BUSM 411: Derivatives and Fixed Income
|
|
- Dennis Hampton
- 5 years ago
- Views:
Transcription
1 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 ways to model the uncertainty inherent in financial markets (probability distributions), as well as ways to quantify how we feel about that risk (utility functions). For this we look to the tools of probability and statistics, treating outcomes like future stock and commodities prices or other economic conditions as random events. If we understood the universe well enough, we might be able to see everything as deterministic and predictable. To the extent that we do not fully understand the workings of the universe, or of financial markets in particular, we use the mathematical concept of probability distributions to describe the seemingly random outcomes of variables that concern us, such as securities prices, interest rates, and so forth. We will use these probability models for the prices of underlying assets in order to derive models and formulae for pricing derivatives Probability distributions Probability distributions are simply mathematical functions for describing random events. At its essence, a probability distribution is simply a list of the possible outcomes of some random variable, together with the probability of each outcome. Perhaps the simplest example is the probability distribution for a coin toss: Outcome Probability Heads 0.5 Tails 0.5 A more general representation of this is the binomial distribution: { 1 with probability p X = 0 with probability 1 p 1
2 Our bread-and-butter probability distribution in this course is the normal distribution % 95.54% 99.74% Figure 1: Standard normal distribution 3.2. A random walk down Wall Street In his 1900 dissertation titled The Theory of Speculation, Louis Bachelier searched for a formula to express the likelihood of a market price fluctuation. He ended up with a mathematical formula that describes what we know call Brownian motion. Einstein came up with the same formula five years later in a different context. In the finance world, Brownian motion came to be known as a random walk, the path a drunken man might follow at night in the light of a lamp post. Using the geometric Brownian motion to describe the random fluctuations in stock prices, Fisher Black, Myron Scholes, and Bob Merton worked out the Black-Scholes option pricing formula, which we ll cover in depth later in the course. 2
3 A simple random walk: S t+1 = µs t + ε where S t is the stock price at time t, µ is a drift term (to allow for an upward trend in prices over time), and ε N(0, 1). Problem: stock prices can t be negative! A multiplicative random walk: S t+1 = u t S t where u t is a random variable that is i.i.d. over time. Taking logarithms gives us: ln S t+1 = ln S t + ln u t. Now we we can assume that ln u t (the log-return!) is normally distributed, which makes successive prices lognormally distributed. The lognormal distribution cannot go below zero, as desired. If you take the limit of this process as the time interval goes toward zero, you get the geometric Brownian motion mentioned above. Implication: if stock prices follow a random walk, then they are essentially unpredictable (at least based on the sequence of past prices). The change in price from today to tomorrow is random and independent of past price changes. There are theoretical reasons why this should be the case in a well-functioning market, and it seems to be a pretty good description of actual stock prices Why the normal distribution? Model the random fluctuation of stock prices using geometric Brownian motion. This implies that (continuously compounded) stock returns as normally distributed. We can conveniently characterize stock returns as being normally distributed with a certain mean (µ) and standard deviation (σ). For annualized S&P 500 stock returns, µ is roughly 8%, while σ is roughly 15%. The latter is also called volatility. Pick a time horizon, say t. The stock return over t is normally distributed with mean µ t and standard deviation σ t. Given the annual statisitcs mentioned above, what is the distribution of daily returns (assuming 252 trading days per year)? 3.4. Events that are not normal A negative surprise: on October 19, 1987, the S&P 500 index dropped more than 23% in one day 3
4 A positive surprise: on January 3, 2001, the Nasdaq composite index gained more than 14% in one day Suppose we use a normal distribution to characterize stock returns. What are the probabilities of such surprises? What is the probability of a Black Monday sized crash? 3.5. What the normal distribution fails to capture... There are large movements (both up and down) in stock prices that cannot be captured at all by the normal distribution In mathematical terms, the tail distribution of a normal random variable is too thin. Historical stock returns exhibit fat tails. If we make financial decisions based on the normal distribution, we underestimate the probability of large movements. The consequences can be catestrophic! This is especially important for leveraged investments over a short time horizon Tail fatness can be an particularly important issue in risk management 3.6. Data analysis Preliminaries for data analysis: When given raw data, first look for trends. If there are any, the first step is always to de-trend the data. Why? This is why we typically work with stock returns rather than stock prices. Unlike prices, returns are reasonably stationary over time (i.e., stable mean and variance) When we estimate average stock returns or return variability, we are implicitly assuming that returns are independent and identically distributed. The longer we observe, the more we know about the probability distribution ldots but do not forget structural changes! (shifts in the return-generating process) Sample statistics: Mean: N ˆµ = 1 N i=1 r i 4
5 Variance (standard deviation is the square root of variance) ˆσ 2 = 1 N N (r i µ) 2 i=1 Skewness (the degree of asymmetry): ˆ skew = 1 N N i=1 (r i µ) 3 σ 3 Kurtosis (the degree to which the distribution has a skinny middle and fat tails : ˆ kurt = 1 N N i=1 (r i µ) 4 σ 2 Covariance (the degree to which two variables, say the returns of stock A and the returns of stock B, move together): ˆ Cov(r A, r B = 1 N N (r A µ A )(r B µ B ) i=1 Standard errors: Take the sample mean as an example: N ˆµ = 1 N i=1 r i We assume that the r i s are random draws from a stationary distribution This implies that the sample mean µ is itself a random variable: if we observed a different sample of returns drawn from the same distribution we would get a different estimate of the mean What is the mean of ˆµ? What is the standard deviation of ˆµ? We call this the standard error. In this case: s.e. = ˆσ N Standard error is a measure of the precision of the estimates 5
6 4. Risk Aversion and Pricing We ve just talked about how we model risk (variability and uncertainty in the future values of assets that are of interest to firms or investors), using probability distributions described by mean and variance. Now we need a way to represent the decision makers preferences. That is, how do the you feel about this risk? 4.1. Utility and Risk Aversion Utility is a concept developed by economists to measure the relative satisfaction from or desirability of the consumption of goods and services Given this measure, one may speak meaningfully of increasing or decreasing utility, and thereby explain economic behavior in terms of attempts to increase one s utility. We will be particularly concerned with utility of money or wealth. Based on some reasonable assumptions about human preferences, derived from observation and introspection: We prefer more wealth to less wealth (non-satiability) We get less satisfaction or utility out of an additional dollar if we are already relatively wealthy, vs. if we are relatively poor (diminishing marginal utility) We prefer a fixed sum of money to a gamble with the same expected payoff (risk aversion) It turns out that we can capture all of these features by thinking of utility as a concave, increasing function of wealth Risk aversion and asset pricing The prices (or equivalently, the expected returns) of the basic financial securities like stocks and bonds are determined by investors risk aversion. This is easily seen in the Capital Asset Pricing Model (CAPM) A quick review of the CAPM For a given set of risky assets, there is a unique portfolio on the efficient frontier that has the highest possible Sharpe ratio. If everyone observes the same set of risky assets and has the same expectations about the returns on those assets (that is, everyone agrees on the expected returns and variances of the risky assets), then everyone chooses the same tangency portfolio. 6
7 If everyone chooses the same risky portfolio, then that risky portfolio must be the market portfolio! In equilibrium, prices adjust until supply equals demand. In this case, the supply is the number of shares of each stock or security existing in the market. The demand is the amount of each risky security that investors want to hold in their portfolio. Furthermore, because the degree of risk aversion varies across investors, some will want to lend (hold some of their wealth in the risk-free asset), while some will want to borrow at the risk-free rate to invest more in the tangency portfolio. The risk-free rate will adjust until the amount of lending and the amount of borrowing cancel out. Thus, the value of the aggregate risky portfolio will equal the entire wealth of the economy! The market price of risk Suppose there are N investors in the economy, and each investor i has utility U i = E[r] 1 2 A iσ 2 where U i is utility (of the ith investor), E[r] is the expected return (of the investor s portfolio), σ 2 is the return variance, and A i is a parameter that captures the investor s degree of risk aversion. For concreteness, suppose that each has $1 to invest How much will each investor put in the market portfolio? y i = E[r M] r f A i σ 2 M If we add up the amount invested in the market portfolio by all investors, we get: $1 E[r M] r f σ 2 M ( ) A 1 A 2 A N In equilibrium, the total wealth invested in the stock market must be $1 N 7
8 This implies where E[r M ] r f = Āσ2 M Ā is the average risk aversion of investors in the market (to be precise, it s actually the inverse of investors average risk tolerance): Ā 1 N ( ) A 1 A 2 A N The main idea In market equilibrium, investors are only rewarded for bearing systematic risk the type of risk that cannot be diversified away. They should not be rewarded for bearing idiosyncratic risk, since this uncertainty can be mitigated through appropriate diversification. William Sharpe (one of the originators of the CAPM and namesake of the Sharpe ratio), in an interview with the Dow Jones Asset Manager: But the fundamental idea remains that there s no reason to expect reward just for bearing risk. Otherwise, you d make a lot of money in Las Vegas. If there s reward for risk, it s got to be special. There s got to be some economics behind it or else the world is a very crazy place. - Sharpe (1998) Risk aversion and derivatives pricing Like the games in Las Vegas, derivatives contracts are essentially bets bets that the value of some underlying asset will be high or low on a given date in the future. As such, they are not special, to use Bill Sharpe s words. Their riskiness is easily eliminated by taking an offsetting position, and is not systematic. Therefore, derivatives prices are NOT driven by risk aversion. That is, derivatives prices do not incorporate any premium for bearing risk. Instead, derivatives prices are determined by the principle of no arbitrage. Because the payoff of a derivative is contingent on the value of some underlying asset, we can replicate the payoff of the derivative using a portfolio of the underlying asset itself together with the risk-free security. By no arbitrage arguments, the price of the derivative should be equal to the cost of the replicating portfolio. 8
9 We will spend the remainder of the course focusing on developing models or formulae for derivatives prices based on this idea of arbitrage-free pricing. 9
INVESTMENTS 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 informationMathematics of Finance Final Preparation December 19. To be thoroughly prepared for the final exam, you should
Mathematics of Finance Final Preparation December 19 To be thoroughly prepared for the final exam, you should 1. know how to do the homework problems. 2. be able to provide (correct and complete!) definitions
More information1.1 Interest rates Time value of money
Lecture 1 Pre- Derivatives Basics Stocks and bonds are referred to as underlying basic assets in financial markets. Nowadays, more and more derivatives are constructed and traded whose payoffs depend on
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 informationRisk and Return: Past and Prologue
Chapter 5 Risk and Return: Past and Prologue Bodie, Kane, and Marcus Essentials of Investments Tenth Edition 5.1 Rates of Return Holding-Period Return (HPR) Rate of return over given investment period
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 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 informationHow Much Should You Pay For a Financial Derivative?
City University of New York (CUNY) CUNY Academic Works Publications and Research New York City College of Technology Winter 2-26-2016 How Much Should You Pay For a Financial Derivative? Boyan Kostadinov
More informationRisk and Return and Portfolio Theory
Risk and Return and Portfolio Theory Intro: Last week we learned how to calculate cash flows, now we want to learn how to discount these cash flows. This will take the next several weeks. We know discount
More informationRisk and Return: Past and Prologue
Chapter 5 Risk and Return: Past and Prologue Bodie, Kane, and Marcus Essentials of Investments Tenth Edition What is in Chapter 5 5.1 Rates of Return HPR, arithmetic, geometric, dollar-weighted, APR, EAR
More informationMath 5760/6890 Introduction to Mathematical Finance
Math 5760/6890 Introduction to Mathematical Finance Instructor: Jingyi Zhu Office: LCB 335 Telephone:581-3236 E-mail: zhu@math.utah.edu Class web page: www.math.utah.edu/~zhu/5760_12f.html What you should
More informationPeriodic Returns, and Their Arithmetic Mean, Offer More Than Researchers Expect
Periodic Returns, and Their Arithmetic Mean, Offer More Than Researchers Expect Entia non sunt multiplicanda praeter necessitatem, Things should not be multiplied without good reason. Occam s Razor Carl
More informationCHAPTER 5. Introduction to Risk, Return, and the Historical Record INVESTMENTS BODIE, KANE, MARCUS. McGraw-Hill/Irwin
CHAPTER 5 Introduction to Risk, Return, and the Historical Record McGraw-Hill/Irwin Copyright 2011 by The McGraw-Hill Companies, Inc. All rights reserved. 5-2 Interest Rate Determinants Supply Households
More informationThe mean-variance portfolio choice framework and its generalizations
The mean-variance portfolio choice framework and its generalizations Prof. Massimo Guidolin 20135 Theory of Finance, Part I (Sept. October) Fall 2014 Outline and objectives The backward, three-step solution
More informationThe Black-Scholes Model
IEOR E4706: Foundations of Financial Engineering c 2016 by Martin Haugh The Black-Scholes Model In these notes we will use Itô s Lemma and a replicating argument to derive the famous Black-Scholes formula
More informationEconomics 430 Handout on Rational Expectations: Part I. Review of Statistics: Notation and Definitions
Economics 430 Chris Georges Handout on Rational Expectations: Part I Review of Statistics: Notation and Definitions Consider two random variables X and Y defined over m distinct possible events. Event
More informationECO 317 Economics of Uncertainty Fall Term 2009 Tuesday October 6 Portfolio Allocation Mean-Variance Approach
ECO 317 Economics of Uncertainty Fall Term 2009 Tuesday October 6 ortfolio Allocation Mean-Variance Approach Validity of the Mean-Variance Approach Constant absolute risk aversion (CARA): u(w ) = exp(
More informationValuing Investments A Statistical Perspective. Bob Stine Department of Statistics Wharton, University of Pennsylvania
Valuing Investments A Statistical Perspective Bob Stine, University of Pennsylvania Overview Principles Focus on returns, not cumulative value Remove market performance (CAPM) Watch for unseen volatility
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 informationStatistics and Finance
David Ruppert Statistics and Finance An Introduction Springer Notation... xxi 1 Introduction... 1 1.1 References... 5 2 Probability and Statistical Models... 7 2.1 Introduction... 7 2.2 Axioms of Probability...
More informationThe Merton Model. A Structural Approach to Default Prediction. Agenda. Idea. Merton Model. The iterative approach. Example: Enron
The Merton Model A Structural Approach to Default Prediction Agenda Idea Merton Model The iterative approach Example: Enron A solution using equity values and equity volatility Example: Enron 2 1 Idea
More informationOverview of Concepts and Notation
Overview of Concepts and Notation (BUSFIN 4221: Investments) - Fall 2016 1 Main Concepts This section provides a list of questions you should be able to answer. The main concepts you need to know are embedded
More informationLecture 1 Definitions from finance
Lecture 1 s from finance Financial market instruments can be divided into two types. There are the underlying stocks shares, bonds, commodities, foreign currencies; and their derivatives, claims that promise
More informationBasic Concepts in Mathematical Finance
Chapter 1 Basic Concepts in Mathematical Finance In this chapter, we give an overview of basic concepts in mathematical finance theory, and then explain those concepts in very simple cases, namely in the
More informationFinancial Time Series and Their Characteristics
Financial Time Series and Their Characteristics Egon Zakrajšek Division of Monetary Affairs Federal Reserve Board Summer School in Financial Mathematics Faculty of Mathematics & Physics University of Ljubljana
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 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 informationModeling via Stochastic Processes in Finance
Modeling via Stochastic Processes in Finance Dimbinirina Ramarimbahoaka Department of Mathematics and Statistics University of Calgary AMAT 621 - Fall 2012 October 15, 2012 Question: What are appropriate
More informationPrinciples of Finance Risk and Return. Instructor: Xiaomeng Lu
Principles of Finance Risk and Return Instructor: Xiaomeng Lu 1 Course Outline Course Introduction Time Value of Money DCF Valuation Security Analysis: Bond, Stock Capital Budgeting (Fundamentals) Portfolio
More informationLECTURE 3. Market Efficiency & Investment Valuation - EMH and Behavioral Analysis. The Quants Book Eugene Fama and Cliff Asnes
Baruch College Executive MS in Financial Statement Analysis CHAPTER 6 (PARTIAL) LECTURE 3 Market Efficiency & Investment Valuation - EMH and Behavioral Analysis Professor s Notes Are markets efficient?????
More informationRisk Neutral Valuation, the Black-
Risk Neutral Valuation, the Black- Scholes Model and Monte Carlo Stephen M Schaefer London Business School Credit Risk Elective Summer 01 C = SN( d )-PV( X ) N( ) N he Black-Scholes formula 1 d (.) : cumulative
More informationLecture 5 Theory of Finance 1
Lecture 5 Theory of Finance 1 Simon Hubbert s.hubbert@bbk.ac.uk January 24, 2007 1 Introduction In the previous lecture we derived the famous Capital Asset Pricing Model (CAPM) for expected asset returns,
More informationHANDBOOK OF. Market Risk CHRISTIAN SZYLAR WILEY
HANDBOOK OF Market Risk CHRISTIAN SZYLAR WILEY Contents FOREWORD ACKNOWLEDGMENTS ABOUT THE AUTHOR INTRODUCTION XV XVII XIX XXI 1 INTRODUCTION TO FINANCIAL MARKETS t 1.1 The Money Market 4 1.2 The Capital
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 informationFinancial Mathematics III Theory summary
Financial Mathematics III Theory summary Table of Contents Lecture 1... 7 1. State the objective of modern portfolio theory... 7 2. Define the return of an asset... 7 3. How is expected return defined?...
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 information1) Understanding Equity Options 2) Setting up Brokerage Systems
1) Understanding Equity Options 2) Setting up Brokerage Systems M. Aras Orhan, 12.10.2013 FE 500 Intro to Financial Engineering 12.10.2013, ARAS ORHAN, Intro to Fin Eng, Boğaziçi University 1 Today s agenda
More informationBARUCH COLLEGE DEPARTMENT OF ECONOMICS & FINANCE Professor Chris Droussiotis LECTURE 6. Modern Portfolio Theory (MPT): The Keynesian Animal Spirits
LECTURE 6 Modern Portfolio Theory (MPT): CHALLENGED BY BEHAVIORAL ECONOMICS Efficient Frontier is the intersection of the Set of Portfolios with Minimum Variance (MVS) and set of portfolios with Maximum
More informationAsset Allocation. Cash Flow Matching and Immunization CF matching involves bonds to match future liabilities Immunization involves duration matching
Asset Allocation Strategic Asset Allocation Combines investor s objectives, risk tolerance and constraints with long run capital market expectations to establish asset allocations Create the policy portfolio
More informationFatness of Tails in Risk Models
Fatness of Tails in Risk Models By David Ingram ALMOST EVERY BUSINESS DECISION MAKER IS FAMILIAR WITH THE MEANING OF AVERAGE AND STANDARD DEVIATION WHEN APPLIED TO BUSINESS STATISTICS. These commonly used
More informationCHAPTER 5. Introduction to Risk, Return, and the Historical Record INVESTMENTS BODIE, KANE, MARCUS
CHAPTER 5 Introduction to Risk, Return, and the Historical Record INVESTMENTS BODIE, KANE, MARCUS McGraw-Hill/Irwin Copyright 2011 by The McGraw-Hill Companies, Inc. All rights reserved. 5-2 Supply Interest
More informationChapter Ten. The Efficient Market Hypothesis
Chapter Ten The Efficient Market Hypothesis Slide 10 3 Topics Covered We Always Come Back to NPV What is an Efficient Market? Random Walk Efficient Market Theory The Evidence on Market Efficiency Puzzles
More informationFNCE 4030 Fall 2012 Roberto Caccia, Ph.D. Midterm_2a (2-Nov-2012) Your name:
Answer the questions in the space below. Written answers require no more than few compact sentences to show you understood and master the concept. Show your work to receive partial credit. Points are as
More informationMidterm Review. P resent value = P V =
JEM034 Corporate Finance Winter Semester 2017/2018 Instructor: Olga Bychkova Midterm Review F uture value of $100 = $100 (1 + r) t Suppose that you will receive a cash flow of C t dollars at the end of
More informationEcon 422 Eric Zivot Fall 2005 Final Exam
Econ 422 Eric Zivot Fall 2005 Final Exam This is a closed book exam. However, you are allowed one page of notes (double-sided). Answer all questions. For the numerical problems, if you make a computational
More informationRandomness and Fractals
Randomness and Fractals Why do so many physicists become traders? Gregory F. Lawler Department of Mathematics Department of Statistics University of Chicago September 25, 2011 1 / 24 Mathematics and the
More informationLecture 10-12: CAPM.
Lecture 10-12: CAPM. I. Reading II. Market Portfolio. III. CAPM World: Assumptions. IV. Portfolio Choice in a CAPM World. V. Minimum Variance Mathematics. VI. Individual Assets in a CAPM World. VII. Intuition
More informationSome history. The random walk model. Lecture notes on forecasting Robert Nau Fuqua School of Business Duke University
Lecture notes on forecasting Robert Nau Fuqua School of Business Duke University http://people.duke.edu/~rnau/forecasting.htm The random walk model Some history Brownian motion is a random walk in continuous
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 informationMean Variance Analysis and CAPM
Mean Variance Analysis and CAPM Yan Zeng Version 1.0.2, last revised on 2012-05-30. Abstract A summary of mean variance analysis in portfolio management and capital asset pricing model. 1. Mean-Variance
More informationSample Midterm Questions Foundations of Financial Markets Prof. Lasse H. Pedersen
Sample Midterm Questions Foundations of Financial Markets Prof. Lasse H. Pedersen 1. Security A has a higher equilibrium price volatility than security B. Assuming all else is equal, the equilibrium bid-ask
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 informationFinance Concepts I: Present Discounted Value, Risk/Return Tradeoff
Finance Concepts I: Present Discounted Value, Risk/Return Tradeoff Federal Reserve Bank of New York Central Banking Seminar Preparatory Workshop in Financial Markets, Instruments and Institutions Anthony
More informationThe Black-Scholes PDE from Scratch
The Black-Scholes PDE from Scratch chris bemis November 27, 2006 0-0 Goal: Derive the Black-Scholes PDE To do this, we will need to: Come up with some dynamics for the stock returns Discuss Brownian motion
More informationQR43, Introduction to Investments Class Notes, Fall 2003 IV. Portfolio Choice
QR43, Introduction to Investments Class Notes, Fall 2003 IV. Portfolio Choice A. Mean-Variance Analysis 1. Thevarianceofaportfolio. Consider the choice between two risky assets with returns R 1 and R 2.
More informationOPTIONS. Options: Definitions. Definitions (Cont) Price of Call at Maturity and Payoff. Payoff from Holding Stock and Riskfree Bond
OPTIONS Professor Anant K. Sundaram THUNERBIR Spring 2003 Options: efinitions Contingent claim; derivative Right, not obligation when bought (but, not when sold) More general than might first appear Calls,
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 informationMeasuring Risk. Expected value and expected return 9/4/2018. Possibilities, Probabilities and Expected Value
Chapter Five Understanding Risk Introduction Risk cannot be avoided. Everyday decisions involve financial and economic risk. How much car insurance should I buy? Should I refinance my mortgage now or later?
More informationECMC49F Midterm. Instructor: Travis NG Date: Oct 26, 2005 Duration: 1 hour 50 mins Total Marks: 100. [1] [25 marks] Decision-making under certainty
ECMC49F Midterm Instructor: Travis NG Date: Oct 26, 2005 Duration: 1 hour 50 mins Total Marks: 100 [1] [25 marks] Decision-making under certainty (a) [5 marks] Graphically demonstrate the Fisher Separation
More informationINTRODUCTION TO THE ECONOMICS AND MATHEMATICS OF FINANCIAL MARKETS. Jakša Cvitanić and Fernando Zapatero
INTRODUCTION TO THE ECONOMICS AND MATHEMATICS OF FINANCIAL MARKETS Jakša Cvitanić and Fernando Zapatero INTRODUCTION TO THE ECONOMICS AND MATHEMATICS OF FINANCIAL MARKETS Table of Contents PREFACE...1
More informationTwo Hours. Mathematical formula books and statistical tables are to be provided THE UNIVERSITY OF MANCHESTER. 22 January :00 16:00
Two Hours MATH38191 Mathematical formula books and statistical tables are to be provided THE UNIVERSITY OF MANCHESTER STATISTICAL MODELLING IN FINANCE 22 January 2015 14:00 16:00 Answer ALL TWO questions
More informationLecture 2: Stochastic Discount Factor
Lecture 2: Stochastic Discount Factor Simon Gilchrist Boston Univerity and NBER EC 745 Fall, 2013 Stochastic Discount Factor (SDF) A stochastic discount factor is a stochastic process {M t,t+s } such that
More informationRisk and Return (Introduction) Professor: Burcu Esmer
Risk and Return (Introduction) Professor: Burcu Esmer 1 Overview Rates of Return: A Review A Century of Capital Market History Measuring Risk Risk & Diversification Thinking About Risk Measuring Market
More informationMBF2263 Portfolio Management. Lecture 8: Risk and Return in Capital Markets
MBF2263 Portfolio Management Lecture 8: Risk and Return in Capital Markets 1. A First Look at Risk and Return We begin our look at risk and return by illustrating how the risk premium affects investor
More informationMonetary Economics Risk and Return, Part 2. Gerald P. Dwyer Fall 2015
Monetary Economics Risk and Return, Part 2 Gerald P. Dwyer Fall 2015 Reading Malkiel, Part 2, Part 3 Malkiel, Part 3 Outline Returns and risk Overall market risk reduced over longer periods Individual
More informationMaking Hard Decision. ENCE 627 Decision Analysis for Engineering. Identify the decision situation and understand objectives. Identify alternatives
CHAPTER Duxbury Thomson Learning Making Hard Decision Third Edition RISK ATTITUDES A. J. Clark School of Engineering Department of Civil and Environmental Engineering 13 FALL 2003 By Dr. Ibrahim. Assakkaf
More informationImportant Concepts LECTURE 3.2: OPTION PRICING MODELS: THE BLACK-SCHOLES-MERTON MODEL. Applications of Logarithms and Exponentials in Finance
Important Concepts The Black Scholes Merton (BSM) option pricing model LECTURE 3.2: OPTION PRICING MODELS: THE BLACK-SCHOLES-MERTON MODEL Black Scholes Merton Model as the Limit of the Binomial Model Origins
More informationChapter 15 Trade-offs Involving Time and Risk. Outline. Modeling Time and Risk. The Time Value of Money. Time Preferences. Probability and Risk
Involving Modeling The Value Part VII: Equilibrium in the Macroeconomy 23. Employment and Unemployment 15. Involving Web 1. Financial Decision Making 24. Credit Markets 25. The Monetary System 1 / 36 Involving
More informationPortfolio Management
MCF 17 Advanced Courses Portfolio Management Final Exam Time Allowed: 60 minutes Family Name (Surname) First Name Student Number (Matr.) Please answer all questions by choosing the most appropriate alternative
More informationModels and Decision with Financial Applications UNIT 1: Elements of Decision under Uncertainty
Models and Decision with Financial Applications UNIT 1: Elements of Decision under Uncertainty We always need to make a decision (or select from among actions, options or moves) even when there exists
More informationLecture Notes: Basic Concepts in Option Pricing - The Black and Scholes Model (Continued)
Brunel University Msc., EC5504, Financial Engineering Prof Menelaos Karanasos Lecture Notes: Basic Concepts in Option Pricing - The Black and Scholes Model (Continued) In previous lectures we saw that
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 informationInternational Finance. Investment Styles. Campbell R. Harvey. Duke University, NBER and Investment Strategy Advisor, Man Group, plc.
International Finance Investment Styles Campbell R. Harvey Duke University, NBER and Investment Strategy Advisor, Man Group, plc February 12, 2017 2 1. Passive Follow the advice of the CAPM Most influential
More informationMathematics in Finance
Mathematics in Finance Steven E. Shreve Department of Mathematical Sciences Carnegie Mellon University Pittsburgh, PA 15213 USA shreve@andrew.cmu.edu A Talk in the Series Probability in Science and Industry
More informationChapter 13 Return, Risk, and Security Market Line
1 Chapter 13 Return, Risk, and Security Market Line Konan Chan Financial Management, Spring 2018 Topics Covered Expected Return and Variance Portfolio Risk and Return Risk & Diversification Systematic
More informationCHAPTER 5. Introduction to Risk, Return, and the Historical Record INVESTMENTS BODIE, KANE, MARCUS. McGraw-Hill/Irwin
CHAPTER 5 Introduction to Risk, Return, and the Historical Record McGraw-Hill/Irwin Copyright 2011 by The McGraw-Hill Companies, Inc. All rights reserved. 5-2 Interest Rate Determinants Supply Households
More informationModels of Asset Pricing
appendix1 to chapter 5 Models of Asset Pricing In Chapter 4, we saw that the return on an asset (such as a bond) measures how much we gain from holding that asset. When we make a decision to buy an asset,
More information29 Week 10. Portfolio theory Overheads
29 Week 1. Portfolio theory Overheads 1. Outline (a) Mean-variance (b) Multifactor portfolios (value etc.) (c) Outside income, labor income. (d) Taking advantage of predictability. (e) Options (f) Doubts
More informationOne-Period Valuation Theory
One-Period Valuation Theory Part 2: Chris Telmer March, 2013 1 / 44 1. Pricing kernel and financial risk 2. Linking state prices to portfolio choice Euler equation 3. Application: Corporate financial leverage
More informationPortfolio Performance Measurement
Portfolio Performance Measurement Eric Zivot December 8, 2009 1 Investment Styles 1.1 Passive Management Believe that markets are in equilibrium Assets are correctly priced Hold securities for relatively
More informationUniversity 18 Lessons Financial Management. Unit 12: Return, Risk and Shareholder Value
University 18 Lessons Financial Management Unit 12: Return, Risk and Shareholder Value Risk and Return Risk and Return Security analysis is built around the idea that investors are concerned with two principal
More informationPart 1 In which we meet the law of averages. The Law of Averages. The Expected Value & The Standard Error. Where Are We Going?
1 The Law of Averages The Expected Value & The Standard Error Where Are We Going? Sums of random numbers The law of averages Box models for generating random numbers Sums of draws: the Expected Value Standard
More informationAssicurazioni Generali: An Option Pricing Case with NAGARCH
Assicurazioni Generali: An Option Pricing Case with NAGARCH Assicurazioni Generali: Business Snapshot Find our latest analyses and trade ideas on bsic.it Assicurazioni Generali SpA is an Italy-based insurance
More informationFinance 100: Corporate Finance. Professor Michael R. Roberts Quiz 3 November 8, 2006
Finance 100: Corporate Finance Professor Michael R. Roberts Quiz 3 November 8, 006 Name: Solutions Section ( Points...no joke!): Question Maximum Student Score 1 30 5 3 5 4 0 Total 100 Instructions: Please
More informationWhy Indexing Works. October Abstract
Why Indexing Works J. B. Heaton N. G. Polson J. H. Witte October 2015 arxiv:1510.03550v1 [q-fin.pm] 13 Oct 2015 Abstract We develop a simple stock selection model to explain why active equity managers
More informationP s =(0,W 0 R) safe; P r =(W 0 σ,w 0 µ) risky; Beyond P r possible if leveraged borrowing OK Objective function Mean a (Std.Dev.
ECO 305 FALL 2003 December 2 ORTFOLIO CHOICE One Riskless, One Risky Asset Safe asset: gross return rate R (1 plus interest rate) Risky asset: random gross return rate r Mean µ = E[r] >R,Varianceσ 2 =
More informationCredit Risk Modelling: A Primer. By: A V Vedpuriswar
Credit Risk Modelling: A Primer By: A V Vedpuriswar September 8, 2017 Market Risk vs Credit Risk Modelling Compared to market risk modeling, credit risk modeling is relatively new. Credit risk is more
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 informationMA 1125 Lecture 14 - Expected Values. Wednesday, October 4, Objectives: Introduce expected values.
MA 5 Lecture 4 - Expected Values Wednesday, October 4, 27 Objectives: Introduce expected values.. Means, Variances, and Standard Deviations of Probability Distributions Two classes ago, we computed the
More informationProblem set 1 Answers: 0 ( )= [ 0 ( +1 )] = [ ( +1 )]
Problem set 1 Answers: 1. (a) The first order conditions are with 1+ 1so 0 ( ) [ 0 ( +1 )] [( +1 )] ( +1 ) Consumption follows a random walk. This is approximately true in many nonlinear models. Now we
More informationEnergy Price Processes
Energy Processes Used for Derivatives Pricing & Risk Management In this first of three articles, we will describe the most commonly used process, Geometric Brownian Motion, and in the second and third
More informationA Scholar s Introduction to Stocks, Bonds and Derivatives
A Scholar s Introduction to Stocks, Bonds and Derivatives Martin V. Day June 8, 2004 1 Introduction This course concerns mathematical models of some basic financial assets: stocks, bonds and derivative
More informationOptions Markets: Introduction
17-2 Options Options Markets: Introduction Derivatives are securities that get their value from the price of other securities. Derivatives are contingent claims because their payoffs depend on the value
More 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 to Real Options
IEOR E4706: Foundations of Financial Engineering c 2016 by Martin Haugh Introduction to Real Options We introduce real options and discuss some of the issues and solution methods that arise when tackling
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 informationWashington University Fall Economics 487
Washington University Fall 2009 Department of Economics James Morley Economics 487 Project Proposal due Tuesday 11/10 Final Project due Wednesday 12/9 (by 5:00pm) (20% penalty per day if the project is
More informationThe misleading nature of correlations
The misleading nature of correlations In this note we explain certain subtle features of calculating correlations between time-series. Correlation is a measure of linear co-movement, to be contrasted with
More informationChapter 10. Chapter 10 Topics. What is Risk? The big picture. Introduction to Risk, Return, and the Opportunity Cost of Capital
1 Chapter 10 Introduction to Risk, Return, and the Opportunity Cost of Capital Chapter 10 Topics Risk: The Big Picture Rates of Return Risk Premiums Expected Return Stand Alone Risk Portfolio Return and
More informationDr. 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 information