Name: 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.
|
|
- Jayson Cobb
- 6 years ago
- Views:
Transcription
1 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 maximal score on this exam is 100 points. Time: 50 minutes MULTIPLE CHOICE 1 (5) a b c d e TRUE/FALSE 1 (2) TRUE FALSE 2 (2) TRUE FALSE 3 (2) TRUE FALSE 4 (2) TRUE FALSE 5 (2) TRUE FALSE 2 (5) a b c d e 3 (5) a b c d e 4 (5) a b c d e 5 (5) a b c d e 6 (5) a b c d e 7 (5) a b c d e 8 (5) a b c d e
2 2 FREE-RESPONSE PROBLEMS Problem 2.1. (5 points) An investor wants to hold 200 euros two years from today. The spot exchange rate is $1.31 per euro. If the euro denominated annual interest rate is 3.0% what is the price of a currency prepaid forward? Solution: F P 0,T (x) = 200e = Problem 2.2. Consider a two-period forward binomial tree, where the length of each period is 6 months. Assume the stock price is $50.00, σ = 0.20, r = 0.06 and the dividend yield δ = What is the lowest strike price for which early exercise could occur after the first time step and with an American put option? Solution: With the given data, we get that approximately e σ h = 1.15 and e (r δ)h = Hence, u = 1.16 and d = S uu = ; S ud = S du = ; S dd = Let us denote the strike by K. The payoffs, if there is no early exercise, as a function of K are V uu = (K 62.28) +, V ud = (K 51.04) +, and V dd = (K 38.72) +. The risk-neutral probability is p = = So, the continuation values of the put after taking a single step back in the binomial tree are V cont u = e 0.03 [0.46 (K 62.28) (K 51.04) + ] ; V cont d = e 0.03 [0.46 (K 51.04) (K 38.72) + ]. On the other hand, the stock prices at these two nodes are So, the values of immediate/early exercise are S u = and S d = V imm u = (K 58) + and V imm d = (K 44) +. At any of these two nodes, early exercise will happen if V imm node > V node cont. Hence, we must have that K > 44 otherwise, no early exercise should happen at either of the two nodes. Then, for K > 44, we have that V cont u = 0 and V cont d = (K 38.72).
3 3 So, we need to figure out for which K the inequality K 44 > (K 38.72) K( ) > holds. We get K >
4 4 MULTIPLE CHOICE QUESTIONS Please, circle the correct answer on the front page of this exam. Problem 2.3. The current price of a non-dividend-paying stock is $100 per share and its volatility is given to be The continuously compounded, risk-free interest rate equals Consider a $110-strike, one-year American put on the above stock. Use a two-period forward binomial stock-price tree to calculate the current price of the American put. (a) $20.03 (b) $15.41 (c) $13.38 (d) $11.11 Solution: (c) By the definition of the forward binomial tree, we obtain u = e (r δ)h+σ h = e , d = e (r δ)h σ h = e in our usual notation. The binomial tree modeling the stock price is The risk-neutral probability of the stock price going up in a single period equals p = e(r δ)h d u d = e = Should the American option not be exercised early the possible payoffs would be V uu = 0, V ud = = 3.82, V dd = = It is not sensible to exercise the American put at the up node, so the value of the American put equals the continuation value at the up node. We get V A u = CV u = e 0.03 ( ) 3.82 = At the down node, the value of immediate exercise is IE d = = On the other hand, the continuation value at the down node equals CV d = e 0.03 [ ( ) 35.44] = We conclude that the American put s value at the down node equals the value of immediate exervise, i.e., V A d =
5 Should the option be exercised at time 0, the payoff would be 10. The continuation value at the root node is CV 0 = e 0.03 [ ( ) 23.65] = So, the price we were looking for is $ Problem 2.4. The current price of a continuous-dividend-paying stock is $100 per share. Its volatility is given to be 0.2 and its dividend yield is The continuously compounded risk-free interest rate equals Consider a $95-strike European put option on the above stock with nine months to expiration. Using a three-period forward binomial tree, find the price of this put option. (a) $2.97 (b) $3.06 (c) $3.59 (d) $3.70 Solution: (c) The up and down factors in the above model are u = e = , d = e = The relevant possible stock prices at the leaves of the binomial tree are S ddd = d 3 S(0) = 100(0.9116) 3 = , S ddu = d 2 us(0) = The remaining two final states of the world result in the put option being out-of-the-money at expiration. The risk-neutral probability of the stock price moving up in a single period is p 1 = 1 + e = So, the price of the European put option equals V P (0) = e 0.06(3/4) [ ( )( ) 3 + ( )(3)( ) 2 (0.475) ] = Problem 2.5. (5 points) Assume that the current exchange rate is $1.3 per euro. The continuously compounded interest rate for the euro is 0.03, while continuously compounded interest rate for the USD is Let the price of an at-the-money USD-denominated European call on on the euro with exercise date in 6 months be equal to What is the price of an at-the-money Euro-denominated put on the USD with the exercise date in 6 months? (a) About
6 6 (b) About (c) About (d) About (e) None of the above Solution: (b) Let x denote the exchange rate from euros to dollars. We are given that x(0) = 1.3. Using the put-call symmetry/duality for options on currencies, we get VP Euro (0, 1/x(0)) = (1/x(0) 2 VC USD (0, x(0)) Problem 2.6. The following two one-year European put options on the same asset are available in the market: a $50-strike put with the premium of $5, a $55-strike put with the premium of $10. The continuously compounded, risk-free interest rate is Which of the following positions certainly exploits the arbitrage opportunity caused by the above put premia? (a) Put bull spread. (b) Put bear spread. (c) Both of the above positions. (d) There is no arbitrage opportunity. Solution: (a) Problem 2.7. You are given that the price of: a $50-strike, one-year European call equals $8, a $65-strike, one-year European call equals $2. Both options have the same underlying asset. What is the maximal price of a $56-strike, one-year European call such that there is no arbitrage in our market model? (a) $4.40 (b) $5 (c) $5.60 (d) $6.02 Solution: (c) Using the convexity of call price with respect to the strike, we get the following answer: = = 5.60.
7 Problem 2.8. (5 points) Consider a non-dividend-paying stock with the initial price of S(0) = 100. Assume that the annual risk-free continuously compounded interest rate equals r = Let the annualized standard deviation of the sontinuously compounded stock return, i.e., the volatility be σ = Using a one-period forward binomial tree, calculate the price of a one-year at-the-money European call on this underlying asset. (a) $11.07 (b) $12.46 (c) $13.38 (d) $14.58 Solution: (d) By the definition of the forward binomial tree, with the given data, u = e (r δ)h+σ h = e = e , d = e (r δ)h σ h = e 0.2 < 1. We do not care about the actual value of d since the option is at-the-money and we get the payoff of 0 at the lower node regardless of the actual value. Also, we do not need d explicitly to calculate the risk-neutral probability in this model, since Finally, by risk-neutral pricing p = e = 1 = σ h 1 + e0.25 V C (0) = e 0.05 ( ) TRUE/FALSE QUESTIONS. Problem 2.9. (2 points) Let the continuously compounded interest rate be denoted by r. Consider a futures contract for delivery at time T of a market index with the continuous dividend yield δ. As a function of time, the price of this contract at time t is denoted by F t,t. Denote the time t price of a European call on the futures contract with strike K and exercise date T < T by V C (t), and denote the time t price of a European put on the same futures contract with the same strike price and the same exercise date by V P (t). Then, the following equality is always true V C (t) V P (t) = F t,t e δ(t t) Ke rt. Solution: FALSE There are many things amiss with the right-hand side of the above expression. The correct put-call parity for options on futures reads as V C (t) V P (t) = e r(t t) (F t,t K).
8 8 Problem (2 points) It is never optimal to exercise an American call option on a nondividend paying stock early. Solution: TRUE Problem (2 points) In the setting of the one-period binomial model, denote by i the effective interest rate per period. Let u denote the up factor and let d denote the down factor in the stock-price model. If d < u 1 + i then there certainly is no possibility for arbitrage. Solution: FALSE Problem (2 points) You are using a binomial asset-pricing model to model the evolution of the price of a particular stock. Then, the in the replicating portfolio of a single call option on that stock never exceeds 1. Solution: TRUE The call s will always be between 0 and 1. Problem The expiration date of a futures option cannot exceed the delivery date of the underlying futures contract. Solution: TRUE Problem (2 points) The price of a European call option on a non-dividend-paying stock is equal to the price of an otherwise identical American call option. Solution: TRUE
Notes: 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 informationName: 2.2. MULTIPLE CHOICE QUESTIONS. Please, circle the correct answer on the front page of this exam.
Name: M339D=M389D Introduction to Actuarial Financial Mathematics University of Texas at Austin In-Term Exam II Extra problems Instructor: Milica Čudina Notes: This is a closed book and closed notes exam.
More informationMULTIPLE 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. 4 (5) a b c d e 3 (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 information= e S u S(0) From the other component of the call s replicating portfolio, we get. = e 0.015
Name: M339D=M389D Introduction to Actuarial Financial Mathematics University of Texas at Austin In-Term Exam II Extra problems Instructor: Milica Čudina Notes: This is a closed book and closed notes exam.
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 informationName: T/F 2.13 M.C. Σ
Name: M339D=M389D Introduction to Actuarial Financial Mathematics University of Texas at Austin In-Term Exam II Instructor: Milica Čudina Notes: This is a closed book and closed notes exam. The maximal
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 informationNotes: This is a closed book and closed notes exam. The maximal score on this exam is 100 points. Time: 75 minutes
M339D/M389D Introduction to Financial Mathematics for Actuarial Applications University of Texas at Austin Sample In-Term Exam II - Solutions Instructor: Milica Čudina Notes: This is a closed book and
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 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 informationTRUE/FALSE 1 (2) TRUE FALSE 2 (2) TRUE FALSE. MULTIPLE CHOICE 1 (5) a b c d e 3 (2) TRUE FALSE 4 (2) TRUE FALSE. 2 (5) a b c d e 5 (2) TRUE FALSE
Tuesday, February 26th M339W/389W Financial Mathematics for Actuarial Applications Spring 2013, University of Texas at Austin In-Term Exam I Instructor: Milica Čudina Notes: This is a closed book and closed
More informationName: Def n T/F?? 1.17 M.C. Σ
Name: M339D=M389D Introduction to Actuarial Financial Mathematics University of Texas at Austin Sample In-Term Exam I Instructor: Milica Čudina Notes: This is a closed book and closed notes exam. The maximal
More information1.15 (5) a b c d e. ?? (5) a b c d e. ?? (5) a b c d e. ?? (5) a b c d e FOR GRADER S USE ONLY: DEF T/F ?? M.C.
Name: M339W/389W Financial Mathematics for Actuarial Applications University of Texas at Austin The Prerequisite In-Term Exam Instructor: Milica Čudina Notes: This is a closed book and closed notes exam.
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 Midterm Exam - Solutions Instructor: Milica Čudina Notes: This is a closed book and closed notes exam.
More information.5 M339W/389W Financial Mathematics for Actuarial Applications University of Texas at Austin Sample In-Term Exam 2.5 Instructor: Milica Čudina
.5 M339W/389W Financial Mathematics for Actuarial Applications University of Texas at Austin Sample In-Term Exam 2.5 Instructor: Milica Čudina Notes: This is a closed book and closed notes exam. Time:
More informationUniversity of Texas at Austin. HW Assignment 5. Exchange options. Bull/Bear spreads. Properties of European call/put prices.
HW: 5 Course: M339D/M389D - Intro to Financial Math Page: 1 of 5 University of Texas at Austin HW Assignment 5 Exchange options. Bull/Bear spreads. Properties of European call/put prices. 5.1. Exchange
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 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 informationUniversity of Texas at Austin. Problem Set #4
Problem set: 4 Course: M339D/M389D - Intro to Financial Math Page: 1 of 5 University of Texas at Austin Problem Set #4 Problem 4.1. The current price of a non-dividend-paying stock is $80 per share. You
More informationIntroduction to Binomial Trees. Chapter 12
Introduction to Binomial Trees Chapter 12 1 A Simple Binomial Model l A stock price is currently $20 l In three months it will be either $22 or $18 Stock Price = $22 Stock price = $20 Stock Price = $18
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 informationLecture 17 Option pricing in the one-period binomial model.
Lecture: 17 Course: M339D/M389D - Intro to Financial Math Page: 1 of 9 University of Texas at Austin Lecture 17 Option pricing in the one-period binomial model. 17.1. Introduction. Recall the one-period
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 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 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 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 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 informationThe Multistep Binomial Model
Lecture 10 The Multistep Binomial Model Reminder: Mid Term Test Friday 9th March - 12pm Examples Sheet 1 4 (not qu 3 or qu 5 on sheet 4) Lectures 1-9 10.1 A Discrete Model for Stock Price Reminder: The
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 informationFinancial Derivatives Section 3
Financial Derivatives Section 3 Introduction to Option Pricing Michail Anthropelos anthropel@unipi.gr http://web.xrh.unipi.gr/faculty/anthropelos/ University of Piraeus Spring 2018 M. Anthropelos (Un.
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 informationnon linear Payoffs Markus K. Brunnermeier
Institutional Finance Lecture 10: Dynamic Arbitrage to Replicate non linear Payoffs Markus K. Brunnermeier Preceptor: Dong Beom Choi Princeton University 1 BINOMIAL OPTION PRICING Consider a European call
More informationErrata and updates for ASM Exam MFE/3F (Ninth Edition) sorted by page.
Errata for ASM Exam MFE/3F Study Manual (Ninth Edition) Sorted by Page 1 Errata and updates for ASM Exam MFE/3F (Ninth Edition) sorted by page. Note the corrections to Practice Exam 6:9 (page 613) and
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 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 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 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 informationOutline One-step model Risk-neutral valuation Two-step model Delta u&d Girsanov s Theorem. Binomial Trees. Haipeng Xing
Haipeng Xing Department of Applied Mathematics and Statistics Outline 1 An one-step Bionomial model and a no-arbitrage argument 2 Risk-neutral valuation 3 Two-step Binomial trees 4 Delta 5 Matching volatility
More informationMulti-Period Binomial Option Pricing - Outline
Multi-Period Binomial Option - Outline 1 Multi-Period Binomial Basics Multi-Period Binomial Option European Options American Options 1 / 12 Multi-Period Binomials To allow for more possible stock prices,
More informationChapter 2 Questions Sample Comparing Options
Chapter 2 Questions Sample Comparing Options Questions 2.16 through 2.21 from Chapter 2 are provided below as a Sample of our Questions, followed by the corresponding full Solutions. At the beginning of
More informationMATH 476/567 ACTUARIAL RISK THEORY FALL 2016 PROFESSOR WANG
MATH 476/567 ACTUARIAL RISK THEORY FALL 206 PROFESSOR WANG Homework 5 (max. points = 00) Due at the beginning of class on Tuesday, November 8, 206 You are encouraged to work on these problems in groups
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 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 informationP-7. Table of Contents. Module 1: Introductory Derivatives
Preface P-7 Table of Contents Module 1: Introductory Derivatives Lesson 1: Stock as an Underlying Asset 1.1.1 Financial Markets M1-1 1.1. Stocks and Stock Indexes M1-3 1.1.3 Derivative Securities M1-9
More informationMATH 361: Financial Mathematics for Actuaries I
MATH 361: Financial Mathematics for Actuaries I Albert Cohen Actuarial Sciences Program Department of Mathematics Department of Statistics and Probability C336 Wells Hall Michigan State University East
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 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 informationOption Valuation with Binomial Lattices corrected version Prepared by Lara Greden, Teaching Assistant ESD.71
Option Valuation with Binomial Lattices corrected version Prepared by Lara Greden, Teaching Assistant ESD.71 Note: corrections highlighted in bold in the text. To value options using the binomial lattice
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 informationRisk-neutral Binomial Option Valuation
Risk-neutral Binomial Option Valuation Main idea is that the option price now equals the expected value of the option price in the future, discounted back to the present at the risk free rate. Assumes
More informationOption Properties Liuren Wu
Option Properties Liuren Wu Options Markets (Hull chapter: 9) Liuren Wu ( c ) Option Properties Options Markets 1 / 17 Notation c: European call option price. C American call price. p: European put option
More informationS 0 C (30, 0.5) + P (30, 0.5) e rt 30 = PV (dividends) PV (dividends) = = $0.944.
Chapter 9 Parity and Other Option Relationships Question 9.1 This problem requires the application of put-call-parity. We have: Question 9.2 P (35, 0.5) = C (35, 0.5) e δt S 0 + e rt 35 P (35, 0.5) = $2.27
More informationUNIVERSITY OF AGDER EXAM. Faculty of Economicsand Social Sciences. Exam code: Exam name: Date: Time: Number of pages: Number of problems: Enclosure:
UNIVERSITY OF AGDER Faculty of Economicsand Social Sciences Exam code: Exam name: Date: Time: Number of pages: Number of problems: Enclosure: Exam aids: Comments: EXAM BE-411, ORDINARY EXAM Derivatives
More informationOutline One-step model Risk-neutral valuation Two-step model Delta u&d Girsanov s Theorem. Binomial Trees. Haipeng Xing
Haipeng Xing Department of Applied Mathematics and Statistics Outline 1 An one-step Bionomial model and a no-arbitrage argument 2 Risk-neutral valuation 3 Two-step Binomial trees 4 Delta 5 Matching volatility
More informationHelp Session 2. David Sovich. Washington University in St. Louis
Help Session 2 David Sovich Washington University in St. Louis TODAY S AGENDA 1. Refresh the concept of no arbitrage and how to bound option prices using just the principle of no arbitrage 2. Work on applying
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 informationStochastic Calculus for Finance
Stochastic Calculus for Finance Albert Cohen Actuarial Sciences Program Department of Mathematics Department of Statistics and Probability A336 Wells Hall Michigan State University East Lansing MI 48823
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 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 informationLecture 10 An introduction to Pricing Forward Contracts.
Lecture: 10 Course: M339D/M389D - Intro to Financial Math Page: 1 of 5 University of Texas at Austin Lecture 10 An introduction to Pricing Forward Contracts 101 Different ways to buy an asset (1) Outright
More informationMS-E2114 Investment Science Lecture 10: Options pricing in binomial lattices
MS-E2114 Investment Science Lecture 10: Options pricing in binomial lattices A. Salo, T. Seeve Systems Analysis Laboratory Department of System Analysis and Mathematics Aalto University, School of Science
More informationSolutions of Exercises on Black Scholes model and pricing financial derivatives MQF: ACTU. 468 S you can also use d 2 = d 1 σ T
1 KING SAUD UNIVERSITY Academic year 2016/2017 College of Sciences, Mathematics Department Module: QMF Actu. 468 Bachelor AFM, Riyadh Mhamed Eddahbi Solutions of Exercises on Black Scholes model and pricing
More informationUNIVERSITY OF OSLO. Faculty of Mathematics and Natural Sciences
UNIVERSITY OF OSLO Faculty of Mathematics and Natural Sciences Examination in MAT2700 Introduction to mathematical finance and investment theory. Day of examination: Monday, December 14, 2015. Examination
More informationd St+ t u. With numbers e q = The price of the option in three months is
Exam in SF270 Financial Mathematics. Tuesday June 3 204 8.00-3.00. Answers and brief solutions.. (a) This exercise can be solved in two ways. i. Risk-neutral valuation. The martingale measure should satisfy
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 informationUNIVERSITY OF OSLO. Faculty of Mathematics and Natural Sciences
UNIVERSITY OF OSLO Faculty of Mathematics and Natural Sciences Examination in MAT2700 Introduction to mathematical finance and investment theory. Day of examination: November, 2015. Examination hours:??.????.??
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 informationUniversity of Texas at Austin. Problem Set 2. Collars. Ratio spreads. Box spreads.
In-Class: 2 Course: M339D/M389D - Intro to Financial Math Page: 1 of 7 2.1. Collars in hedging. University of Texas at Austin Problem Set 2 Collars. Ratio spreads. Box spreads. Definition 2.1. A collar
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 informationFinancial Markets & Risk
Financial Markets & Risk Dr Cesario MATEUS Senior Lecturer in Finance and Banking Room QA259 Department of Accounting and Finance c.mateus@greenwich.ac.uk www.cesariomateus.com Session 3 Derivatives Binomial
More informationMATH 425 EXERCISES G. BERKOLAIKO
MATH 425 EXERCISES G. BERKOLAIKO 1. Definitions and basic properties of options and other derivatives 1.1. Summary. Definition of European call and put options, American call and put option, forward (futures)
More information3 + 30e 0.10(3/12) > <
Millersville University Department of Mathematics MATH 472, Financial Mathematics, Homework 06 November 8, 2011 Please answer the following questions. Partial credit will be given as appropriate, do not
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 informationRMSC 2001 Introduction to Risk Management
RMSC 2001 Introduction to Risk Management Tutorial 6 (2011/12) 1 March 19, 2012 Outline: 1. Option Strategies 2. Option Pricing - Binomial Tree Approach 3. More about Option ====================================================
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 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 informationSOCIETY OF ACTUARIES EXAM IFM INVESTMENT AND FINANCIAL MARKETS EXAM IFM SAMPLE QUESTIONS AND SOLUTIONS DERIVATIVES
SOCIETY OF ACTUARIES EXAM IFM INVESTMENT AND FINANCIAL MARKETS EXAM IFM SAMPLE QUESTIONS AND SOLUTIONS DERIVATIVES These questions and solutions are based on the readings from McDonald and are identical
More informationHelp Session 4. David Sovich. Washington University in St. Louis
Help Session 4 David Sovich Washington University in St. Louis TODAY S AGENDA More on no-arbitrage bounds for calls and puts Some discussion of American options Replicating complex payoffs Pricing in the
More informationActuarialBrew.com. Exam MFE / 3F. Actuarial Models Financial Economics Segment. Solutions 2014, 2nd edition
ActuarialBrew.com Exam MFE / 3F Actuarial Models Financial Economics Segment Solutions 04, nd edition www.actuarialbrew.com Brewing Better Actuarial Exam Preparation Materials ActuarialBrew.com 04 Please
More informationActuarialBrew.com. Exam MFE / 3F. Actuarial Models Financial Economics Segment. Solutions 2014, 1 st edition
ActuarialBrew.com Exam MFE / 3F Actuarial Models Financial Economics Segment Solutions 04, st edition www.actuarialbrew.com Brewing Better Actuarial Exam Preparation Materials ActuarialBrew.com 04 Please
More informationOptions. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan Options
Options An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2014 Definitions and Terminology Definition An option is the right, but not the obligation, to buy or sell a security such
More informationIntroduction. Financial Economics Slides
Introduction. Financial Economics Slides Howard C. Mahler, FCAS, MAAA These are slides that I have presented at a seminar or weekly class. The whole syllabus of Exam MFE is covered. At the end is my section
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 informationMotivating example: MCI
Real Options - intro Real options concerns using option pricing like thinking in situations where one looks at investments in real assets. This is really a matter of creative thinking, playing the game
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 informationErrata and updates for ASM Exam MFE (Tenth Edition) sorted by page.
Errata for ASM Exam MFE Study Manual (Tenth Edition) Sorted by Page 1 Errata and updates for ASM Exam MFE (Tenth Edition) sorted by page. Practice Exam 9:18 and 10:26 are defective. [4/3/2017] On page
More informationThe exam will be closed book and notes; only the following calculators will be permitted: TI-30X IIS, TI-30X IIB, TI-30Xa.
21-270 Introduction to Mathematical Finance D. Handron Exam #1 Review The exam will be closed book and notes; only the following calculators will be permitted: TI-30X IIS, TI-30X IIB, TI-30Xa. 1. (25 points)
More informationNPTEL INDUSTRIAL AND MANAGEMENT ENGINEERING DEPARTMENT, IIT KANPUR QUANTITATIVE FINANCE ASSIGNMENT-5 (2015 JULY-AUG ONLINE COURSE)
NPTEL INDUSTRIAL AND MANAGEMENT ENGINEERING DEPARTMENT, IIT KANPUR QUANTITATIVE FINANCE ASSIGNMENT-5 (2015 JULY-AUG ONLINE COURSE) NOTE THE FOLLOWING 1) There are five questions and you are required to
More informationExercise 14 Interest Rates in Binomial Grids
Exercise 4 Interest Rates in Binomial Grids Financial Models in Excel, F65/F65D Peter Raahauge December 5, 2003 The objective with this exercise is to introduce the methodology needed to price callable
More information(1) Consider a European call option and a European put option on a nondividend-paying stock. You are given:
(1) Consider a European call option and a European put option on a nondividend-paying stock. You are given: (i) The current price of the stock is $60. (ii) The call option currently sells for $0.15 more
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 informationDerivative Instruments
Derivative Instruments Paris Dauphine University - Master I.E.F. (272) Autumn 2016 Jérôme MATHIS jerome.mathis@dauphine.fr (object: IEF272) http://jerome.mathis.free.fr/ief272 Slides on book: John C. Hull,
More informationUCLA Anderson School of Management Daniel Andrei, Option Markets 232D, Fall MBA Midterm. November Date:
UCLA Anderson School of Management Daniel Andrei, Option Markets 232D, Fall 2013 MBA Midterm November 2013 Date: Your Name: Your Equiz.me email address: Your Signature: 1 This exam is open book, open notes.
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 informationOptions Markets: Introduction
17-2 Options Options Markets: Introduction Derivatives are securities that get their value from the price of other securities. Derivatives are contingent claims because their payoffs depend on the value
More informationThe Johns Hopkins Carey Business School. Derivatives. Spring Final Exam
The Johns Hopkins Carey Business School Derivatives Spring 2010 Instructor: Bahattin Buyuksahin Final Exam Final DUE ON WEDNESDAY, May 19th, 2010 Late submissions will not be graded. Show your calculations.
More informationBasics of Derivative Pricing
Basics o Derivative Pricing 1/ 25 Introduction Derivative securities have cash ows that derive rom another underlying variable, such as an asset price, interest rate, or exchange rate. The absence o arbitrage
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 informationMS-E2114 Investment Science Exercise 10/2016, Solutions
A simple and versatile model of asset dynamics is the binomial lattice. In this model, the asset price is multiplied by either factor u (up) or d (down) in each period, according to probabilities p and
More informationChapter 5. Financial Forwards and Futures. Copyright 2009 Pearson Prentice Hall. All rights reserved.
Chapter 5 Financial Forwards and Futures Introduction Financial futures and forwards On stocks and indexes On currencies On interest rates How are they used? How are they priced? How are they hedged? 5-2
More information