Dr. Maddah ENMG 625 Financial Eng g II 10/16/06. Chapter 11 Models of Asset Dynamics (1)
|
|
- Tobias Watkins
- 5 years ago
- Views:
Transcription
1 Dr Maddah ENMG 65 Financial Eng g II 0/6/06 Chater Models of Asset Dynamics () Overview Stock rice evolution over time is commonly modeled with one of two rocesses: The binomial lattice and geometric Brownian motion Both of these are stochastic rocesses that have the Markovian roerty Geometric Brownian motion is a secial case of Ito rocess Binomial lattices are discrete-time, discrete-value Ie, o There are limited time instants where the stock rice can change; o There are limited values that the stock rice can take On the other hand geometric Brownian motion is continuous in time and in values With small time intervals for stock rice changes and arameters chosen aroriately, a binomial lattice can reasonably aroximate a geometric Brownian motion Binomial lattices are analytically simler than Geometric Brownian motion rocesses They are of great use in ractical comutational works
2 Binomial lattice model A binomial lattice model is defined over a basic eriod length t (eg, t = day =/365 years) where the rice can change If S is the rice at the beginning of a eriod, the rice at the end of the eriod is us w and ds w, where u > and d < This model can be extended for any number of eriods A roer choice of the arameters, u, and d is such that the binomial lattice aroximates a geometric Brownian motion Define the mean and variance of the yearly growth rate as v = E[ln(S T /S 0 )] and σ = var[ln(s T /S 0 )], where S 0 is the initial stock rice and S T is the rice at the end of year Then, the binomial lattice arameters are chosen as v σ t = + t, u = e, d = e σ σ t
3 The additive model This model allows the stock rice to change over a continuum of values but at discrete times The stock rice at time k +, S(k+) is given function of the stock rice at time k, S(k), as S( k + ) = S( k) + ε k, where ε k are indeendent and identically distributed normal random variables with mean 0 and variance σ It can be easily shown that k = + i i= S( k) S(0) ε It follows that S(k) is a normal random variable with E[S(k)] = S(0) and k σ kσ i= var[ S ] = = k The additive model is simle and easy to work with However, it lacks realism because (i) The normal distribution allows negative stock rices; (ii) The robability that the rice changes by a certain amount is the same regardless of the rice level 3
4 Lognormal random variables A random variable Y is said to be lognormal if X = ln(y) is a normal random variable Alternatively, Y is a lognormal r v if Y = e X, where X is a normal r v If X = ln(y) is normal with mean λ and variance σ, then the density function of Y is (ln y λ ) fy y e y y πσ σ ( ) =, > 0 dlnorm y, 0, dlnorm( y, 0, ) 3 dlnorm y, 0, The mean and variance of Y are given by E Y e Y e e λ+ σ / λ+ σ σ [ ] =, var[ ] = ( ) The mode of the distribution is m = e λ σ A good book to learn about lognormal and other distributions is Simulation Modeling and Analysis by Law and Kelton 4
5 The multilicative model Similar to the additive model, this model is continuous in value but discrete in time The stock rice at time k +, S(k+) is given function of the stock rice at time k, S(k), as S( k + ) = S( k) ε k, where ε k are iid lognormal random variables with arameters v and σ The logarithmic of the stock rice in a multilicative model follows an additive model ln S( k + ) = ln S( k) + ln( ε k ) Using the additive model results, we find that ln(s(k)) is a normal random variable with mean and variance E S k S vk S k kσ [ln ( )] = ln (0) +, var[ln ( )] = Therefore, the stock rice, S(k) is a lognormal rv Note that in this model the stock rice cannot be negative In addition, the roblem with stock rice difference over a eriod being indeendent of the rice level is eliminated With the multilicative model ln S(k+)/ S(k) has the same normal distribution This imlies that the robability of a given relative (ercent) change of the rice is the same at all rice levels 5
6 Real stock distribution To validate the multilicative model, test the goodness of fit of a normal distribution to lns(k+)/s(k) Chooses a eriod length (eg week, day), and then note the stock rice at the beginning of each eriod Then, form the histogram of the logarithmic of the ratio of two consecutive observations, ln S(k+)/S(k) It is observed (according to our text) that most stocks are actually close to the multilicative model Eg, the histogram below is for American Airlines stock, ln S(k+)/S(k), from 98 to 99 with a eriod length of week A normal fit with the same vaeiance is also shown Two tyical differences between fitted and actual distribution are that the actual distribution is skinnier around the mean and has a fatter tail 6
7 Tyical values and estimation of arameters Tyical annual values for the arameters of the multilicative models are v = E[ln(S T /S 0 )] = % and σ = stdev[ln(s T /S 0 )] = 5% If the observation eriod length is art of the year then these values scale down to v = v and σ = σ Eg, tyical weekly arameters values are v /5 = 03% and σ /5 = 08% Estimate v and σ based on observations from n+ eriods of length each are as follows n k = 0 [ ] vˆ = ln S( k + ) ln S( k) = ln S( n) / S(0), n n ˆ σ n = { ln [ S( k + ) / S( k) ] vˆ } n k = 0 The error on these estimates are characterized by ˆ var[ v ] = / n, ˆ σ n var[ σ ] = σ /( ) With the tyical values of v % and σ 5% It turns out that estimating σ accurately requires much less data than estimating v accurately (Remember Mean Blur?) 7
Dr. Maddah ENMG 625 Financial Eng g II 10/16/06
Dr. Maddah ENMG 65 Financial Eng g II 10/16/06 Chapter 11 Models of Asset Dynamics () Random Walk A random process, z, is an additive process defined over times t 0, t 1,, t k, t k+1,, such that z( t )
More information1 < = α σ +σ < 0. Using the parameters and h = 1/365 this is N ( ) = If we use h = 1/252, the value would be N ( ) =
Chater 6 Value at Risk Question 6.1 Since the rice of stock A in h years (S h ) is lognormal, 1 < = α σ +σ < 0 ( ) P Sh S0 P h hz σ α σ α = P Z < h = N h. σ σ (1) () Using the arameters and h = 1/365 this
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 informationLecture 3. Sergei Fedotov Introduction to Financial Mathematics. Sergei Fedotov (University of Manchester) / 6
Lecture 3 Sergei Fedotov 091 - Introduction to Financial Mathematics Sergei Fedotov (University of Manchester) 091 010 1 / 6 Lecture 3 1 Distribution for lns(t) Solution to Stochastic Differential Equation
More informationA random variable X is a function that assigns (real) numbers to the elements of the sample space S of a random experiment.
RANDOM VARIABLES and PROBABILITY DISTRIBUTIONS A random variable X is a function that assigns (real) numbers to the elements of the samle sace S of a random exeriment. The value sace V of a random variable
More informationChapter 1: Stochastic Processes
Chater 1: Stochastic Processes 4 What are Stochastic Processes, and how do they fit in? STATS 210 Foundations of Statistics and Probability Tools for understanding randomness (random variables, distributions)
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 informationReading: You should read Hull chapter 12 and perhaps the very first part of chapter 13.
FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 Asset Price Dynamics Introduction These notes give assumptions of asset price returns that are derived from the efficient markets hypothesis. Although a hypothesis,
More informationValuation of Asian Option. Qi An Jingjing Guo
Valuation of Asian Option Qi An Jingjing Guo CONTENT Asian option Pricing Monte Carlo simulation Conclusion ASIAN OPTION Definition of Asian option always emphasizes the gist that the payoff depends on
More informationMonte Carlo Simulation of Stochastic Processes
Monte Carlo Simulation of Stochastic Processes Last update: January 10th, 2004. In this section is presented the steps to perform the simulation of the main stochastic processes used in real options applications,
More information2/20/2013. of Manchester. The University COMP Building a yes / no classifier
COMP4 Lecture 6 Building a yes / no classifier Buildinga feature-basedclassifier Whatis a classifier? What is an information feature? Building a classifier from one feature Probability densities and the
More informationLecture 2. Main Topics: (Part II) Chapter 2 (2-7), Chapter 3. Bayes Theorem: Let A, B be two events, then. The probabilities P ( B), probability of B.
STT315, Section 701, Summer 006 Lecture (Part II) Main Toics: Chater (-7), Chater 3. Bayes Theorem: Let A, B be two events, then B A) = A B) B) A B) B) + A B) B) The robabilities P ( B), B) are called
More informationValue at Risk Ch.12. PAK Study Manual
Value at Risk Ch.12 Related Learning Objectives 3a) Apply and construct risk metrics to quantify major types of risk exposure such as market risk, credit risk, liquidity risk, regulatory risk etc., and
More informationModeling Uncertainty in Financial Markets
Modeling Uncertainty in Financial Markets Peter Ritchken 1 Modeling Uncertainty in Financial Markets In this module we review the basic stochastic model used to represent uncertainty in the equity markets.
More informationFinancial Engineering. Craig Pirrong Spring, 2006
Financial Engineering Craig Pirrong Spring, 2006 March 8, 2006 1 Levy Processes Geometric Brownian Motion is very tractible, and captures some salient features of speculative price dynamics, but it is
More informationMarket Risk: FROM VALUE AT RISK TO STRESS TESTING. Agenda. Agenda (Cont.) Traditional Measures of Market Risk
Market Risk: FROM VALUE AT RISK TO STRESS TESTING Agenda The Notional Amount Approach Price Sensitivity Measure for Derivatives Weakness of the Greek Measure Define Value at Risk 1 Day to VaR to 10 Day
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 informationM.I.T Fall Practice Problems
M.I.T. 15.450-Fall 2010 Sloan School of Management Professor Leonid Kogan Practice Problems 1. Consider a 3-period model with t = 0, 1, 2, 3. There are a stock and a risk-free asset. The initial stock
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 informationDiscounting a mean reverting cash flow
Discounting a mean reverting cash flow Marius Holtan Onward Inc. 6/26/2002 1 Introduction Cash flows such as those derived from the ongoing sales of particular products are often fluctuating in a random
More informationQuantitative Aggregate Effects of Asymmetric Information
Quantitative Aggregate Effects of Asymmetric Information Pablo Kurlat February 2012 In this note I roose a calibration of the model in Kurlat (forthcoming) to try to assess the otential magnitude of the
More informationLecture Note 8 of Bus 41202, Spring 2017: Stochastic Diffusion Equation & Option Pricing
Lecture Note 8 of Bus 41202, Spring 2017: Stochastic Diffusion Equation & Option Pricing We shall go over this note quickly due to time constraints. Key concept: Ito s lemma Stock Options: A contract giving
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 information3.1 Itô s Lemma for Continuous Stochastic Variables
Lecture 3 Log Normal Distribution 3.1 Itô s Lemma for Continuous Stochastic Variables Mathematical Finance is about pricing (or valuing) financial contracts, and in particular those contracts which depend
More information1 Geometric Brownian motion
Copyright c 05 by Karl Sigman Geometric Brownian motion Note that since BM can take on negative values, using it directly for modeling stock prices is questionable. There are other reasons too why BM is
More informationOrdering a deck of cards... Lecture 3: Binomial Distribution. Example. Permutations & Combinations
Ordering a dec of cards... Lecture 3: Binomial Distribution Sta 111 Colin Rundel May 16, 2014 If you have ever shuffled a dec of cards you have done something no one else has ever done before or will ever
More informationand their probabilities p
AP Statistics Ch. 6 Notes Random Variables A variable is any characteristic of an individual (remember that individuals are the objects described by a data set and may be eole, animals, or things). Variables
More informationImportance Sampling for Option Pricing. Steven R. Dunbar. Put Options. Monte Carlo Method. Importance. Sampling. Examples.
for for January 25, 2016 1 / 26 Outline for 1 2 3 4 2 / 26 Put Option for A put option is the right to sell an asset at an established price at a certain time. The established price is the strike price,
More informationLattice Model of System Evolution. Outline
Lattice Model of System Evolution Richard de Neufville Professor of Engineering Systems and of Civil and Environmental Engineering MIT Massachusetts Institute of Technology Lattice Model Slide 1 of 32
More informationStochastic modelling of skewed data exhibiting long range dependence
IUGG XXIV General Assembly 27 Perugia, Italy, 2 3 July 27 International Association of Hydrological Sciences, Session HW23 Analysis of Variability in Hydrological Data Series Stochastic modelling of skewed
More informationINDIAN INSTITUTE OF SCIENCE STOCHASTIC HYDROLOGY. Lecture -5 Course Instructor : Prof. P. P. MUJUMDAR Department of Civil Engg., IISc.
INDIAN INSTITUTE OF SCIENCE STOCHASTIC HYDROLOGY Lecture -5 Course Instructor : Prof. P. P. MUJUMDAR Department of Civil Engg., IISc. Summary of the previous lecture Moments of a distribubon Measures of
More informationBounds for Stop-Loss Premiums of Life Annuities with Random Interest Rates
Bounds for Stop-Loss Premiums of Life Annuities with Random Interest Rates Tom Hoedemakers (K.U.Leuven) Grzegorz Darkiewicz (K.U.Leuven) Griselda Deelstra (ULB) Jan Dhaene (K.U.Leuven) Michèle Vanmaele
More informationMFE/3F Questions Answer Key
MFE/3F Questions Download free full solutions from www.actuarialbrew.com, or purchase a hard copy from www.actexmadriver.com, or www.actuarialbookstore.com. Chapter 1 Put-Call Parity and Replication 1.01
More informationA NOTE ON SKEW-NORMAL DISTRIBUTION APPROXIMATION TO THE NEGATIVE BINOMAL DISTRIBUTION
A NOTE ON SKEW-NORMAL DISTRIBUTION APPROXIMATION TO THE NEGATIVE BINOMAL DISTRIBUTION JYH-JIUAN LIN 1, CHING-HUI CHANG * AND ROSEMARY JOU 1 Deartment of Statistics Tamkang University 151 Ying-Chuan Road,
More informationContinuous random variables
Continuous random variables probability density function (f(x)) the probability distribution function of a continuous random variable (analogous to the probability mass function for a discrete random variable),
More informationStatistics and Probability Letters. Variance stabilizing transformations of Poisson, binomial and negative binomial distributions
Statistics and Probability Letters 79 (9) 6 69 Contents lists available at ScienceDirect Statistics and Probability Letters journal homeage: www.elsevier.com/locate/staro Variance stabilizing transformations
More informationIEOR 3106: Introduction to OR: Stochastic Models. Fall 2013, Professor Whitt. Class Lecture Notes: Tuesday, September 10.
IEOR 3106: Introduction to OR: Stochastic Models Fall 2013, Professor Whitt Class Lecture Notes: Tuesday, September 10. The Central Limit Theorem and Stock Prices 1. The Central Limit Theorem (CLT See
More informationDescribing Uncertain Variables
Describing Uncertain Variables L7 Uncertainty in Variables Uncertainty in concepts and models Uncertainty in variables Lack of precision Lack of knowledge Variability in space/time Describing Uncertainty
More informationRisk Neutral Modelling Exercises
Risk Neutral Modelling Exercises Geneviève Gauthier Exercise.. Assume that the rice evolution of a given asset satis es dx t = t X t dt + X t dw t where t = ( + sin (t)) and W = fw t : t g is a (; F; P)
More informationMath489/889 Stochastic Processes and Advanced Mathematical Finance Solutions to Practice Problems
Math489/889 Stochastic Processes and Advanced Mathematical Finance Solutions to Practice Problems Steve Dunbar No Due Date: Practice Only. Find the mode (the value of the independent variable with the
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 informationSimulation Analysis of Option Buying
Mat-.108 Sovelletun Matematiikan erikoistyöt Simulation Analysis of Option Buying Max Mether 45748T 04.0.04 Table Of Contents 1 INTRODUCTION... 3 STOCK AND OPTION PRICING THEORY... 4.1 RANDOM WALKS AND
More informationVolumetric Hedging in Electricity Procurement
Volumetric Hedging in Electricity Procurement Yumi Oum Deartment of Industrial Engineering and Oerations Research, University of California, Berkeley, CA, 9472-777 Email: yumioum@berkeley.edu Shmuel Oren
More information4-2 Probability Distributions and Probability Density Functions. Figure 4-2 Probability determined from the area under f(x).
4-2 Probability Distributions and Probability Density Functions Figure 4-2 Probability determined from the area under f(x). 4-2 Probability Distributions and Probability Density Functions Definition 4-2
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 informationINDEX NUMBERS. Introduction
INDEX NUMBERS Introduction Index numbers are the indicators which reflect changes over a secified eriod of time in rices of different commodities industrial roduction (iii) sales (iv) imorts and exorts
More informationObjectives. 3.3 Toward statistical inference
Objectives 3.3 Toward statistical inference Poulation versus samle (CIS, Chater 6) Toward statistical inference Samling variability Further reading: htt://onlinestatbook.com/2/estimation/characteristics.html
More information***SECTION 7.1*** Discrete and Continuous Random Variables
***SECTION 7.*** Discrete and Continuous Random Variables Samle saces need not consist of numbers; tossing coins yields H s and T s. However, in statistics we are most often interested in numerical outcomes
More informationLecture Stat 302 Introduction to Probability - Slides 15
Lecture Stat 30 Introduction to Probability - Slides 15 AD March 010 AD () March 010 1 / 18 Continuous Random Variable Let X a (real-valued) continuous r.v.. It is characterized by its pdf f : R! [0, )
More informationSubject CT8 Financial Economics Core Technical Syllabus
Subject CT8 Financial Economics Core Technical Syllabus for the 2018 exams 1 June 2017 Aim The aim of the Financial Economics subject is to develop the necessary skills to construct asset liability models
More informationBUSM 411: Derivatives and Fixed Income
BUSM 411: Derivatives and Fixed Income 3. Uncertainty and Risk Uncertainty and risk lie at the core of everything we do in finance. In order to make intelligent investment and hedging decisions, we need
More informationECON 214 Elements of Statistics for Economists 2016/2017
ECON 214 Elements of Statistics for Economists 2016/2017 Topic The Normal Distribution Lecturer: Dr. Bernardin Senadza, Dept. of Economics bsenadza@ug.edu.gh College of Education School of Continuing and
More informationCS522 - Exotic and Path-Dependent Options
CS522 - Exotic and Path-Deendent Otions Tibor Jánosi May 5, 2005 0. Other Otion Tyes We have studied extensively Euroean and American uts and calls. The class of otions is much larger, however. A digital
More informationOPTIMAL PORTFOLIO CONTROL WITH TRADING STRATEGIES OF FINITE
Proceedings of the 44th IEEE Conference on Decision and Control, and the European Control Conference 005 Seville, Spain, December 1-15, 005 WeA11.6 OPTIMAL PORTFOLIO CONTROL WITH TRADING STRATEGIES OF
More informationAdvanced Topics in Derivative Pricing Models. Topic 4 - Variance products and volatility derivatives
Advanced Topics in Derivative Pricing Models Topic 4 - Variance products and volatility derivatives 4.1 Volatility trading and replication of variance swaps 4.2 Volatility swaps 4.3 Pricing of discrete
More informationCounterparty Credit Risk Simulation
Counterparty Credit Risk Simulation Alex Yang FinPricing http://www.finpricing.com Summary Counterparty Credit Risk Definition Counterparty Credit Risk Measures Monte Carlo Simulation Interest Rate Curve
More informationMixing Di usion and Jump Processes
Mixing Di usion and Jump Processes Mixing Di usion and Jump Processes 1/ 27 Introduction Using a mixture of jump and di usion processes can model asset prices that are subject to large, discontinuous changes,
More informationCalculation of Volatility in a Jump-Diffusion Model
Calculation of Volatility in a Jump-Diffusion Model Javier F. Navas 1 This Draft: October 7, 003 Forthcoming: The Journal of Derivatives JEL Classification: G13 Keywords: jump-diffusion process, option
More informationMonte Carlo Methods in Financial Engineering
Paul Glassennan Monte Carlo Methods in Financial Engineering With 99 Figures
More informationRisk and Return. Calculating Return - Single period. Calculating Return - Multi periods. Uncertainty of Investment.
Chater 10, 11 Risk and Return Chater 13 Cost of Caital Konan Chan, 018 Risk and Return Return measures Exected return and risk? Portfolio risk and diversification CPM (Caital sset Pricing Model) eta Calculating
More information( ) since this is the benefit of buying the asset at the strike price rather
Review of some financial models for MAT 483 Parity and Other Option Relationships The basic parity relationship for European options with the same strike price and the same time to expiration is: C( KT
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 informationChapter 3 Statistical Quality Control, 7th Edition by Douglas C. Montgomery. Copyright (c) 2013 John Wiley & Sons, Inc.
1 3.1 Describing Variation Stem-and-Leaf Display Easy to find percentiles of the data; see page 69 2 Plot of Data in Time Order Marginal plot produced by MINITAB Also called a run chart 3 Histograms Useful
More information**BEGINNING OF EXAMINATION** A random sample of five observations from a population is:
**BEGINNING OF EXAMINATION** 1. You are given: (i) A random sample of five observations from a population is: 0.2 0.7 0.9 1.1 1.3 (ii) You use the Kolmogorov-Smirnov test for testing the null hypothesis,
More informationFinance: A Quantitative Introduction Chapter 8 Option Pricing in Continuous Time
Finance: A Quantitative Introduction Chapter 8 Option Pricing in Continuous Time Nico van der Wijst 1 Finance: A Quantitative Introduction c Cambridge University Press 1 Modelling stock returns in continuous
More informationPolicyholder Outcome Death Disability Neither Payout, x 10,000 5, ,000
Two tyes of Random Variables: ) Discrete random variable has a finite number of distinct outcomes Examle: Number of books this term. ) Continuous random variable can take on any numerical value within
More informationOne note for Session Two
ESD.70J Engineering Economy Module Fall 2004 Session Three Link for PPT: http://web.mit.edu/tao/www/esd70/s3/p.ppt ESD.70J Engineering Economy Module - Session 3 1 One note for Session Two If you Excel
More informationFinancial Risk Management and Governance Other VaR methods. Prof. Hugues Pirotte
Financial Risk Management and Governance Other VaR methods Prof. ugues Pirotte Idea of historical simulations Why rely on statistics and hypothetical distribution?» Use the effective past distribution
More informationBROWNIAN MOTION Antonella Basso, Martina Nardon
BROWNIAN MOTION Antonella Basso, Martina Nardon basso@unive.it, mnardon@unive.it Department of Applied Mathematics University Ca Foscari Venice Brownian motion p. 1 Brownian motion Brownian motion plays
More informationChapter 7 1. Random Variables
Chapter 7 1 Random Variables random variable numerical variable whose value depends on the outcome of a chance experiment - discrete if its possible values are isolated points on a number line - continuous
More informationLecture 8: The Black-Scholes theory
Lecture 8: The Black-Scholes theory Dr. Roman V Belavkin MSO4112 Contents 1 Geometric Brownian motion 1 2 The Black-Scholes pricing 2 3 The Black-Scholes equation 3 References 5 1 Geometric Brownian motion
More informationInvestment Guarantees Chapter 7. Investment Guarantees Chapter 7: Option Pricing Theory. Key Exam Topics in This Lesson.
Investment Guarantees Chapter 7 Investment Guarantees Chapter 7: Option Pricing Theory Mary Hardy (2003) Video By: J. Eddie Smith, IV, FSA, MAAA Investment Guarantees Chapter 7 1 / 15 Key Exam Topics in
More informationChapter 7: Point Estimation and Sampling Distributions
Chapter 7: Point Estimation and Sampling Distributions Seungchul Baek Department of Statistics, University of South Carolina STAT 509: Statistics for Engineers 1 / 20 Motivation In chapter 3, we learned
More informationINVESTMENTS Class 2: Securities, Random Walk on Wall Street
15.433 INVESTMENTS Class 2: Securities, Random Walk on Wall Street Reto R. Gallati MIT Sloan School of Management Spring 2003 February 5th 2003 Outline Probability Theory A brief review of probability
More information5 Probability densities
ENGG450 robability and Statistics or Engineers Introduction 3 robability 4 robability distributions 5 robability Densities Organization and description o data 6 Sampling distributions 7 Inerences concerning
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 informationDrawdowns Preceding Rallies in the Brownian Motion Model
Drawdowns receding Rallies in the Brownian Motion Model Olympia Hadjiliadis rinceton University Department of Electrical Engineering. Jan Večeř Columbia University Department of Statistics. This version:
More information2 f. f t S 2. Delta measures the sensitivityof the portfolio value to changes in the price of the underlying
Sensitivity analysis Simulating the Greeks Meet the Greeks he value of a derivative on a single underlying asset depends upon the current asset price S and its volatility Σ, the risk-free interest rate
More informationSeminar 2 A Model of the Behavior of Stock Prices. Miloslav S. Vosvrda UTIA AV CR
Seminar A Model of the Behavior of Stock Prices Miloslav S. Vosvrda UTIA AV CR The Black-Scholes Analysis Ito s lemma The lognormal property of stock prices The distribution of the rate of return Estimating
More informationMarket Volatility and Risk Proxies
Market Volatility and Risk Proxies... an introduction to the concepts 019 Gary R. Evans. This slide set by Gary R. Evans is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International
More informationThe histogram should resemble the uniform density, the mean should be close to 0.5, and the standard deviation should be close to 1/ 12 =
Chapter 19 Monte Carlo Valuation Question 19.1 The histogram should resemble the uniform density, the mean should be close to.5, and the standard deviation should be close to 1/ 1 =.887. Question 19. The
More informationEcon 250 Fall Due at November 16. Assignment 2: Binomial Distribution, Continuous Random Variables and Sampling
Econ 250 Fall 2010 Due at November 16 Assignment 2: Binomial Distribution, Continuous Random Variables and Sampling 1. Suppose a firm wishes to raise funds and there are a large number of independent financial
More informationUniversity of California, Los Angeles Department of Statistics. Final exam 07 June 2013
University of California, Los Angeles Department of Statistics Statistics C183/C283 Instructor: Nicolas Christou Final exam 07 June 2013 Name: Problem 1 (20 points) a. Suppose the variable X follows the
More informationAspects of Financial Mathematics:
Aspects of Financial Mathematics: Options, Derivatives, Arbitrage, and the Black-Scholes Pricing Formula J. Robert Buchanan Millersville University of Pennsylvania email: Bob.Buchanan@millersville.edu
More informationChapter 4: Commonly Used Distributions. Statistics for Engineers and Scientists Fourth Edition William Navidi
Chapter 4: Commonly Used Distributions Statistics for Engineers and Scientists Fourth Edition William Navidi 2014 by Education. This is proprietary material solely for authorized instructor use. Not authorized
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 informationCVE SOME DISCRETE PROBABILITY DISTRIBUTIONS
CVE 472 2. SOME DISCRETE PROBABILITY DISTRIBUTIONS Assist. Prof. Dr. Bertuğ Akıntuğ Civil Engineering Program Middle East Technical University Northern Cyprus Campus CVE 472 Statistical Techniques in Hydrology.
More informationInformation and uncertainty in a queueing system
Information and uncertainty in a queueing system Refael Hassin December 7, 7 Abstract This aer deals with the effect of information and uncertainty on rofits in an unobservable single server queueing system.
More information1 The continuous time limit
Derivative Securities, Courant Institute, Fall 2008 http://www.math.nyu.edu/faculty/goodman/teaching/derivsec08/index.html Jonathan Goodman and Keith Lewis Supplementary notes and comments, Section 3 1
More informationManagement of Pricing Policies and Financial Risk as a Key Element for Short Term Scheduling Optimization
Ind. Eng. Chem. Res. 2005, 44, 557-575 557 Management of Pricing Policies and Financial Risk as a Key Element for Short Term Scheduling Otimization Gonzalo Guillén, Miguel Bagajewicz, Sebastián Eloy Sequeira,
More informationJDEP 384H: Numerical Methods in Business
Chapter 4: Numerical Integration: Deterministic and Monte Carlo Methods Chapter 8: Option Pricing by Monte Carlo Methods JDEP 384H: Numerical Methods in Business Instructor: Thomas Shores Department of
More informationUsing the Standard Ultimate Life Table, determine the following: 41,841.1 l 75,
Chater 8 1. For a multile state model where there are two states: i. State is a erson is alive ii. State 1 is a erson is dead Further you are given that a erson can transition from State to State 1 but
More informationPricing of Stochastic Interest Bonds using Affine Term Structure Models: A Comparative Analysis
Dottorato di Ricerca in Matematica er l Analisi dei Mercati Finanziari - Ciclo XXII - Pricing of Stochastic Interest Bonds using Affine Term Structure Models: A Comarative Analysis Dott.ssa Erica MASTALLI
More informationLattice (Binomial Trees) Version 1.2
Lattice (Binomial Trees) Version 1. 1 Introduction This plug-in implements different binomial trees approximations for pricing contingent claims and allows Fairmat to use some of the most popular binomial
More informationFinancial Risk Management
Extensions of the Black-Scholes model University of Oulu - Department of Finance Spring 018 Dividend payment Extensions of the Black-Scholes model ds = rsdt + σsdz ÊS = S 0 e r S 0 he risk-neutral price
More informationUnit 5: Sampling Distributions of Statistics
Unit 5: Sampling Distributions of Statistics Statistics 571: Statistical Methods Ramón V. León 6/12/2004 Unit 5 - Stat 571 - Ramon V. Leon 1 Definitions and Key Concepts A sample statistic used to estimate
More informationUnit 5: Sampling Distributions of Statistics
Unit 5: Sampling Distributions of Statistics Statistics 571: Statistical Methods Ramón V. León 6/12/2004 Unit 5 - Stat 571 - Ramon V. Leon 1 Definitions and Key Concepts A sample statistic used to estimate
More informationMarket Risk Analysis Volume I
Market Risk Analysis Volume I Quantitative Methods in Finance Carol Alexander John Wiley & Sons, Ltd List of Figures List of Tables List of Examples Foreword Preface to Volume I xiii xvi xvii xix xxiii
More informationQueens College, CUNY, Department of Computer Science Computational Finance CSCI 365 / 765 Fall 2017 Instructor: Dr. Sateesh Mane.
Queens College, CUNY, Department of Computer Science Computational Finance CSCI 365 / 765 Fall 217 Instructor: Dr. Sateesh Mane c Sateesh R. Mane 217 13 Lecture 13 November 15, 217 Derivation of the Black-Scholes-Merton
More information