Notes: This is a closed book and closed notes exam. The maximal score on this exam is 100 points. Time: 75 minutes
|
|
- Isaac Green
- 5 years ago
- Views:
Transcription
1 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 75 minute class! Instructor: Milica Čudina Notes: This is a closed book and closed notes exam. The maximal score on this exam is 100 points. Time: 75 minutes Part I. TRUE/FALSE QUESTIONS 1. (2 pts) All else being equal, American put options are at least as valuable as European put options. TRUE 2. (2 points) In our usual notation, let S(0) = 40, r = 0.08, σ = 0.3, δ = 0. You need to construct a 2 period forward binomial tree for the above stock with every period in the tree of length h = 0.5. Then, u > FALSE u = exp{(0.08 0) } (2 points) In the usual notation for the binomial asset pricing model, we always have FALSE 1 < d < u. 4. (2 pts) In the setting of the Binomial Model, with i denoting the effective interest rate per period and assuming that the underlying asset pays no dividends: If then there is no possibility for arbitrage. FALSE d < u 1 + i 5. (2 pts) An American call option on a non-dividend paying stock should never be exercised early. More precisely, it is never more profitable to early exercise an American call option on a stock which pays no dividends prior to the expiry date of the option. TRUE 6. (2 pts) Suppose that the European options with the same maturity and the same underlying assets have the following prices: (1) a 50 strike call costs $9; (2) a 55 strike call costs $10;
2 2 Then, some of the monotonicity conditions for no-arbitrage are violated by the above premiums. TRUE We know that for strikes K 1 < K 2, the price of a call with strike K 1 should be greater than or equal to the price of a call with strike K 2 (see equation (9.13) in the book). This condition is violated. 7. (2 pts) Let V A (0, T ) denote the price at time 0 of an American option with expiration date T. Then, we always have TRUE V A (0, T ) V A (0, 2T ). 8. (2 pts) In the Cox-Ross-Rubinstein tree, we always have u = 1/d. TRUE Part II. FREE-RESPONSE PROBLEMS Please, explain carefully all your statements and assumptions. Numerical results or singleword answers without an explanation (even if they re correct) are worth 0 points. 1. (20 points) 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? 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 Vu cont = e 0.03 [0.46 (K 62.28) (K 51.04) + ] ; Vd cont = e 0.03 [0.46 (K 51.04) (K 38.72) + ]. On the other hand, the stock prices at these two nodes are S u = and S d =
3 3 So, the values of immediate/early exercise are 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). So, we need to figure out for which K the inequality K 44 > (K 38.72) K( ) > holds. We get K > (30 points) Consider a two-period binomial model with S(0) = $50, u = 2 and d = 0.5. (i) (5 points) Draw the binomial tree modeling the future evolution of this stock price with the given u and d. Your goal is to price an at-the-money European call option with two periods to maturity under the following assumptions: the underlying stock does not pay any dividends, the effective risk-free interest rate per period equals i = 25%. (ii) (5 pts) Find the risk-neutral probability. (iii) (8 pts) Find the fair price of the call using the risk-neutral pricing formula. (iv) (12 pts) Find the which should be used at every node in the tree in order to form a replicating portfolio. More precisely, in the notation used in class, calculate u, d and 0.
4 4 The binomial tree looks like this: The risk-neutral probability of the stock price going up in a single period is simply: p (1 + i) d = = 0.5. u d We proceed backwards through the inner nodes of the tree. At the up node, we have that the value of the call is V u = 1 1 [ ] = On the other hand, the remaining two final payoffs are both zero, which yields that the value of the call at the down node equals V d = 0. We have now reduced the pricing problem to a one-step binomial tree. The usual calculation gives us that the fair price of the above call is V C (0) = 1 1 [60 + 0] = With the usual notation, we have u = = 1; d = 0; 0 = = (20 points) Consider a one-period Binomial model with S(0) = $105, S u = $130 and S d = $80. Your goal is to determine if there is an arbitrage opportunity in a market in which a European call option on S with strike of K = $120 and exercise date T = 1
5 year is sold for $5. Assume that the continuously compounded risk-free interest rate equals r = 10%. If you believe that there is an arbitrage opportunity, describe the arbitrage portfolio and show that it is, indeed, an arbitrage portfolio. If you believe that there is no arbitrage opportunity, explain your reasoning. First, we need to figure out the no-arbitrage price of the given call option. Since no dividends are mentioned, we set δ = 0. The risk-neutral probability of the stock price going up is p = e(r δ)h d u d = e0.1 (80/105) (130 80)/105 = So, using the risk-neutral pricing formula, we get V C (0) = e ( ) = The observed price of the call is given to be C = 5. We conclude that there is an arbitrage opportunity since the no-arbitrage price (i.e., the value at time 0 of the replicating portfolio for the call) is different from the observed call premium. Moreover, since the observed premium is lower than the no-arbitrage price, we conclude that he observed call is relatively cheap when comapred to its replicating portfolio. To take advantage of this situation, we construct the following arbitrage portfolio: (1) long call, (2) short = Vu V d S u S d = 0.2 shares of stock, (3) invest B = V C (0) S(0) = = in the money market at the risk-free interest rate r. Note: Since B is a negative number, the interpretation is that we lend B = B to the money market; this can be interpreted as purchasing zero-coupon bonds in the required amout to be redeemed at time T. The initial cost of this portfolio is C S(0) + ( (V C (0) S(0))) = C V C (0) < 0. So, there is an initial inflow of money. At the expiration date, the call and the short replicating portfolio have payoffs which exactly cancel out (by construction!). Since there was an initial inflow of money, we indeed constructed an arbitrage portfolio. The arbitrage portfolio above is just an example of an arbitrage portfolio it is by no means unique! 4. (10 points) Let S(0) = 40, r = 0.08, σ = 0.3, δ = 0. You need to find the up and down factors in a 2 period forward binomial tree modeling the price of this stock during the following year. (a) (4 pts) What are u and d? (b) (6 pts) What is the risk-neutral probability of the stock price going up in a single period? 5
6 6 (a) (b) u = e (r δ)h+σ h = e = 1.29; d = e (r δ)h σ h = e = p = e σ h = Part III. MULTIPLE CHOICE QUESTIONS 1. (5 pts) Let S be a non-dividend paying stock with a current price equal to S 0. You know that in the one-period binomial tree for this stock, S u = 150 and S d = 120. An actuary calculates the volatility of S based on the provided values and gets σ Which of the models we covered in class did this actuary use to obtain the volatility? (a) The forward binomial tree. (b) The Cox-Ross-Rubinstein model. (c) The lognormal tree. (d) Any of the three models. (e) None of the three models. (d) The ratio S u /S d = u/d is always e 2σ for all of the models we covered. In this case, u = 1.25 = d e2σ σ = 1 ln(1.25)
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.
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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 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 informationBond Future Option Valuation Guide
Valuation Guide David Lee FinPricing http://www.finpricing.com Summary Bond Future Option Introduction The Use of Bond Future Options Valuation European Style Valuation American Style Practical Guide A
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 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 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 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 informationLecture 16: Delta Hedging
Lecture 16: Delta Hedging We are now going to look at the construction of binomial trees as a first technique for pricing options in an approximative way. These techniques were first proposed in: J.C.
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 informationAdvanced Numerical Methods
Advanced Numerical Methods Solution to Homework One Course instructor: Prof. Y.K. Kwok. When the asset pays continuous dividend yield at the rate q the expected rate of return of the asset is r q under
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 informationBinomial Trees. Liuren Wu. Zicklin School of Business, Baruch College. Options Markets
Binomial Trees Liuren Wu Zicklin School of Business, Baruch College Options Markets Binomial tree represents a simple and yet universal method to price options. I am still searching for a numerically efficient,
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 informationS u =$55. S u =S (1+u) S=$50. S d =$48.5. S d =S (1+d) C u = $5 = Max{55-50,0} $1.06. C u = Max{Su-X,0} (1+r) (1+r) $1.06. C d = $0 = Max{48.
Fi8000 Valuation of Financial Assets Spring Semester 00 Dr. Isabel katch Assistant rofessor of Finance Valuation of Options Arbitrage Restrictions on the Values of Options Quantitative ricing Models Binomial
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 informationOption Pricing Models. c 2013 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 205
Option Pricing Models c 2013 Prof. Yuh-Dauh Lyuu, National Taiwan University Page 205 If the world of sense does not fit mathematics, so much the worse for the world of sense. Bertrand Russell (1872 1970)
More informationThe Binomial Model. Chapter 3
Chapter 3 The Binomial Model In Chapter 1 the linear derivatives were considered. They were priced with static replication and payo tables. For the non-linear derivatives in Chapter 2 this will not work
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 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 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 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 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 informationP2.T5. Tuckman Chapter 7 The Science of Term Structure Models. Bionic Turtle FRM Video Tutorials. By: David Harper CFA, FRM, CIPM
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
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 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 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 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 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 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 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 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 informationUniversity of Waterloo Final Examination
University of Waterloo Final Examination Term: Fall 2008 Last Name First Name UW Student ID Number Course Abbreviation and Number AFM 372 Course Title Math Managerial Finance 2 Instructor Alan Huang Date
More informationFinal Exam. 5. (21 points) Short Questions. Parts (i)-(v) are multiple choice: in each case, only one answer is correct.
Final Exam Spring 016 Econ 180-367 Closed Book. Formula Sheet Provided. Calculators OK. Time Allowed: 3 hours Please write your answers on the page below each question 1. (10 points) What is the duration
More informationCourse MFE/3F Practice Exam 1 Solutions
Course MFE/3F Practice Exam 1 Solutions he chapter references below refer to the chapters of the ActuraialBrew.com Study Manual. Solution 1 C Chapter 16, Sharpe Ratio If we (incorrectly) assume that the
More informationPage 1. Real Options for Engineering Systems. Financial Options. Leverage. Session 4: Valuation of financial options
Real Options for Engineering Systems Session 4: Valuation of financial options Stefan Scholtes Judge Institute of Management, CU Slide 1 Financial Options Option: Right (but not obligation) to buy ( call
More informationI. Reading. A. BKM, Chapter 20, Section B. BKM, Chapter 21, ignore Section 21.3 and skim Section 21.5.
Lectures 23-24: Options: Valuation. I. Reading. A. BKM, Chapter 20, Section 20.4. B. BKM, Chapter 21, ignore Section 21.3 and skim Section 21.5. II. Preliminaries. A. Up until now, we have been concerned
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 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 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 17. Options and Corporate Finance. Key Concepts and Skills
Chapter 17 Options and Corporate Finance Prof. Durham Key Concepts and Skills Understand option terminology Be able to determine option payoffs and profits Understand the major determinants of option prices
More informationSolutions FINAL EXAM 2002 SPRING Sridhar Seshadri B
Solutions FINAL EXAM 22 SPRING Sridhar Seshadri B9.238. Answer all questions. Answer questions and 2 before attempting question 3. The exam is closed book and closed notes (except for 4 pages of notes).
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 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 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 informationPerpetual Option Pricing Revision of the NPV Rule, Application in C++
Utah State University DigitalCommons@USU All Graduate Plan B and other Reports Graduate Studies 5-2015 Perpetual Option Pricing Revision of the NPV Rule, Application in C++ Andy Ferguson Utah State University
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 informationANALYSIS OF THE BINOMIAL METHOD
ANALYSIS OF THE BINOMIAL METHOD School of Mathematics 2013 OUTLINE 1 CONVERGENCE AND ERRORS OUTLINE 1 CONVERGENCE AND ERRORS 2 EXOTIC OPTIONS American Options Computational Effort OUTLINE 1 CONVERGENCE
More informationComputational Finance Binomial Trees Analysis
Computational Finance Binomial Trees Analysis School of Mathematics 2018 Review - Binomial Trees Developed a multistep binomial lattice which will approximate the value of a European option Extended the
More informationSuper-replicating portfolios
Super-replicating portfolios 1. Introduction Assume that in one year from now the price for a stock X may take values in the set. Consider four derivative instruments and their payoffs which depends on
More informationOne Period Binomial Model: The risk-neutral probability measure assumption and the state price deflator approach
One Period Binomial Model: The risk-neutral probability measure assumption and the state price deflator approach Amir Ahmad Dar Department of Mathematics and Actuarial Science B S AbdurRahmanCrescent University
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 informationBarrier Option Valuation with Binomial Model
Division of Applied Mathmethics School of Education, Culture and Communication Box 833, SE-721 23 Västerås Sweden MMA 707 Analytical Finance 1 Teacher: Jan Röman Barrier Option Valuation with Binomial
More informationOption Pricing Models for European Options
Chapter 2 Option Pricing Models for European Options 2.1 Continuous-time Model: Black-Scholes Model 2.1.1 Black-Scholes Assumptions We list the assumptions that we make for most of this notes. 1. The underlying
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 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 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 informationECO OPTIONS AND FUTURES SPRING Options
ECO-30004 OPTIONS AND FUTURES SPRING 2008 Options These notes describe the payoffs to European and American put and call options the so-called plain vanilla options. We consider the payoffs to these options
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 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 informationINSTITUTE OF ACTUARIES OF INDIA
INSTITUTE OF ACTUARIES OF INDIA EXAMINATIONS 10 th November 2008 Subject CT8 Financial Economics Time allowed: Three Hours (14.30 17.30 Hrs) Total Marks: 100 INSTRUCTIONS TO THE CANDIDATES 1) Please read
More informationOption Models for Bonds and Interest Rate Claims
Option Models for Bonds and Interest Rate Claims Peter Ritchken 1 Learning Objectives We want to be able to price any fixed income derivative product using a binomial lattice. When we use the lattice to
More information2 The binomial pricing model
2 The binomial pricing model 2. Options and other derivatives A derivative security is a financial contract whose value depends on some underlying asset like stock, commodity (gold, oil) or currency. The
More informationBinomial Option Pricing
Binomial Option Pricing The wonderful Cox Ross Rubinstein model Nico van der Wijst 1 D. van der Wijst Finance for science and technology students 1 Introduction 2 3 4 2 D. van der Wijst Finance for science
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 informationFinancial Derivatives Section 5
Financial Derivatives Section 5 The Black and Scholes Model Michail Anthropelos anthropel@unipi.gr http://web.xrh.unipi.gr/faculty/anthropelos/ University of Piraeus Spring 2018 M. Anthropelos (Un. of
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 information