Hull, Options, Futures, and Other Derivatives, 9 th Edition
|
|
- Jonah Henry
- 5 years ago
- Views:
Transcription
1 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
2 Hull, Chapter 13: Binomial Trees CALCULATE THE VALUE OF A EUROPEAN CALL OR PUT OPTION USING THE ONE STEP AND TWO STEP BINOMIAL MODEL
3 Hull, Chapter 13: Binomial Trees Calculate the value of an American and a European call or put option using a one-step and two-step binomial model. Describe how volatility is captured in the binomial model. Describe how the value calculated using a binomial model converges as time periods are added. Explain how the binomial model can be altered to price options on: stocks with dividends, stock indices, currencies, and futures. The basic approaches to option valuation are binomial (a simulation) and Black- Scholes (analytical) The two basic approaches to option valuation are Black-Scholes (analytical or closed-form) and Binomial (simulation or open lattice). The binomial is a tree of future price possibilities; it is very flexible and highly intuitive but tedious to generate. The Black-Scholes is efficient, elegant, logical, robust and super-quick, but for most people it remains cryptic. c S N( d ) Ke N( d ) rt Binomial (discrete time, aka lattice) Black-Scholes (continuous time & closed form) Calculate the value of a European call or put option using the onestep and two step binomial model. A binomial tree is a useful and popular technique for pricing an option, which is a diagram representing different possible random paths that might be followed by the stock price over the life of an option. In each time step, the stock has a certain probability of moving up by a certain percentage amount and a certain probability of moving down by a certain percentage amount. 3
4 To price an option in a binomial model, we assume that arbitrage opportunities do not exist. We set up a portfolio of the stock and the option in such a way that there is no uncertainty about the value of the portfolio at the end of the time period. We then argue that, because the portfolio has no risk, the return it earns must equal the risk-free interest rate. This enables us to work out the cost of the setting up the riskless portfolio and the option s price. We use the following notations: = price/value of option = proportional up jump ( >1) = stock price = proportional down jump ( <1) = number of shares of stock = option payoff if stock jumps up t = length of the time step = option payoff if stock jumps down Let us consider a portfolio consisting of a long position in shares of a stock and a short position in one option. The portfolio is riskless if the value of is chosen so that the final value of the portfolio is the same for both alternatives. = This equation shows that is the ratio of the change in the option price to the change in the stock price as we move between the nodes at time T. Risk-neutral Valuation In a risk neutral world all individuals are indifferent to risk, and investors would require no compensation for risk as against the real world, where investors require higher returns for assuming higher risks. Though options are risky investments, the principle of pricing an option under the risk-neutral assumption is possible in the real-world because when pricing an option relatively in terms of the price of the underlying stock, risk preferences are unimportant. In a risk-neutral world, the expected return on a stock (or any other instrument) would be the risk- free rate. ( ) = If the risk-neutral probability of an up movement is denoted by so that 1 is the riskneutral probability of a down movement. We assume > so that 0 < < 1. = = where = The expected future payoff from the option in a risk-neutral world is + (1 ). This probability then plugs into the equation that solves for the option price: = [ + (1 ) ] This equation states that the value of the option today is its expected future payoff in a riskneutral world discounted at the risk-free rate. This is an application of risk-neutral valuation. 4
5 Binomial Tree In any binomial pricing model, we need to calculate the option price at the initial node of the tree. This can be done by (i) building the paths forward and (ii) backward induction. The option prices at the final nodes of the tree are first calculated. For this we need to build the build the paths forward, where forward stock prices are a function of their respective up or down movements. Stock prices can move up from an initial level of to a new level, or down to in a single time step. As time steps increases they may move to new levels like, or. Then the option values at final nodes are calculated as the payoffs from the option, such as: for a call option: Max (0, Stock price at final node strike price) for a put option: Max (0, Strike price stock price at the final node) If at any node, the option is out of the money, then its value is zero. Then, working backward by discounting the value of the two subsequent nodes at the riskfree rate, (that is, weighing the final node option values by their respective risk neutral probabilities and then discounting them back for a single time period at the risk-free rate), we calculate the option price ( ) at the initial node. A one-step binomial model (as shown above) will have a single node from which stock prices will move to either up or down state at the end of the given time period one. This can be extended to a two-step model wherein from each branch node at time period one, the stock prices may move up or down by the end of the time period two. For example, after two movements, the value of the option can be. If is the length of the time step, then the value at the initial node, can be found by backward induction (as above) or by directly using the formula below, which discounts the value for two periods: = 2 [ (1 ) + (1 ) ] where, 2 (1 ) (1 ) 2 are the probabilities that the upper, middle and final nodes will be reached, respectively. 5
6 Two step Binomial Trees Below is the two-step binomial for a European call option on a stock. The assumptions are: Current asset price = $20, Strike = $21, Time step= 0.25 years, Volatility = n.a, Riskless rate = 12%, and Dividend Yield = 0%. We consider a 6-month option, so each time step is three months long, or 0.25 years. Here we assume the stock price (at each of the two time steps) will either go up by 10% or down by 10%. The stock prices at the end of period one are $22.00 and $ The possible final stock prices are: $24.20, $19.80, and $ The risk-neutral probability ( = ) is found first. The option prices at the final nodes are calculated as payoffs from the option. At the topmost node of time period two (T=0.5), the option value ( ) is 3.2 (stock price of 24.2 minus strike price of 21). In the middle and lowermost nodes, the option is out of the money and so its value is zero (for and ). Moving backwards to time period one (T=0.25), we find that at the lowermost node the option price ( ) is zero, because this node leads to forward nodes where the option is out of money ( and ). The value of the topmost node ( ) during this time period is found to be by discounting back the risk neutral probability weighted option values at top and middle nodes ( and ) in time period two. Repeatedly moving one step back to time period zero(t=0) at the initial node, the value of the call ( =1.2822) can once again be found by discounting the risk neutral probability weighted option values obtained from the two nodes ( = and = 0) found at time period one. Hull s Example 13.4: Two-step European call option, with up and down simply given as inputs. Here volatility does not inform up and down. Consequently, this model does not implicitly assume lognormal prices. Here the assumption is simply that the jump is+/- 10%. Asset price $20.00 Params Strike price $21.00 u % given (normal arithmetic) Time/step, Δt 0.25 yrs d % given (normal arithmetic) Volatility, σ n.a. a a = exp[(r-q)*δt] Riskfree rate, r 12.0% p < probability of up jump Dividend, q 0.0% 1-p < probability of down jump Node Time (yrs) S = c = Stock Option $
Risk-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 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 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 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 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 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 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 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 informationLecture 16. Options and option pricing. Lecture 16 1 / 22
Lecture 16 Options and option pricing Lecture 16 1 / 22 Introduction One of the most, perhaps the most, important family of derivatives are the options. Lecture 16 2 / 22 Introduction One of the most,
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 informationP1.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 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 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 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 informationP2.T8. Risk Management & Investment Management. Jorion, Value at Risk: The New Benchmark for Managing Financial Risk, 3rd Edition.
P2.T8. Risk Management & Investment Management Jorion, Value at Risk: The New Benchmark for Managing Financial Risk, 3rd Edition. Bionic Turtle FRM Study Notes By David Harper, CFA FRM CIPM and Deepa Raju
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 informationValuing Stock Options: The Black-Scholes-Merton Model. Chapter 13
Valuing Stock Options: The Black-Scholes-Merton Model Chapter 13 1 The Black-Scholes-Merton Random Walk Assumption l Consider a stock whose price is S l In a short period of time of length t the return
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 informationPricing Options on Dividend paying stocks, FOREX, Futures, Consumption Commodities
Pricing Options on Dividend paying stocks, FOREX, Futures, Consumption Commodities The Black-Scoles Model The Binomial Model and Pricing American Options Pricing European Options on dividend paying stocks
More information15 American. Option Pricing. Answers to Questions and Problems
15 American Option Pricing Answers to Questions and Problems 1. Explain why American and European calls on a nondividend stock always have the same value. An American option is just like a European option,
More informationAnthony Saunders and Marcia Millon Cornett, Financial Institutions Management: A Risk Management Approach
P1.T3. Financial Markets & Products Anthony Saunders and Marcia Millon Cornett, Financial Institutions Management: A Risk Management Approach Bionic Turtle FRM Study Notes Sample By David Harper, CFA FRM
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 informationValuation of Options: Theory
Valuation of Options: Theory Valuation of Options:Theory Slide 1 of 49 Outline Payoffs from options Influences on value of options Value and volatility of asset ; time available Basic issues in valuation:
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 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 informationThe Pennsylvania State University. The Graduate School. Department of Industrial Engineering AMERICAN-ASIAN OPTION PRICING BASED ON MONTE CARLO
The Pennsylvania State University The Graduate School Department of Industrial Engineering AMERICAN-ASIAN OPTION PRICING BASED ON MONTE CARLO SIMULATION METHOD A Thesis in Industrial Engineering and Operations
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 informationTwo Types of Options
FIN 673 Binomial Option Pricing Professor Robert B.H. Hauswald Kogod School of Business, AU Two Types of Options An option gives the holder the right, but not the obligation, to buy or sell a given quantity
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 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 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 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 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 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 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 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 informationAN IMPROVED BINOMIAL METHOD FOR PRICING ASIAN OPTIONS
Commun. Korean Math. Soc. 28 (2013), No. 2, pp. 397 406 http://dx.doi.org/10.4134/ckms.2013.28.2.397 AN IMPROVED BINOMIAL METHOD FOR PRICING ASIAN OPTIONS Kyoung-Sook Moon and Hongjoong Kim Abstract. We
More informationThe Binomial Approach
W E B E X T E N S I O N 6A The Binomial Approach See the Web 6A worksheet in IFM10 Ch06 Tool Kit.xls for all calculations. The example in the chapter illustrated the binomial approach. This extension explains
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 informationOption Valuation (Lattice)
Page 1 Option Valuation (Lattice) Richard de Neufville Professor of Systems Engineering and of Civil and Environmental Engineering MIT Massachusetts Institute of Technology Option Valuation (Lattice) Slide
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 information4. Black-Scholes Models and PDEs. Math6911 S08, HM Zhu
4. Black-Scholes Models and PDEs Math6911 S08, HM Zhu References 1. Chapter 13, J. Hull. Section.6, P. Brandimarte Outline Derivation of Black-Scholes equation Black-Scholes models for options Implied
More informationUtility Indifference Pricing and Dynamic Programming Algorithm
Chapter 8 Utility Indifference ricing and Dynamic rogramming Algorithm In the Black-Scholes framework, we can perfectly replicate an option s payoff. However, it may not be true beyond the Black-Scholes
More informationCash Flows on Options strike or exercise price
1 APPENDIX 4 OPTION PRICING In general, the value of any asset is the present value of the expected cash flows on that asset. In this section, we will consider an exception to that rule when we will look
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 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 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 informationP2.T5. Market Risk Measurement & Management
P2.T5. Market Risk Measurement & Management Kevin Dowd, Measuring Market Risk Bionic Turtle FRM Study Notes By David Harper, CFA FRM CIPM and Deepa Raju www.bionicturtle.com Dowd Chapter 3: Estimating
More informationECON4510 Finance Theory Lecture 10
ECON4510 Finance Theory Lecture 10 Diderik Lund Department of Economics University of Oslo 11 April 2016 Diderik Lund, Dept. of Economics, UiO ECON4510 Lecture 10 11 April 2016 1 / 24 Valuation of options
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 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 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 informationComputational Finance. Computational Finance p. 1
Computational Finance Computational Finance p. 1 Outline Binomial model: option pricing and optimal investment Monte Carlo techniques for pricing of options pricing of non-standard options improving accuracy
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 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 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 informationMASM006 UNIVERSITY OF EXETER SCHOOL OF ENGINEERING, COMPUTER SCIENCE AND MATHEMATICS MATHEMATICAL SCIENCES FINANCIAL MATHEMATICS.
MASM006 UNIVERSITY OF EXETER SCHOOL OF ENGINEERING, COMPUTER SCIENCE AND MATHEMATICS MATHEMATICAL SCIENCES FINANCIAL MATHEMATICS May/June 2006 Time allowed: 2 HOURS. Examiner: Dr N.P. Byott This is a CLOSED
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 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 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 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 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 informationFixed-Income Securities Lecture 5: Tools from Option Pricing
Fixed-Income Securities Lecture 5: Tools from Option Pricing Philip H. Dybvig Washington University in Saint Louis Review of binomial option pricing Interest rates and option pricing Effective duration
More informationThe Black-Scholes Equation
The Black-Scholes Equation MATH 472 Financial Mathematics J. Robert Buchanan 2018 Objectives In this lesson we will: derive the Black-Scholes partial differential equation using Itô s Lemma and no-arbitrage
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 informationEdgeworth Binomial Trees
Mark Rubinstein Paul Stephens Professor of Applied Investment Analysis University of California, Berkeley a version published in the Journal of Derivatives (Spring 1998) Abstract This paper develops a
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 informationWeb Extension: The Binomial Approach
19878_06W_p001-009.qxd 3/10/06 9:53 AM Page 1 C H A P T E R 6 Web Extension: The Binomial Approach The example in the chapter illustrated the binomial approach. This extension explains the approach in
More informationFINITE DIFFERENCE METHODS
FINITE DIFFERENCE METHODS School of Mathematics 2013 OUTLINE Review 1 REVIEW Last time Today s Lecture OUTLINE Review 1 REVIEW Last time Today s Lecture 2 DISCRETISING THE PROBLEM Finite-difference approximations
More informationAn Adjusted Trinomial Lattice for Pricing Arithmetic Average Based Asian Option
American Journal of Applied Mathematics 2018; 6(2): 28-33 http://www.sciencepublishinggroup.com/j/ajam doi: 10.11648/j.ajam.20180602.11 ISSN: 2330-0043 (Print); ISSN: 2330-006X (Online) An Adjusted Trinomial
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 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 informationThe Black-Scholes Model
The Black-Scholes Model Liuren Wu Options Markets Liuren Wu ( c ) The Black-Merton-Scholes Model colorhmoptions Markets 1 / 18 The Black-Merton-Scholes-Merton (BMS) model Black and Scholes (1973) and Merton
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 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 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 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 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 informationEconomic Risk and Decision Analysis for Oil and Gas Industry CE School of Engineering and Technology Asian Institute of Technology
Economic Risk and Decision Analysis for Oil and Gas Industry CE81.98 School of Engineering and Technology Asian Institute of Technology January Semester Presented by Dr. Thitisak Boonpramote Department
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 informationPricing theory of financial derivatives
Pricing theory of financial derivatives One-period securities model S denotes the price process {S(t) : t = 0, 1}, where S(t) = (S 1 (t) S 2 (t) S M (t)). Here, M is the number of securities. At t = 1,
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 informationCrashcourse Interest Rate Models
Crashcourse Interest Rate Models Stefan Gerhold August 30, 2006 Interest Rate Models Model the evolution of the yield curve Can be used for forecasting the future yield curve or for pricing interest rate
More informationComputational Finance
Path Dependent Options Computational Finance School of Mathematics 2018 The Random Walk One of the main assumption of the Black-Scholes framework is that the underlying stock price follows a random walk
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 and Deepa Raju
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 informationLecture 6: Option Pricing Using a One-step Binomial Tree. Thursday, September 12, 13
Lecture 6: Option Pricing Using a One-step Binomial Tree An over-simplified model with surprisingly general extensions a single time step from 0 to T two types of traded securities: stock S and a bond
More informationComputational Finance Finite Difference Methods
Explicit finite difference method Computational Finance Finite Difference Methods School of Mathematics 2018 Today s Lecture We now introduce the final numerical scheme which is related to the PDE solution.
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 informationAs we saw in Chapter 12, one of the many uses of Monte Carlo simulation by
Financial Modeling with Crystal Ball and Excel, Second Edition By John Charnes Copyright 2012 by John Charnes APPENDIX C Variance Reduction Techniques As we saw in Chapter 12, one of the many uses of Monte
More informationLECTURE 2: MULTIPERIOD MODELS AND TREES
LECTURE 2: MULTIPERIOD MODELS AND TREES 1. Introduction One-period models, which were the subject of Lecture 1, are of limited usefulness in the pricing and hedging of derivative securities. In real-world
More informationOptions Pricing Using Combinatoric Methods Postnikov Final Paper
Options Pricing Using Combinatoric Methods 18.04 Postnikov Final Paper Annika Kim May 7, 018 Contents 1 Introduction The Lattice Model.1 Overview................................ Limitations of the Lattice
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 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 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 informationOption Pricing With Dividends
Option Pricing With Dividends Betuel Canhanga. Carolyne Ogutu. Analytical Finance I Seminar Report October, 13 Contents 1 Introduction Solution One: Include Any Dividends After Expiration 3.1 Expiry before
More informationDynamic Portfolio Choice II
Dynamic Portfolio Choice II Dynamic Programming Leonid Kogan MIT, Sloan 15.450, Fall 2010 c Leonid Kogan ( MIT, Sloan ) Dynamic Portfolio Choice II 15.450, Fall 2010 1 / 35 Outline 1 Introduction to Dynamic
More informationKevin Dowd, Measuring Market Risk, 2nd Edition
P1.T4. Valuation & Risk Models Kevin Dowd, Measuring Market Risk, 2nd Edition Bionic Turtle FRM Study Notes By David Harper, CFA FRM CIPM www.bionicturtle.com Dowd, Chapter 2: Measures of Financial Risk
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 information