %
5 Calculate the expected discounted value of a zero coupon bond using a binomial tree. One-year interest rate tree T = 1.0 year and dt or Δt= 0.5 years T Exp Disc Value (Zero) Spot rate term structure s(0.5) s(1.0) s(1.5) Spot (0,T) 5.00% 5.15% 5.25% PV of $1,000 Par Interest Rate Process T= 0.0 T= 0.5 T= 1.0 T= % 5.50% 5.0% 5.0% 4.50% 4.0% 5
6 Calculate the expected discounted value of a zero coupon bond using a binomial tree. Expected discounted value is the weighted the future nodes, weighted by price, discounted to present value: Interest Rate Binomial Tree T= 0.0 T= 0.5 T= % T Exp Disc Value (Zero) p % 1-p % $1, % 2 1 $ $ $ % 2 $1, % 2 $ $ Price of Zero (semi-annual compounding) T= 0.0 T= 0.5 T= 1.0 $1,000 Par $1,000 $ $ $1,000 $ $1,000 Same, but discount factors instead T= 0.0 T= 0.5 T=
7 Using replicating portfolios, construct and apply an arbitrage argument to price a call option on a zero-coupon security. The payoff of the call option (i.e., $0 or $3) can be replicated by the portfolio that is long the one-year bond and short the six month bond. Therefore, the value of the option must equal the cost of the replicating portfolio T Replicate Port (Option) Interest Rate Binomial Tree T= 0.0 T= 0.5 p 50% 5.50% 5.0% 1-p 50% 4.50% II. Replicating Portfolio T= 0.0 T= 0.5 Bond $ Call $0.00 In this (Tuckman s) example, long $ face amount of the one-year bond plus short -$ face amount of the sixmonth bond replicates option payout and has an cost of $0.58 Bond Par: $1,000 Option $975 Long B (1.0) $ Short B (0.5) ($612.50) $0.00 Replicating Face(1.0) $ Portfolio Face(0.5) -$ Bond $ Cost: $0.58 Call $3.00 Long B (1.0) $ Short B (0.5) ($612.50) $3.00 7
8 Using replicating portfolios, construct and apply an arbitrage argument to price a call option on a zero-coupon security. Tuckman s prior example has three basic steps: 1. Specify the interest rate assumptions which includes both an interest rate tree (50% probability of an up-jump from current 5.0% to 5.5% and 50% probability of down-jump to 4.5%) and a one-year rate of 5.15%. 2. Assume the derivative instrument: in this case, a call option with a strike price of $975 (on a bond with face value of $1,000). Find the replicating portfolio (II.). This is the combination of long position in a one-year bond plus a short position in a six-month bond that produces a payoff identical to the derivative. The cost of the portfolio is $0.58, which therefore must be the price of the derivative. 3. Compare the expected discounted value of $1.46, which discounts with the true (or real-world) probabilities (p = 50% and 1-p = 50%), to the arbitrage price of $0.58, which discounts with the risk-neutral probabilities (p = 80.09% and 1-p = 19.91%). 8
9 Explain why a call option on a zero-coupon security cannot be properly priced using expected discounted values. T Replicate Port (Option) The true value ($0.58) is less than the discounted expected value ($1.46) Investors dislike the risk of the call option: risk-aversion insists on paying less than expected discounted value. The risk penalty implicit in the call option price is inherited from the risk penalty of the one-year zero, that is, from the property that the price of the one-year zero is less than its expected discounted value III. Discounting at true and risk-neutral probabilities T= 0.0 T= 0.5 Exp. Discount Value: $1.46 Risk Neutral Price: $ % $ % 50.00% $ % 9
10 T Risk Neutral Pricing Explain the role of up state and down state probabilities in the option valuation. A remarkable feature of arbitrage pricing is that the up/down probabilities never enter into the calculation of the arbitrage (risk-neutral) price 10
11 Define risk neutral pricing and explain how it is used in option pricing. Risk-neutral pricing modifies an assumed interest rate process so that any contingent claim can be priced without having to construct and price its replicating portfolio. Since the original interest rate process has to be modified only once, and since this modification requires no more effort than pricing a single contingent claim by arbitrage, risk-neutral pricing is an extremely efficient way to price many contingent claims under the same assumed rate process. 11
12 T Risk Neutral Pricing Define risk neutral pricing and explain how it is used in option pricing. Spot rates 0 s(0.5) s(1.0) Spot (0,T) 5.00% 5.15% PV of $1,000 Par Interest Rate Binomial Tree T= 0.0 T= 0.5 p % 5.0% 1-p % "Real-world" probabilities T= 0.0 T= 0.5 T= 1.0 T= 0.0 T= 0.5 T= 1.0 Par: $1,000 $1,000 50% $ $ $ $1,000 $ % $ $ $1,000 "Risk neutral" probabilities T= 0.0 T= 0.5 T= % $ $ % $
13 T Risk Neutral Pricing Define risk neutral pricing and explain how it is used in option pricing. "Real-world" probabilities T= 0.0 T= 0.5 The expected discounted value $ = [(50%)(973.24)+(50%)(978)]/(1+5%/2) 50% $ $ % $ "Risk neutral" probabilities T= 0.0 T= 0.5 The market price $ = $1,000 / ( %/2)^2 80.1% $ $ % $
14 Define risk neutral pricing and explain how it is used in option pricing. Step 1: Given trees for the underlying securities, the price of a security that is priced by arbitrage does not depend on investors risk preferences. 14
15 Define risk neutral pricing and explain how it is used in option pricing. Step 2: Imagine an economy identical to the true economy with respect to current bond prices and the possible value of the six-month rate over time but different in that the investors in the imaginary economy are risk neutral. Unlike investors in the true economy, investors in the imaginary economy do not penalize securities for risk and, therefore, price securities by expected discounted value. It follows that, under the probabilities in the imaginary economy, the expected discounted value of the one-year zero equals its market price. But these probabilities satisfy equation (7.8), namely the risk-neutral probabilities of.8024 and
16 Define risk neutral pricing and explain how it is used in option pricing. Step 3: The price of the option in the imaginary economy, like any other security in that economy, is computed by expected discounted value. Since the probability of the up state in that economy is.8024, the price of the option in that economy is given by equation (7.9) and is, therefore, $
17 Define risk neutral pricing and explain how it is used in option pricing. Step 4: Step 1 implies that given the prices of the six-month and one-year zeros, as well as possible values of the six-month rate, the price of an option does not depend on investor risk preferences. It follows that since the real and imaginary economies have the same bond prices and the same possible values for the six-month rate, the option price must be the same in both economies. In particular, the option price in the real economy must equal $.58, the option price in the imaginary economy. More generally, the price of a derivative in the real economy may be computed by expected discounted value under the risk-neutral probabilities 17
18 Relate the difference between true and risk neutral probabilities to interest rate drift. Under true probabilities there is a 50% chance that the six-month rate rises from 5% to 5.50% and a 50% chance that it falls from 5% to 4.50%. Hence the expected change in the six-month rate, or the drift of the sixmonth rate, is zero. Under risk-neutral probabilities there is an 80.24% chance of a 50 basis point increase in the six-month rate and a 19.76% chance of a 50 basis point decline. Hence the drift of the six-month rate under these probabilities is basis points. 18
19 T Three Steps Explain how the principles of arbitrage pricing of derivatives on fixed income securities can be extended over multiple periods. Spot rate term structure s(0.5) s(1.0) s(1.5) Spot (0,T) 5.00% 5.15% 5.25% PV of $1,000 Par Interest Rate Process T= 0.0 T= 0.5 T= 1.0 T= % 5.50% 5.0% 5.0% 4.50% 4.0% Risk-neutral probabilities p p q q Binomial Price Tree Par: $1, $1, $ $ $1, $ $ $ $1, $ $1,
20 Describe the rationale behind the use of non recombining trees in option pricing. If ud du, tree is non-recombining Nothing wrong with non-recombining tree, from economic perspective For example, to justify this particular tree, could argue that when short rates are 5% or higher they tend to change in increments of 50 basis points. But when rates fall below 5%, the size of the change starts to decrease. In particular, at a rate of 4.50% the short rate may change by only 45 basis points. A volatility process that depends on the level of rates exhibits state-dependent volatility. 20
21 Describe the rationale behind the use of non recombining trees in option pricing. But practitioners tend to avoid non-recombining trees because they are difficult or impossible to implement After six months there are two possible rates, after one year there are four, and after N semiannual periods there are 2N possibilities. For example, a tree with semiannual steps large enough to price 10-year securities will, in its rightmost column alone, have over 500,000 nodes, while a tree used to price 20-year securities will in its rightmost column have over 500 billion nodes. 21
22 T CMS Calculate the value of a constant maturity Treasure swap, given an interest rate tree and the risk neutral probabilities. Pricing a constant maturity Treasury (CMT) swap Pays notional/2 * (sovereign Treasury yield - 5%) every six months Notional 1,000,000 Fixed rate 5.0% Interest Rate Process T= 0.0 T= 0.5 T= 1.0 q 64.89% 6.0% p 80.24% 5.50% 5.0% q % 5.0% p % 4.50% 4.0% Expected discounted value T= 0.0 T= 0.5 T= 1.0 $5,000 Payoff: $2,500 PV: $5,658 PV= $3,616 $0 Payoff: -$2,500 PV: -$4,217 -$5,000 22
23 Describe the advantages and disadvantages of reducing the size of the time steps on the pricing of derivatives on fixed income securities. If cash flows do not occur at given intervals, reducing time step will get closer (more proximate) to actual cash flows More attention must be paid to numerical issues like round-off error Six-month rate trees can too coarse. Reducing the step size (i.e., increasing the granularity) fills the tree with enough rates to price with greater accuracy Decreasing the time step increases computation time
24 Explain why the Black Scholes Merton model to value equity derivatives is not appropriate to value derivatives on fixed income securities. The price of a bond must converge to its face value at maturity But the stochastic process underlying the stock price assumption in the Black-Scholes is not so constrained. Due of the maturity constraint, the volatility of a bond s price must eventually get smaller as the bond approaches maturity. But the Black-Scholes makes the simpler assumption that stock volatility is constant: not appropriate for bonds. 24
25 Describe the impact of embedded options on the value of fixed income securities. European call option on a 5-year bond paying 6% coupon (each step = six months) Price 0 = $ r 0 = 5.0% Option 0 = $ Price u = $ r u = 5.5% Option u = $2.09 Price d = $ r d = 4.5% Option d = $5.67 Price uu = $ r uu = 6.0% Option uu = $0.00 Price ud = $ r ud = 5.0% Option ud = $3.59 Price dd = $ r dd = 4.0% Option dd = $7.33
26 End of P2.T5. Tuckman, Chapter 7, The Science of Term Structure Models Visit us on the
P1.T3. Hull, Chapter 10. Bionic Turtle FRM Video Tutorials. By: David Harper CFA, FRM, CIPM
P1.T3. Hull, Chapter 1 Bionic Turtle FRM Video Tutorials By: David Harper CFA, FRM, CIPM Note: This tutorial is for paid members only. You know who you are. Anybody else is using an illegal copy and also
More informationP2.T5. Tuckman Chapter 9. Bionic Turtle FRM Video Tutorials. By: David Harper CFA, FRM, CIPM
P2.T5. Tuckman Chapter 9 Bionic Turtle FRM Video Tutorials By: David Harper CFA, FRM, CIPM Note: This tutorial is for paid members only. You know who you are. Anybody else is using an illegal copy and
More informationP1.T3. Hull, Chapter 5. Bionic Turtle FRM Video Tutorials. By: David Harper CFA, FRM, CIPM
P1.T3. Hull, Chapter 5 Bionic Turtle FRM Video Tutorials By: David Harper CFA, FRM, CIPM Note: This tutorial is for paid members only. You know who you are. Anybody else is using an illegal copy and also
More informationP1.T3. Hull, Chapter 3. Bionic Turtle FRM Video Tutorials. By: David Harper CFA, FRM, CIPM
P1.T3. Hull, Chapter 3 Bionic Turtle FRM Video Tutorials By: David Harper CFA, FRM, CIPM Note: This tutorial is for paid members only. You know who you are. Anybody else is using an illegal copy and also
More informationP1.T4.Valuation Tuckman, Chapter 5. Bionic Turtle FRM Video Tutorials
P1.T4.Valuation Tuckman, Chapter 5 Bionic Turtle FRM Video Tutorials By: David Harper CFA, FRM, CIPM Note: This tutorial is for paid members only. You know who you are. Anybody else is using an illegal
More informationSpread Risk and Default Intensity Models
P2.T6. Malz Chapter 7 Spread Risk and Default Intensity Models Bionic Turtle FRM Video Tutorials By: David Harper CFA, FRM, CIPM Note: This tutorial is for paid members only. You know who you are. Anybody
More informationHull, Options, Futures, and Other Derivatives, 9 th Edition
P1.T4. Valuation & Risk Models Hull, Options, Futures, and Other Derivatives, 9 th Edition Bionic Turtle FRM Study Notes By David Harper, CFA FRM CIPM and Deepa Sounder www.bionicturtle.com Hull, Chapter
More informationDerivative Securities Fall 2012 Final Exam Guidance Extended version includes full semester
Derivative Securities Fall 2012 Final Exam Guidance Extended version includes full semester Our exam is Wednesday, December 19, at the normal class place and time. You may bring two sheets of notes (8.5
More informationCorporate Finance, Module 21: Option Valuation. Practice Problems. (The attached PDF file has better formatting.) Updated: July 7, 2005
Corporate Finance, Module 21: Option Valuation Practice Problems (The attached PDF file has better formatting.) Updated: July 7, 2005 {This posting has more information than is needed for the corporate
More informationFixed Income Financial Engineering
Fixed Income Financial Engineering Concepts and Buzzwords From short rates to bond prices The simple Black, Derman, Toy model Calibration to current the term structure Nonnegativity Proportional volatility
More informationNotes: This is a closed book and closed notes exam. The maximal score on this exam is 100 points. Time: 75 minutes
M375T/M396C Introduction to Financial Mathematics for Actuarial Applications Spring 2013 University of Texas at Austin Sample In-Term Exam II - Solutions This problem set is aimed at making up the lost
More informationAppendix A Financial Calculations
Derivatives Demystified: A Step-by-Step Guide to Forwards, Futures, Swaps and Options, Second Edition By Andrew M. Chisholm 010 John Wiley & Sons, Ltd. Appendix A Financial Calculations TIME VALUE OF MONEY
More informationP1.T6. Credit Risk Measurement & Management
Bionic Turtle FRM Practice Questions P1.T6. Credit Risk Measurement & Management Global Topic Drill By David Harper, CFA FRM CIPM www.bionicturtle.com GLOBAL TOPIC DRILL: CREDIT RISK MEASUREMENT & MANAGEMENT...
More informationOPTION VALUATION Fall 2000
OPTION VALUATION Fall 2000 2 Essentially there are two models for pricing options a. Black Scholes Model b. Binomial option Pricing Model For equities, usual model is Black Scholes. For most bond options
More informationB6302 Sample Placement Exam Academic Year
Revised June 011 B630 Sample Placement Exam Academic Year 011-01 Part 1: Multiple Choice Question 1 Consider the following information on three mutual funds (all information is in annualized units). Fund
More informationFixed Income and Risk Management
Fixed Income and Risk Management Fall 2003, Term 2 Michael W. Brandt, 2003 All rights reserved without exception Agenda and key issues Pricing with binomial trees Replication Risk-neutral pricing Interest
More informationReview of Derivatives I. Matti Suominen, Aalto
Review of Derivatives I Matti Suominen, Aalto 25 SOME STATISTICS: World Financial Markets (trillion USD) 2 15 1 5 Securitized loans Corporate bonds Financial institutions' bonds Public debt Equity market
More informationUniversity of Colorado at Boulder Leeds School of Business MBAX-6270 MBAX Introduction to Derivatives Part II Options Valuation
MBAX-6270 Introduction to Derivatives Part II Options Valuation Notation c p S 0 K T European call option price European put option price Stock price (today) Strike price Maturity of option Volatility
More informationFixed-Income Analysis. Assignment 7
FIN 684 Professor Robert B.H. Hauswald Fixed-Income Analysis Kogod School of Business, AU Assignment 7 Please be reminded that you are expected to use contemporary computer software to solve the following
More informationHull, Options, Futures & Other Derivatives Exotic Options
P1.T3. Financial Markets & Products Hull, Options, Futures & Other Derivatives Exotic Options Bionic Turtle FRM Video Tutorials By David Harper, CFA FRM 1 Exotic Options Define and contrast exotic derivatives
More informationM339W/389W Financial Mathematics for Actuarial Applications University of Texas at Austin Sample In-Term Exam I Instructor: Milica Čudina
Notes: This is a closed book and closed notes exam. Time: 50 minutes M339W/389W Financial Mathematics for Actuarial Applications University of Texas at Austin Sample In-Term Exam I Instructor: Milica Čudina
More informationFinal Exam. Please answer all four questions. Each question carries 25% of the total grade.
Econ 174 Financial Insurance Fall 2000 Allan Timmermann UCSD Final Exam Please answer all four questions. Each question carries 25% of the total grade. 1. Explain the reasons why you agree or disagree
More informationPricing Options with Binomial Trees
Pricing Options with Binomial Trees MATH 472 Financial Mathematics J. Robert Buchanan 2018 Objectives In this lesson we will learn: a simple discrete framework for pricing options, how to calculate risk-neutral
More informationFINANCIAL OPTION ANALYSIS HANDOUTS
FINANCIAL OPTION ANALYSIS HANDOUTS 1 2 FAIR PRICING There is a market for an object called S. The prevailing price today is S 0 = 100. At this price the object S can be bought or sold by anyone for any
More informationHull, Options, Futures & Other Derivatives
P1.T3. Financial Markets & Products Hull, Options, Futures & Other Derivatives Bionic Turtle FRM Study Notes Sample By David Harper, CFA FRM CIPM and Deepa Raju www.bionicturtle.com Hull, Chapter 1: Introduction
More informationB6302 B7302 Sample Placement Exam Answer Sheet (answers are indicated in bold)
B6302 B7302 Sample Placement Exam Answer Sheet (answers are indicated in bold) Part 1: Multiple Choice Question 1 Consider the following information on three mutual funds (all information is in annualized
More informationThe Binomial Lattice Model for Stocks: Introduction to Option Pricing
1/33 The Binomial Lattice Model for Stocks: Introduction to Option Pricing Professor Karl Sigman Columbia University Dept. IEOR New York City USA 2/33 Outline The Binomial Lattice Model (BLM) as a Model
More informationP2.T5. Market Risk Measurement & Management. Hull, Options, Futures, and Other Derivatives, 9th Edition.
P2.T5. Market Risk Measurement & Management Hull, Options, Futures, and Other Derivatives, 9th Edition. Bionic Turtle FRM Study Notes By David Harper, CFA FRM CIPM www.bionicturtle.com Hull, Chapter 9:
More information1b. Write down the possible payoffs of each of the following instruments separately, and of the portfolio of all three:
Fi8000 Quiz #3 - Example Part I Open Questions 1. The current price of stock ABC is $25. 1a. Write down the possible payoffs of a long position in a European put option on ABC stock, which expires in one
More informationBinomial model: numerical algorithm
Binomial model: numerical algorithm S / 0 C \ 0 S0 u / C \ 1,1 S0 d / S u 0 /, S u 3 0 / 3,3 C \ S0 u d /,1 S u 5 0 4 0 / C 5 5,5 max X S0 u,0 S u C \ 4 4,4 C \ 3 S u d / 0 3, C \ S u d 0 S u d 0 / C 4
More informationInterest-Sensitive Financial Instruments
Interest-Sensitive Financial Instruments Valuing fixed cash flows Two basic rules: - Value additivity: Find the portfolio of zero-coupon bonds which replicates the cash flows of the security, the price
More informationM339W/M389W Financial Mathematics for Actuarial Applications University of Texas at Austin In-Term Exam I Instructor: Milica Čudina
M339W/M389W Financial Mathematics for Actuarial Applications University of Texas at Austin In-Term Exam I Instructor: Milica Čudina Notes: This is a closed book and closed notes exam. Time: 50 minutes
More informationFRM Markets & Products Saunders & Cornett, Chapter 14: Foreign Exchange Risk
FRM Markets & Products Saunders & Cornett, Chapter 14: Foreign Exchange Risk Hosted by David Harper CFA, FRM, CIPM Published April 14, 2012 Brought to you by bionicturtle.com This tutorial is for paid
More informationInvestment Guarantees Chapter 7. Investment Guarantees Chapter 7: Option Pricing Theory. Key Exam Topics in This Lesson.
Investment Guarantees Chapter 7 Investment Guarantees Chapter 7: Option Pricing Theory Mary Hardy (2003) Video By: J. Eddie Smith, IV, FSA, MAAA Investment Guarantees Chapter 7 1 / 15 Key Exam Topics in
More informationName: MULTIPLE CHOICE. 1 (5) a b c d e. 2 (5) a b c d e TRUE/FALSE 1 (2) TRUE FALSE. 3 (5) a b c d e 2 (2) TRUE FALSE.
Name: M339D=M389D Introduction to Actuarial Financial Mathematics University of Texas at Austin Sample In-Term Exam II Instructor: Milica Čudina Notes: This is a closed book and closed notes exam. The
More informationCHAPTER 10 OPTION PRICING - II. Derivatives and Risk Management By Rajiv Srivastava. Copyright Oxford University Press
CHAPTER 10 OPTION PRICING - II Options Pricing II Intrinsic Value and Time Value Boundary Conditions for Option Pricing Arbitrage Based Relationship for Option Pricing Put Call Parity 2 Binomial Option
More informationMULTIPLE CHOICE QUESTIONS
Name: M375T=M396D Introduction to Actuarial Financial Mathematics Spring 2013 University of Texas at Austin Sample In-Term Exam Two: Pretest Instructor: Milica Čudina Notes: This is a closed book and closed
More informationEcon 174 Financial Insurance Fall 2000 Allan Timmermann. Final Exam. Please answer all four questions. Each question carries 25% of the total grade.
Econ 174 Financial Insurance Fall 2000 Allan Timmermann UCSD Final Exam Please answer all four questions. Each question carries 25% of the total grade. 1. Explain the reasons why you agree or disagree
More informationChapter 24 Interest Rate Models
Chapter 4 Interest Rate Models Question 4.1. a F = P (0, /P (0, 1 =.8495/.959 =.91749. b Using Black s Formula, BSCall (.8495,.9009.959,.1, 0, 1, 0 = $0.0418. (1 c Using put call parity for futures options,
More informationChapter 14. Exotic Options: I. Question Question Question Question The geometric averages for stocks will always be lower.
Chapter 14 Exotic Options: I Question 14.1 The geometric averages for stocks will always be lower. Question 14.2 The arithmetic average is 5 (three 5s, one 4, and one 6) and the geometric average is (5
More informationHomework Assignments
Homework Assignments Week 1 (p 57) #4.1, 4., 4.3 Week (pp 58-6) #4.5, 4.6, 4.8(a), 4.13, 4.0, 4.6(b), 4.8, 4.31, 4.34 Week 3 (pp 15-19) #1.9, 1.1, 1.13, 1.15, 1.18 (pp 9-31) #.,.6,.9 Week 4 (pp 36-37)
More informationP1.T3. Financial Markets & Products. Hull, Options, Futures & Other Derivatives. Trading Strategies Involving Options
P1.T3. Financial Markets & Products Hull, Options, Futures & Other Derivatives Trading Strategies Involving Options Bionic Turtle FRM Video Tutorials By David Harper, CFA FRM 1 Trading Strategies Involving
More informationP2.T5. Market Risk Measurement & Management. Bruce Tuckman, Fixed Income Securities, 3rd Edition
P2.T5. Market Risk Measurement & Management Bruce Tuckman, Fixed Income Securities, 3rd Edition Bionic Turtle FRM Study Notes Reading 40 By David Harper, CFA FRM CIPM www.bionicturtle.com TUCKMAN, CHAPTER
More information1. In this exercise, we can easily employ the equations (13.66) (13.70), (13.79) (13.80) and
CHAPTER 13 Solutions Exercise 1 1. In this exercise, we can easily employ the equations (13.66) (13.70), (13.79) (13.80) and (13.82) (13.86). Also, remember that BDT model will yield a recombining binomial
More informationChapter 14 Exotic Options: I
Chapter 14 Exotic Options: I Question 14.1. The geometric averages for stocks will always be lower. Question 14.2. The arithmetic average is 5 (three 5 s, one 4, and one 6) and the geometric average is
More informationMartingale Pricing Theory in Discrete-Time and Discrete-Space Models
IEOR E4707: Foundations of Financial Engineering c 206 by Martin Haugh Martingale Pricing Theory in Discrete-Time and Discrete-Space Models These notes develop the theory of martingale pricing in a discrete-time,
More information************************
Derivative Securities Options on interest-based instruments: pricing of bond options, caps, floors, and swaptions. The most widely-used approach to pricing options on caps, floors, swaptions, and similar
More informationChapter 9 - Mechanics of Options Markets
Chapter 9 - Mechanics of Options Markets Types of options Option positions and profit/loss diagrams Underlying assets Specifications Trading options Margins Taxation Warrants, employee stock options, and
More informationPut-Call Parity. Put-Call Parity. P = S + V p V c. P = S + max{e S, 0} max{s E, 0} P = S + E S = E P = S S + E = E P = E. S + V p V c = (1/(1+r) t )E
Put-Call Parity l The prices of puts and calls are related l Consider the following portfolio l Hold one unit of the underlying asset l Hold one put option l Sell one call option l The value of the portfolio
More information1. 2 marks each True/False: briefly explain (no formal proofs/derivations are required for full mark).
The University of Toronto ACT460/STA2502 Stochastic Methods for Actuarial Science Fall 2016 Midterm Test You must show your steps or no marks will be awarded 1 Name Student # 1. 2 marks each True/False:
More informationDerivative Securities Section 9 Fall 2004 Notes by Robert V. Kohn, Courant Institute of Mathematical Sciences.
Derivative Securities Section 9 Fall 2004 Notes by Robert V. Kohn, Courant Institute of Mathematical Sciences. Futures, and options on futures. Martingales and their role in option pricing. A brief introduction
More information1 Parameterization of Binomial Models and Derivation of the Black-Scholes PDE.
1 Parameterization of Binomial Models and Derivation of the Black-Scholes PDE. Previously we treated binomial models as a pure theoretical toy model for our complete economy. We turn to the issue of how
More informationP2.T5. Market Risk Measurement & Management. Bruce Tuckman, Fixed Income Securities, 3rd Edition
P2.T5. Market Risk Measurement & Management Bruce Tuckman, Fixed Income Securities, 3rd Edition Bionic Turtle FRM Study Notes By David Harper, CFA FRM CIPM www.bionicturtle.com Tuckman, Chapter 6: Empirical
More informationIntroduction to Binomial Trees. Chapter 12
Introduction to Binomial Trees Chapter 12 Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright John C. Hull 2013 1 A Simple Binomial Model A stock price is currently $20. In three months
More informationBrooks, Introductory Econometrics for Finance, 3rd Edition
P1.T2. Quantitative Analysis Brooks, Introductory Econometrics for Finance, 3rd Edition Bionic Turtle FRM Study Notes Sample By David Harper, CFA FRM CIPM and Deepa Raju www.bionicturtle.com Chris Brooks,
More informationSAMPLE FINAL QUESTIONS. William L. Silber
SAMPLE FINAL QUESTIONS William L. Silber HOW TO PREPARE FOR THE FINAL: 1. Study in a group 2. Review the concept questions in the Before and After book 3. When you review the questions listed below, make
More informationA NOVEL BINOMIAL TREE APPROACH TO CALCULATE COLLATERAL AMOUNT FOR AN OPTION WITH CREDIT RISK
A NOVEL BINOMIAL TREE APPROACH TO CALCULATE COLLATERAL AMOUNT FOR AN OPTION WITH CREDIT RISK SASTRY KR JAMMALAMADAKA 1. KVNM RAMESH 2, JVR MURTHY 2 Department of Electronics and Computer Engineering, Computer
More informationB. Combinations. 1. Synthetic Call (Put-Call Parity). 2. Writing a Covered Call. 3. Straddle, Strangle. 4. Spreads (Bull, Bear, Butterfly).
1 EG, Ch. 22; Options I. Overview. A. Definitions. 1. Option - contract in entitling holder to buy/sell a certain asset at or before a certain time at a specified price. Gives holder the right, but not
More information******************************* The multi-period binomial model generalizes the single-period binomial model we considered in Section 2.
Derivative Securities Multiperiod Binomial Trees. We turn to the valuation of derivative securities in a time-dependent setting. We focus for now on multi-period binomial models, i.e. binomial trees. This
More informationB8.3 Week 2 summary 2018
S p VT u = f(su ) S T = S u V t =? S t S t e r(t t) 1 p VT d = f(sd ) S T = S d t T time Figure 1: Underlying asset price in a one-step binomial model B8.3 Week 2 summary 2018 The simplesodel for a random
More informationDERIVATIVE SECURITIES Lecture 5: Fixed-income securities
DERIVATIVE SECURITIES Lecture 5: Fixed-income securities Philip H. Dybvig Washington University in Saint Louis Interest rates Interest rate derivative pricing: general issues Bond and bond option pricing
More informationP2.T5. Market Risk Measurement & Management. Bionic Turtle FRM Practice Questions Sample
P2.T5. Market Risk Measurement & Management Bionic Turtle FRM Practice Questions Sample Hull, Options, Futures & Other Derivatives By David Harper, CFA FRM CIPM www.bionicturtle.com HULL, CHAPTER 20: VOLATILITY
More informationP2.T6. Credit Risk Measurement & Management. Jon Gregory, The xva Challenge: Counterparty Credit Risk, Funding, Collateral, and Capital
P2.T6. Credit Risk Measurement & Management Jon Gregory, The xva Challenge: Counterparty Credit Risk, Funding, Collateral, and Capital Bionic Turtle FRM Study Notes Sample By David Harper, CFA FRM CIPM
More informationIntroduction Random Walk One-Period Option Pricing Binomial Option Pricing Nice Math. Binomial Models. Christopher Ting.
Binomial Models Christopher Ting Christopher Ting http://www.mysmu.edu/faculty/christophert/ : christopherting@smu.edu.sg : 6828 0364 : LKCSB 5036 October 14, 2016 Christopher Ting QF 101 Week 9 October
More informationStulz, Governance, Risk Management and Risk-Taking in Banks
P1.T1. Foundations of Risk Stulz, Governance, Risk Management and Risk-Taking in Banks Bionic Turtle FRM Study Notes By David Harper, CFA FRM CIPM www.bionicturtle.com Stulz, Governance, Risk Management
More informationThe Binomial Lattice Model for Stocks: Introduction to Option Pricing
1/27 The Binomial Lattice Model for Stocks: Introduction to Option Pricing Professor Karl Sigman Columbia University Dept. IEOR New York City USA 2/27 Outline The Binomial Lattice Model (BLM) as a Model
More informationDowd, Measuring Market Risk, 2nd Edition
P2.T7. Operational & Integrated Risk Management Dowd, Measuring Market Risk, 2nd Edition Bionic Turtle FRM Study Notes Reading 53 By David Harper, CFA FRM CIPM www.bionicturtle.com DOWD CHAPTER 14: ESTIMATING
More informationAppendix: Basics of Options and Option Pricing Option Payoffs
Appendix: Basics of Options and Option Pricing An option provides the holder with the right to buy or sell a specified quantity of an underlying asset at a fixed price (called a strike price or an exercise
More informationMATH 425: BINOMIAL TREES
MATH 425: BINOMIAL TREES G. BERKOLAIKO Summary. These notes will discuss: 1-level binomial tree for a call, fair price and the hedging procedure 1-level binomial tree for a general derivative, fair price
More informationP2.T5. Market Risk Measurement & Management. Kevin Dowd, Measuring Market Risk, 2nd Edition
P2.T5. Market Risk Measurement & Management Kevin Dowd, Measuring Market Risk, 2nd Edition Bionic Turtle FRM Study Notes By David Harper, CFA FRM CIPM www.bionicturtle.com Dowd Chapter 3: Estimating Market
More informationForwards, Futures, Options and Swaps
Forwards, Futures, Options and Swaps A derivative asset is any asset whose payoff, price or value depends on the payoff, price or value of another asset. The underlying or primitive asset may be almost
More informationMORNING SESSION. Date: Wednesday, April 30, 2014 Time: 8:30 a.m. 11:45 a.m. INSTRUCTIONS TO CANDIDATES
SOCIETY OF ACTUARIES Quantitative Finance and Investment Core Exam QFICORE MORNING SESSION Date: Wednesday, April 30, 2014 Time: 8:30 a.m. 11:45 a.m. INSTRUCTIONS TO CANDIDATES General Instructions 1.
More information1 Interest Based Instruments
1 Interest Based Instruments e.g., Bonds, forward rate agreements (FRA), and swaps. Note that the higher the credit risk, the higher the interest rate. Zero Rates: n year zero rate (or simply n-year zero)
More informationIn general, the value of any asset is the present value of the expected cash flows on
ch05_p087_110.qxp 11/30/11 2:00 PM Page 87 CHAPTER 5 Option Pricing Theory and Models In general, the value of any asset is the present value of the expected cash flows on that asset. This section will
More informationAdvanced Corporate Finance. 5. Options (a refresher)
Advanced Corporate Finance 5. Options (a refresher) Objectives of the session 1. Define options (calls and puts) 2. Analyze terminal payoff 3. Define basic strategies 4. Binomial option pricing model 5.
More informationVanilla interest rate options
Vanilla interest rate options Marco Marchioro derivati2@marchioro.org October 26, 2011 Vanilla interest rate options 1 Summary Probability evolution at information arrival Brownian motion and option pricing
More informationBUSM 411: Derivatives and Fixed Income
BUSM 411: Derivatives and Fixed Income 12. Binomial Option Pricing Binomial option pricing enables us to determine the price of an option, given the characteristics of the stock other underlying asset
More informationINSTITUTE OF ACTUARIES OF INDIA
INSTITUTE OF ACTUARIES OF INDIA EXAMINATIONS 23 rd March 2017 Subject CT8 Financial Economics Time allowed: Three Hours (10.30 13.30 Hours) Total Marks: 100 INSTRUCTIONS TO THE CANDIDATES 1. Please read
More informationBruce Tuckman, Angel Serrat, Fixed Income Securities: Tools for Today s Markets, 3rd Edition
P1.T3. Financial Markets & Products Bruce Tuckman, Angel Serrat, Fixed Income Securities: Tools for Today s Markets, 3rd Edition Bionic Turtle FRM Study Notes By David Harper, CFA FRM CIPM www.bionicturtle.com
More informationFixed-Income Options
Fixed-Income Options Consider a two-year 99 European call on the three-year, 5% Treasury. Assume the Treasury pays annual interest. From p. 852 the three-year Treasury s price minus the $5 interest could
More informationIntroduction to Financial Derivatives
55.444 Introduction to Financial Derivatives November 5, 212 Option Analysis and Modeling The Binomial Tree Approach Where we are Last Week: Options (Chapter 9-1, OFOD) This Week: Option Analysis and Modeling:
More informationTerm Structure Lattice Models
IEOR E4706: Foundations of Financial Engineering c 2016 by Martin Haugh Term Structure Lattice Models These lecture notes introduce fixed income derivative securities and the modeling philosophy used to
More informationDerivatives Questions Question 1 Explain carefully the difference between hedging, speculation, and arbitrage.
Derivatives Questions Question 1 Explain carefully the difference between hedging, speculation, and arbitrage. Question 2 What is the difference between entering into a long forward contract when the forward
More informationNotes: This is a closed book and closed notes exam. The maximal score on this exam is 100 points. Time: 75 minutes
M375T/M396C Introduction to Financial Mathematics for Actuarial Applications Spring 2013 University of Texas at Austin Sample In-Term Exam II Post-test Instructor: Milica Čudina Notes: This is a closed
More informationOption pricing models
Option pricing models Objective Learn to estimate the market value of option contracts. Outline The Binomial Model The Black-Scholes pricing model The Binomial Model A very simple to use and understand
More informationFINANCIAL OPTION ANALYSIS HANDOUTS
FINANCIAL OPTION ANALYSIS HANDOUTS 1 2 FAIR PRICING There is a market for an object called S. The prevailing price today is S 0 = 100. At this price the object S can be bought or sold by anyone for any
More informationQueens College, CUNY, Department of Computer Science Computational Finance CSCI 365 / 765 Spring 2018 Instructor: Dr. Sateesh Mane.
Queens College, CUNY, Department of Computer Science Computational Finance CSCI 365 / 765 Spring 218 Instructor: Dr. Sateesh Mane c Sateesh R. Mane 218 19 Lecture 19 May 12, 218 Exotic options The term
More informationPractice of Finance: Advanced Corporate Risk Management
MIT OpenCourseWare http://ocw.mit.edu 15.997 Practice of Finance: Advanced Corporate Risk Management Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
More informationBinomial Trees. Liuren Wu. Options Markets. Zicklin School of Business, Baruch College. Liuren Wu (Baruch ) Binomial Trees Options Markets 1 / 22
Binomial Trees Liuren Wu Zicklin School of Business, Baruch College Options Markets Liuren Wu (Baruch ) Binomial Trees Options Markets 1 / 22 A simple binomial model Observation: The current stock price
More information1.1 Basic Financial Derivatives: Forward Contracts and Options
Chapter 1 Preliminaries 1.1 Basic Financial Derivatives: Forward Contracts and Options A derivative is a financial instrument whose value depends on the values of other, more basic underlying variables
More informationFrom Discrete Time to Continuous Time Modeling
From Discrete Time to Continuous Time Modeling Prof. S. Jaimungal, Department of Statistics, University of Toronto 2004 Arrow-Debreu Securities 2004 Prof. S. Jaimungal 2 Consider a simple one-period economy
More informationThe Black-Scholes PDE from Scratch
The Black-Scholes PDE from Scratch chris bemis November 27, 2006 0-0 Goal: Derive the Black-Scholes PDE To do this, we will need to: Come up with some dynamics for the stock returns Discuss Brownian motion
More informationFINANCIAL MATHEMATICS WITH ADVANCED TOPICS MTHE7013A
UNIVERSITY OF EAST ANGLIA School of Mathematics Main Series UG Examination 2016 17 FINANCIAL MATHEMATICS WITH ADVANCED TOPICS MTHE7013A Time allowed: 3 Hours Attempt QUESTIONS 1 and 2, and THREE other
More informationLecture 17. The model is parametrized by the time period, δt, and three fixed constant parameters, v, σ and the riskless rate r.
Lecture 7 Overture to continuous models Before rigorously deriving the acclaimed Black-Scholes pricing formula for the value of a European option, we developed a substantial body of material, in continuous
More informationModel Calibration and Hedging
Model Calibration and Hedging Concepts and Buzzwords Choosing the Model Parameters Choosing the Drift Terms to Match the Current Term Structure Hedging the Rate Risk in the Binomial Model Term structure
More informationThe Term Structure and Interest Rate Dynamics Cross-Reference to CFA Institute Assigned Topic Review #35
Study Sessions 12 & 13 Topic Weight on Exam 10 20% SchweserNotes TM Reference Book 4, Pages 1 105 The Term Structure and Interest Rate Dynamics Cross-Reference to CFA Institute Assigned Topic Review #35
More informationLattice Model of System Evolution. Outline
Lattice Model of System Evolution Richard de Neufville Professor of Engineering Systems and of Civil and Environmental Engineering MIT Massachusetts Institute of Technology Lattice Model Slide 1 of 32
More informationThe Black-Scholes Model
The Black-Scholes Model Liuren Wu Options Markets (Hull chapter: 12, 13, 14) Liuren Wu ( c ) The Black-Scholes Model colorhmoptions Markets 1 / 17 The Black-Scholes-Merton (BSM) model Black and Scholes
More informationMFE8812 Bond Portfolio Management
MFE8812 Bond Portfolio Management William C. H. Leon Nanyang Business School January 8, 2018 1 / 87 William C. H. Leon MFE8812 Bond Portfolio Management 1 Overview Building an Interest-Rate Tree Calibrating
More informationOptimal Portfolios under a Value at Risk Constraint
Optimal Portfolios under a Value at Risk Constraint Ton Vorst Abstract. Recently, financial institutions discovered that portfolios with a limited Value at Risk often showed returns that were close to
More information