# Discussions of Monte Carlo Simulation in Option Pricing TIANYI SHI, Y LAURENT LIU PROF. RENATO FERES MATH 350 RESEARCH PAPER

Save this PDF as:

Size: px
Start display at page:

Download "Discussions of Monte Carlo Simulation in Option Pricing TIANYI SHI, Y LAURENT LIU PROF. RENATO FERES MATH 350 RESEARCH PAPER"

## Transcription

1 Discussions of Monte Carlo Simulation in Option Pricing TIANYI SHI, Y LAURENT LIU PROF. RENATO FERES MATH 350 RESEARCH PAPER

2 INTRODUCTION Having been exposed to a variety of applications of Monte Carlo methods, including entertainment, engineering and sports, we attempt to spin our interests in financial markets with a mathematical twist. In the realm of finance, the Black-Scholes model is widely applied in calculating prices of European options, which cannot be exercised before the expiration date. This model, first published in 1973 in the paper "The Pricing of Options and Corporate Liabilities", suggests a partial differential equation that governs the price of the option over time. The key idea behind the derivation was to use delta hedging the option through buying and selling the underlying asset in a logical way to mitigate risk. However, this widely acclaimed formula, which helped Scholes to win Nobel Prize in Economics, loses its magic when dealing with American options. American options differ from European style options because the right of exercise for American options is not limited to any specific point of time before maturity. Investors are baffled by the choice of exercising time of American options because there is always uncertainty over whether the underlying assets will go higher or lower in the future. Such uncertainty creates significant complexity in evaluating the value of American options. Monte Carlo simulation is a great method to value American style options because regardless of the future price of an individual option, we should be able to derive the expected return of exercising this American option early, as long as we assume that the underlying assets price will follow a log-normal distribution. According to our research, we find out that in 23.95% of cases, it is optimal for investors to exercise American call options early, contrary to the classical financial theory, which states that one should never exercise the call option before the maturity. Keywords: American options, Monte Carlo simulation, options pricing, early exercise! "!

3 BACKGROUND Option pricing is essential in corporate finance decision making since many corporate liabilities can be expressed in terms of options or combinations of options and such variations help to leverage different methods. A premium is paid or received for purchasing or selling options. This price can be split into two components. Strike price is fixed in the contract and specifies the price at which a specific derivative contract can be exercised. The intrinsic value is the difference between the underlying price and the strike price, to the extent that this is in favor of the option holder. The major difference between American and European options lies in the timing of when the options can be exercised:! American options have a more flexible exercise rule and can be exercised at any time before the expiration date; they usually trade on standardized exchanges.! European options can be exercised only at the end of their lives, which is at its maturity or the expiration date of the option, i.e. at a single pre-defined point in time; it normally trades over the counter. For both, the payoff when it occurs is via: Call option: Max [ (S! K), 0 ] Put option: Max [ (K! S), 0 ], where K is the strike price and S is the spot price of the underlying asset. A large number of exchange-traded stock options, such as those issued for companies like Apple and Wal-Mart, are American-style options. However, financial index options can be issued as either American- or European-style options. For example, according to data from investopedia.com, S&P 100 Index options are traded as American-style options, whereas Nasdaq 100 Index options are traded as European-style options.! #!

4 LITERATURE REVIEW Upon reading a number of research papers, the Monte Carlo approach has proved to be a valuable and flexible computational tool in security pricing. Moreover, due to various reduction methods and models, there are also different ways to prove the efficiency of the Monte Carlo model. Previous literature explores a number of models of option pricing under conditions of general equilibrium as well as exploring the hedging position of logarithmic distributions of different options. Fu (2011) explains different ways to value a put and call option: for a call option, the option is in-the-money if the underlying price is higher than the strike price; then the intrinsic value is the underlying price minus the strike price. For a put option, the option is inthe-money if the strike price is higher than the underlying price; then the intrinsic value is the strike price minus the underlying price. Otherwise the value of the option is zero. Boyle (1997) suggests that the Monte Carlo method simulates the process of generating the returns on the underlying asset and invokes the risk neutrality assumption to derive the value of the option. Fu (2011) also explains several primary methods for pricing American- style options, such as binomial trees and other lattice methods (i.e. trinomial trees), and finite difference methods to solve the associated boundary value partial differential equations (PDEs). However, some of these computational methods can only handle one or two sources of uncertainty, which creates some limitation when it comes to evaluating multiple sources of uncertainty.! \$!

5 DATA COLLECTION Different securities usually do not share the same characteristics, which further complicates the valuation method. To make the scope of the research feasible within the time \$&! #'!!"#\$()*+\$,-.,/.-0,,\$1*\$,-.,/.-0,-\$!"#\$%&\$'\$ #&! "'! "&! '! ()*!+,-./! (0/123/! &! frame that we have in class, we will have to narrow our analysis down to a specific type of underlying assets. After extensive research and comparisons, we chose the Standard & Poor 500 index (Ticker symbol: ^GSPC) 1 as our sample to approximate the equity price. S&P 500 is an index of 500 stocks chosen for market size, liquidity and industry sector. It is an appropriate sample for our study not only because S&P 500 is a leading indicator of U.S. equities, but also because it reflects the risk/return characteristics of the large cap universe (equities that are valued at more than \$10 billion). In order to measure the standard deviation of S&P 500, we used VIX, the Chicago Board Options Exchange Market Volatility Index, which is designed by the Chicago Board Option Exchange to gauge the implied volatility of S&P 500 each day. By checking!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 1 (Data assessed on Dec 19 th 2012)! %!

6 Wolfram Alpha, we also know that the average annual return for the S&P 500 is 9.07% in the last five years. RESEARCH METHODOLOGY Though most path-dependent featured options are easily priced through the simulation of sample paths, pricing American style options generally requires a backward algorithm. Through estimating backwards from the exercise date of the option via dynamic programming, the optimal strategy and option price can be estimated. We use American options expiring one year from today (Dec 19 th 2012), within which there are 252 trading days. This means that one simulation consists of 252 steps. Within each step there are two potential movements: up by!!!!2 2!!!!!!!! or move down by!.!!!!2 2!!!!!!!! This function is derived from the property of lognormal distribution which implies that!!!!!!!!!!!!!!!!!!!!!!!!!!!"#!!!!!!!!!!!! After one simulation is finished, we will determine the number of times it is optimal for investors to exercise the option early. If N simulations are finished, there should be in total 251*N times in which investors can exercise the option early. Using it to divide the actual number of times an investor may optimally engage in early exercise, we can get the percentage of time that an investor should exercise an American option early.! '!

7 SIMULATION PROCESS 1. Create all necessary variables, including total number of simulations N, number of trading days each year n, expected return for S&P 500 using average of the five year realized annual return r, volatility sigma, index value at day 0 (today) iniindex, strike price Strike, risk free rate rf and all matrices we will need to store values. 2. Within each of the 252 steps for one simulation, we will generate a random number with normal distribution first. The variable change is calculated using the aforementioned equation. 3. In order for investors to be indifferent between holding the S&P 500 and another asset generating 9.7% return as well, the probability for S&P 500 to go up is calculated as!!!"#!!"#\$!"!!"#\$ in which up is defined as change when change is greater than 1 and down as 1/change, vice versa. 4. A random number is generated to determine if the index value will go up or down. 5. Repeat step 2-4 for the rest of the 251 steps. 6. Repeat step 2-5 for the rest of the N-1 simulations. 7. After finishing the simulation, we essentially have the performance of the S&P 500 in different worlds. To determine when to exercise early, we will look at the index number at each step and calculate its future value. If the result is higher than the ending index number of that specific world, we call that step as an optimal early exercise time. 8. Divide the total number of optimal exercise opportunities by (n-1)*n to get the percentage amount of time investors will be better off by exercising their American call option early.! 4!

8 CONCLUSION American call option with financial theory According to classical financial theory, for American-style call options without cash dividends, the option should never be exercised before the maturity date. There are a number of reasons to explain this theory. First, for a given movement, i.e. upward or downward, in the price of an underlying asset, the profit from holding an in-the-money call is equivalent to the profit from holding the underlying asset. The call option, nevertheless, has the added benefit of protecting against the risk of a downward price movement below the strike price. Moreover, because of the discount rate, it costs more to exercise the option today at a fixed strike price P than to exercise it in the future at P. Last but not least, there is an intrinsic time value of the option that would be lost by exercising the option prior to the expiration date. Our findings on early exercise of American call option As our program shows, in 23.95% of the time, it is optimal for investors to exercise their American call option early, even after taking into account the time value of money. This surprisingly large value suggests that in the market the right to exercise early does add value to American call options, even when there is no dividend. It explains the discrepancies between prices for American call options and European call options on high-tech companies that rarely pay dividends. The plot below shows the number of optimal early exercises for each of the 252 trading days for simulations. Its shape aligns with our prediction that early exercise gets more! 5!

9 appealing to investors when an American call option approaches its expiration date and uncertainty about the future index values decreases. Finding on exercise of American Put option We applied our algorithm on American put options by just reversing the sign of part of the program. The result suggests that investors should exercise their put option early 80% of the time. However, this is not representative of the true overall picture for two reasons. First, put options are widely used as a hedging instrument. Other than the monetary gain investors may generate by exercising the option early, there are real option values by holding onto it, such as reduced cost of financial distress. Moreover, put options, comparing to calls, are subject to greater changes when volatility increases. The time window will rapidly vanish as time elapses.! 6!

10 FURTHER DISCUSSION We have to point out that despite knowing the percentage amount of time at which early exercise is optimal, the exact date still remains elusive. Without knowledge of higher-level mathematics, such as dynamic simulation, there is no easier way to predict a precise date to exercise early. Should there be more time, we would like to assign a 23.95% chance of early exercise to each trading day regardless of the index value at that day, and calculate the expected payoff for doing so compared to the payoff of exercising at the ending date. Comparison of these two values should give us a clearer picture of the added value provided to American call options by the right of early exercise. REFERENCE Bally, Vlad, Lucia Caramellino, and Antonino Zanette. "Pricing and Hedging American Options by Monte Carlo Methods Using a Malliavin Calculus Approach." Monte Carlo Methods and Applications 11.2 (2005): Boyle, P. "Monte Carlo Methods for Security Pricing." Journal of Economic Dynamics and Control (1997): Fu, Michael C., Scott B. Laprise, Dilip B. Madan, Yi Su, and Rongwen Wu. "Pricing American Options: A Comparison of Monte Carlo Simulation Approaches." Journal of Computational Finance 4.3 (2001): Pagès, Gilles, and Jacques Printems. "Functional Quantization for Numerics with an Application to Option Pricing." Monte Carlo Methods and Applications 11.4 (2005): ! 7!

11 APPENDIX: MATLAB PROGRAM N=10000; %Total number of simulations n=252; %252 trading days for one year r=0.0907; %Realized annual for the last 5 years h=1/n; %Each trading day therefore is 1/252 year; sigma= ; %Average VIX for the last one year iniindex= ; %S&P 500 Index as of 12/19/2012 Strike= ; %Set strike price to a certain value; rf=0.0016; %One year treasury rate indexvalue=zeros(n,n); ee=0; %Set the early exercise count to 0; eedate=zeros(1,n-1); %Set the early exercise date to 0; for o=1:n for i=1:n epsilon=randn(); %Random number with normal distribution change=exp((r-(sigma^2)/2)*h+(sigma*sqrt(h)*epsilon)); %Expected index value change if change>1; up=change; down=1/change; else down=change; up=1/change; end; Prob=(exp(r*h)-down)/(up-down); %Calculate prob with r, h, up and down x=rand; if i==1; if x<prob indexvalue(o,i)=iniindex*up; else indexvalue(o,i)=iniindex*down; end; else if x<prob indexvalue(o,i)=indexvalue(o,i-1)*up; else indexvalue(o,i)=indexvalue(o,i-1)*down; end; end! "&!

12 end; for j=1:n-1 if indexvalue(o,j)*exp(rf*h*(n-j))>indexvalue(o,n) %Check if index value at one specific point is higher than the %ending value; if indexvalue(o,j)>strike; %Check if index value at one specific point is higher than %the strike price; ee=ee+1; %Early exercise count +1 eedate(1,j)=eedate(1,j)+1; %Early exercise date +1 end end end; end; eeratio=ee/((n-1)*n) %Early exercise ratio; plot(1:n-1,eedate); xlabel('trading days'); ylabel('number of times'); title('early exercise opportunity at each trading day'); %Plot the optimal days within a year for early exercise axis([0 n *max(eeDate)]); PROGRAM OUTPUT eeratio = ! ""!

13 PLOTS Random selections of S&P 500 performance from the simulations:! "#!

### Call Price as a Function of the Stock Price

Call Price as a Function of the Stock Price Intuitively, the call price should be an increasing function of the stock price. This relationship allows one to develop a theory of option pricing, derived

### Options: Valuation and (No) Arbitrage

Prof. Alex Shapiro Lecture Notes 15 Options: Valuation and (No) Arbitrage I. Readings and Suggested Practice Problems II. Introduction: Objectives and Notation III. No Arbitrage Pricing Bound IV. The Binomial

### Fundamentals of Futures and Options (a summary)

Fundamentals of Futures and Options (a summary) Roger G. Clarke, Harindra de Silva, CFA, and Steven Thorley, CFA Published 2013 by the Research Foundation of CFA Institute Summary prepared by Roger G.

### Option Valuation. Chapter 21

Option Valuation Chapter 21 Intrinsic and Time Value intrinsic value of in-the-money options = the payoff that could be obtained from the immediate exercise of the option for a call option: stock price

### FINANCIAL ECONOMICS OPTION PRICING

OPTION PRICING Options are contingency contracts that specify payoffs if stock prices reach specified levels. A call option is the right to buy a stock at a specified price, X, called the strike price.

### Options/1. Prof. Ian Giddy

Options/1 New York University Stern School of Business Options Prof. Ian Giddy New York University Options Puts and Calls Put-Call Parity Combinations and Trading Strategies Valuation Hedging Options2

### BINOMIAL OPTION PRICING

Darden Graduate School of Business Administration University of Virginia BINOMIAL OPTION PRICING Binomial option pricing is a simple but powerful technique that can be used to solve many complex option-pricing

### Return to Risk Limited website: www.risklimited.com. Overview of Options An Introduction

Return to Risk Limited website: www.risklimited.com Overview of Options An Introduction Options Definition The right, but not the obligation, to enter into a transaction [buy or sell] at a pre-agreed price,

### Pricing American Options on Leveraged Exchange. Traded Funds in the Binomial Pricing Model

Pricing American Options on Leveraged Exchange Traded Funds in the Binomial Pricing Model By Diana Holmes Wolf A Project Report Submitted to the Faculty of the WORCESTER POLYTECHNIC INSTITUTE In partial

### Finance 436 Futures and Options Review Notes for Final Exam. Chapter 9

Finance 436 Futures and Options Review Notes for Final Exam Chapter 9 1. Options: call options vs. put options, American options vs. European options 2. Characteristics: option premium, option type, underlying

### CS 522 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options

CS 5 Computational Tools and Methods in Finance Robert Jarrow Lecture 1: Equity Options 1. Definitions Equity. The common stock of a corporation. Traded on organized exchanges (NYSE, AMEX, NASDAQ). A common

### CHAPTER 15. Option Valuation

CHAPTER 15 Option Valuation Just what is an option worth? Actually, this is one of the more difficult questions in finance. Option valuation is an esoteric area of finance since it often involves complex

### Option pricing. Vinod Kothari

Option pricing Vinod Kothari Notation we use this Chapter will be as follows: S o : Price of the share at time 0 S T : Price of the share at time T T : time to maturity of the option r : risk free rate

### Lecture 6: Option Pricing Using a One-step Binomial Tree. Friday, September 14, 12

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

### American and European. Put Option

American and European Put Option Analytical Finance I Kinda Sumlaji 1 Table of Contents: 1. Introduction... 3 2. Option Style... 4 3. Put Option 4 3.1 Definition 4 3.2 Payoff at Maturity... 4 3.3 Example

### INTRODUCTION TO OPTIONS MARKETS QUESTIONS

INTRODUCTION TO OPTIONS MARKETS QUESTIONS 1. What is the difference between a put option and a call option? 2. What is the difference between an American option and a European option? 3. Why does an option

### Valuing equity-based payments

E Valuing equity-based payments Executive remuneration packages generally comprise many components. While it is relatively easy to identify how much will be paid in a base salary a fixed dollar amount

### Lecture Notes: Basic Concepts in Option Pricing - The Black and Scholes Model

Brunel University Msc., EC5504, Financial Engineering Prof Menelaos Karanasos Lecture Notes: Basic Concepts in Option Pricing - The Black and Scholes Model Recall that the price of an option is equal to

### Financial Options: Pricing and Hedging

Financial Options: Pricing and Hedging Diagrams Debt Equity Value of Firm s Assets T Value of Firm s Assets T Valuation of distressed debt and equity-linked securities requires an understanding of financial

### Derivative Users Traders of derivatives can be categorized as hedgers, speculators, or arbitrageurs.

OPTIONS THEORY Introduction The Financial Manager must be knowledgeable about derivatives in order to manage the price risk inherent in financial transactions. Price risk refers to the possibility of loss

### Monte Carlo Simulation

Monte Carlo Simulation Palisade User Conference 2007 Real Options Analysis Dr. Colin Beardsley, Managing Director Risk-Return Solutions Pty Limited www.riskreturnsolutions.com The Plot Today We will review

### BUSM 411: Derivatives and Fixed Income

BUSM 411: Derivatives and Fixed Income 2. Forwards, Options, and Hedging This lecture covers the basic derivatives contracts: forwards (and futures), and call and put options. These basic contracts are

OPTIONS CALCULATOR QUICK GUIDE Reshaping Canada s Equities Trading Landscape OCTOBER 2014 Table of Contents Introduction 3 Valuing options 4 Examples 6 Valuing an American style non-dividend paying stock

### Call and Put. Options. American and European Options. Option Terminology. Payoffs of European Options. Different Types of Options

Call and Put Options A call option gives its holder the right to purchase an asset for a specified price, called the strike price, on or before some specified expiration date. A put option gives its holder

### Simplified Option Selection Method

Simplified Option Selection Method Geoffrey VanderPal Webster University Thailand Options traders and investors utilize methods to price and select call and put options. The models and tools range from

### Chapter 8 Financial Options and Applications in Corporate Finance ANSWERS TO END-OF-CHAPTER QUESTIONS

Chapter 8 Financial Options and Applications in Corporate Finance ANSWERS TO END-OF-CHAPTER QUESTIONS 8-1 a. An option is a contract which gives its holder the right to buy or sell an asset at some predetermined

### A Comparison of Option Pricing Models

A Comparison of Option Pricing Models Ekrem Kilic 11.01.2005 Abstract Modeling a nonlinear pay o generating instrument is a challenging work. The models that are commonly used for pricing derivative might

### Chapter 11 Options. Main Issues. Introduction to Options. Use of Options. Properties of Option Prices. Valuation Models of Options.

Chapter 11 Options Road Map Part A Introduction to finance. Part B Valuation of assets, given discount rates. Part C Determination of risk-adjusted discount rate. Part D Introduction to derivatives. Forwards

### S 1 S 2. Options and Other Derivatives

Options and Other Derivatives The One-Period Model The previous chapter introduced the following two methods: Replicate the option payoffs with known securities, and calculate the price of the replicating

### Stock. Call. Put. Bond. Option Fundamentals

Option Fundamentals Payoff Diagrams hese are the basic building blocks of financial engineering. hey represent the payoffs or terminal values of various investment choices. We shall assume that the maturity

### BINOMIAL OPTIONS PRICING MODEL. Mark Ioffe. Abstract

BINOMIAL OPTIONS PRICING MODEL Mark Ioffe Abstract Binomial option pricing model is a widespread numerical method of calculating price of American options. In terms of applied mathematics this is simple

### Option Values. Determinants of Call Option Values. CHAPTER 16 Option Valuation. Figure 16.1 Call Option Value Before Expiration

CHAPTER 16 Option Valuation 16.1 OPTION VALUATION: INTRODUCTION Option Values Intrinsic value - profit that could be made if the option was immediately exercised Call: stock price - exercise price Put:

### 第 9 讲 : 股 票 期 权 定 价 : B-S 模 型 Valuing Stock Options: The Black-Scholes Model

1 第 9 讲 : 股 票 期 权 定 价 : B-S 模 型 Valuing Stock Options: The Black-Scholes Model Outline 有 关 股 价 的 假 设 The B-S Model 隐 性 波 动 性 Implied Volatility 红 利 与 期 权 定 价 Dividends and Option Pricing 美 式 期 权 定 价 American

### CHAPTER 22 Options and Corporate Finance

CHAPTER 22 Options and Corporate Finance Multiple Choice Questions: I. DEFINITIONS OPTIONS a 1. A financial contract that gives its owner the right, but not the obligation, to buy or sell a specified asset

### Management of Asian and Cliquet Option Exposures for Insurance Companies: SPVA applications (I)

Management of Asian and Cliquet Option Exposures for Insurance Companies: SPVA applications (I) Pin Chung and Rachid Lassoued 5th September, 2014, Wicklow, Ireland 0 Agenda 1. Introduction 2. Review of

### Part V: Option Pricing Basics

erivatives & Risk Management First Week: Part A: Option Fundamentals payoffs market microstructure Next 2 Weeks: Part B: Option Pricing fundamentals: intrinsic vs. time value, put-call parity introduction

### Overview. Option Basics. Options and Derivatives. Professor Lasse H. Pedersen. Option basics and option strategies

Options and Derivatives Professor Lasse H. Pedersen Prof. Lasse H. Pedersen 1 Overview Option basics and option strategies No-arbitrage bounds on option prices Binomial option pricing Black-Scholes-Merton

### Valuation of Razorback Executive Stock Options: A Simulation Approach

Valuation of Razorback Executive Stock Options: A Simulation Approach Joe Cheung Charles J. Corrado Department of Accounting & Finance The University of Auckland Private Bag 92019 Auckland, New Zealand.

### Investing In Volatility

Investing In Volatility By: Anish Parvataneni, CFA Portfolio Manager LJM Partners Ltd. LJM Partners, Ltd. is issuing a series a white papers on the subject of investing in volatility as an asset class.

### Futures Price d,f \$ 0.65 = (1.05) (1.04)

24 e. Currency Futures In a currency futures contract, you enter into a contract to buy a foreign currency at a price fixed today. To see how spot and futures currency prices are related, note that holding

### Binary options. Giampaolo Gabbi

Binary options Giampaolo Gabbi Definition In finance, a binary option is a type of option where the payoff is either some fixed amount of some asset or nothing at all. The two main types of binary options

### Lecture 21 Options Pricing

Lecture 21 Options Pricing Readings BM, chapter 20 Reader, Lecture 21 M. Spiegel and R. Stanton, 2000 1 Outline Last lecture: Examples of options Derivatives and risk (mis)management Replication and Put-call

### Black-Scholes-Merton approach merits and shortcomings

Black-Scholes-Merton approach merits and shortcomings Emilia Matei 1005056 EC372 Term Paper. Topic 3 1. Introduction The Black-Scholes and Merton method of modelling derivatives prices was first introduced

### European Options Pricing Using Monte Carlo Simulation

European Options Pricing Using Monte Carlo Simulation Alexandros Kyrtsos Division of Materials Science and Engineering, Boston University akyrtsos@bu.edu European options can be priced using the analytical

### FIN 3710. Final (Practice) Exam 05/23/06

FIN 3710 Investment Analysis Spring 2006 Zicklin School of Business Baruch College Professor Rui Yao FIN 3710 Final (Practice) Exam 05/23/06 NAME: (Please print your name here) PLEDGE: (Sign your name

### Option Premium = Intrinsic. Speculative Value. Value

Chapters 4/ Part Options: Basic Concepts Options Call Options Put Options Selling Options Reading The Wall Street Journal Combinations of Options Valuing Options An Option-Pricing Formula Investment in

### Quantitative Strategies Research Notes

Quantitative Strategies Research Notes December 1992 Valuing Options On Periodically-Settled Stocks Emanuel Derman Iraj Kani Alex Bergier SUMMARY In some countries, for example France, stocks bought or

### Chapter 15 OPTIONS ON MONEY MARKET FUTURES

Page 218 The information in this chapter was last updated in 1993. Since the money market evolves very rapidly, recent developments may have superseded some of the content of this chapter. Chapter 15 OPTIONS

### Options Pricing. This is sometimes referred to as the intrinsic value of the option.

Options Pricing We will use the example of a call option in discussing the pricing issue. Later, we will turn our attention to the Put-Call Parity Relationship. I. Preliminary Material Recall the payoff

### American Options. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan American Options

American Options An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2010 Early Exercise Since American style options give the holder the same rights as European style options plus

### The Black-Scholes Formula

FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 The Black-Scholes Formula These notes examine the Black-Scholes formula for European options. The Black-Scholes formula are complex as they are based on the

### Understanding Stock Options

Understanding Stock Options Introduction...2 Benefits Of Exchange-Traded Options... 4 Options Compared To Common Stocks... 6 What Is An Option... 7 Basic Strategies... 12 Conclusion...20 Glossary...22

### Introduction to Options. Derivatives

Introduction to Options Econ 422: Investment, Capital & Finance University of Washington Summer 2010 August 18, 2010 Derivatives A derivative is a security whose payoff or value depends on (is derived

### Binomial trees and risk neutral valuation

Binomial trees and risk neutral valuation Moty Katzman September 19, 2014 Derivatives in a simple world A derivative is an asset whose value depends on the value of another asset. Call/Put European/American

### Lecture 11: Risk-Neutral Valuation Steven Skiena. skiena

Lecture 11: Risk-Neutral Valuation Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena Risk-Neutral Probabilities We can

### Sensex Realized Volatility Index

Sensex Realized Volatility Index Introduction: Volatility modelling has traditionally relied on complex econometric procedures in order to accommodate the inherent latent character of volatility. Realized

### DERIVATIVE SECURITIES Lecture 2: Binomial Option Pricing and Call Options

DERIVATIVE SECURITIES Lecture 2: Binomial Option Pricing and Call Options Philip H. Dybvig Washington University in Saint Louis review of pricing formulas assets versus futures practical issues call options

### Options pricing in discrete systems

UNIVERZA V LJUBLJANI, FAKULTETA ZA MATEMATIKO IN FIZIKO Options pricing in discrete systems Seminar II Mentor: prof. Dr. Mihael Perman Author: Gorazd Gotovac //2008 Abstract This paper is a basic introduction

### The Binomial Option Pricing Model André Farber

1 Solvay Business School Université Libre de Bruxelles The Binomial Option Pricing Model André Farber January 2002 Consider a non-dividend paying stock whose price is initially S 0. Divide time into small

### EXP 481 -- Capital Markets Option Pricing. Options: Definitions. Arbitrage Restrictions on Call Prices. Arbitrage Restrictions on Call Prices 1) C > 0

EXP 481 -- Capital Markets Option Pricing imple arbitrage relations Payoffs to call options Black-choles model Put-Call Parity Implied Volatility Options: Definitions A call option gives the buyer the

### Chapter 20 Understanding Options

Chapter 20 Understanding Options Multiple Choice Questions 1. Firms regularly use the following to reduce risk: (I) Currency options (II) Interest-rate options (III) Commodity options D) I, II, and III

### On Black-Scholes Equation, Black- Scholes Formula and Binary Option Price

On Black-Scholes Equation, Black- Scholes Formula and Binary Option Price Abstract: Chi Gao 12/15/2013 I. Black-Scholes Equation is derived using two methods: (1) risk-neutral measure; (2) - hedge. II.

### Steve Meizinger. FX Options Pricing, what does it Mean?

Steve Meizinger FX Options Pricing, what does it Mean? For the sake of simplicity, the examples that follow do not take into consideration commissions and other transaction fees, tax considerations, or

### Research on Option Trading Strategies

Research on Option Trading Strategies An Interactive Qualifying Project Report: Submitted to the Faculty of the WORCESTER POLYTECHNIC INSTITUTE In partial fulfillment of the requirements for the Degree

### Conn Valuation Services Ltd.

CONVERTIBLE BONDS: Black Scholes vs. Binomial Models By Richard R. Conn CMA, MBA, CPA, ABV, CFFA, ERP For years I have believed that the Black-Scholes (B-S) option pricing model should not be used in the

### Introduction to Binomial Trees

11 C H A P T E R Introduction to Binomial Trees A useful and very popular technique for pricing an option involves constructing a binomial tree. This is a diagram that represents di erent possible paths

### Chapter 21: Options and Corporate Finance

Chapter 21: Options and Corporate Finance 21.1 a. An option is a contract which gives its owner the right to buy or sell an underlying asset at a fixed price on or before a given date. b. Exercise is the

### FIN 673 Pricing Real Options

FIN 673 Pricing Real Options Professor Robert B.H. Hauswald Kogod School of Business, AU From Financial to Real Options Option pricing: a reminder messy and intuitive: lattices (trees) elegant and mysterious:

### ECMC49F Options Practice Questions Suggested Solution Date: Nov 14, 2005

ECMC49F Options Practice Questions Suggested Solution Date: Nov 14, 2005 Options: General [1] Define the following terms associated with options: a. Option An option is a contract which gives the holder

### New Pricing Formula for Arithmetic Asian Options. Using PDE Approach

Applied Mathematical Sciences, Vol. 5, 0, no. 77, 380-3809 New Pricing Formula for Arithmetic Asian Options Using PDE Approach Zieneb Ali Elshegmani, Rokiah Rozita Ahmad and 3 Roza Hazli Zakaria, School

### Option Theory Basics

Option Basics What is an Option? Option Theory Basics An option is a traded security that is a derivative product. By derivative product we mean that it is a product whose value is based upon, or derived

### Valuation Assistance with the Complexity of ASC 718

Valuation Assistance with the Complexity of ASC 718 Stock-based compensation generally consists of either the transferring of stock or the issuance of stock options to an employee, officer of a company,

### THE OHANA GROUP AT MORGAN STANLEY EMPLOYEE STOCK OPTION VALUATION

THE OHANA GROUP AT MORGAN STANLEY EMPLOYEE STOCK OPTION VALUATION BRYAN GOULD, FINANCIAL ADVISOR, PORTFOLIO MANAGER TELEPHONE: (858) 643-5004 4350 La Jolla Village Drive, Suite 1000, San Diego, CA 92122

### FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008

FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 Options These notes consider the way put and call options and the underlying can be combined to create hedges, spreads and combinations. We will consider the

### Basic Black-Scholes: Option Pricing and Trading

Basic Black-Scholes: Option Pricing and Trading BSc (HONS 1 st Timothy Falcon Crack Class), PGDipCom, MCom, PhD (MIT), IMC Contents Preface Tables Figures ix xi xiii 1 Introduction to Options 1 1.1 Hedging,

### The Intuition Behind Option Valuation: A Teaching Note

The Intuition Behind Option Valuation: A Teaching Note Thomas Grossman Haskayne School of Business University of Calgary Steve Powell Tuck School of Business Dartmouth College Kent L Womack Tuck School

### OPTION PRICING USING MONTE CARLO SIMULATION

University of Salford A Greater Manchester University The General Jonas Žemaitis Military Academy of Lithuania Ministry of National Defence Republic of Lithuania Vilnius Gediminas Technical University

### Introduction to Options

Introduction to Options By: Peter Findley and Sreesha Vaman Investment Analysis Group What Is An Option? One contract is the right to buy or sell 100 shares The price of the option depends on the price

### Lecture 12: The Black-Scholes Model Steven Skiena. http://www.cs.sunysb.edu/ skiena

Lecture 12: The Black-Scholes Model Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena The Black-Scholes-Merton Model

### Volatility as an indicator of Supply and Demand for the Option. the price of a stock expressed as a decimal or percentage.

Option Greeks - Evaluating Option Price Sensitivity to: Price Changes to the Stock Time to Expiration Alterations in Interest Rates Volatility as an indicator of Supply and Demand for the Option Different

### BUS-495 Fall 2011 Final Exam: December 14 Answer Key

Name: Score: BUS-495 Fall 011 Final Exam: December 14 Answer Key 1. (6 points; 1 point each) If you are neutral on a particular stock for the short term and looking to generate some income using options,

### Invesco Great Wall Fund Management Co. Shenzhen: June 14, 2008

: A Stern School of Business New York University Invesco Great Wall Fund Management Co. Shenzhen: June 14, 2008 Outline 1 2 3 4 5 6 se notes review the principles underlying option pricing and some of

### Lecture 4: The Black-Scholes model

OPTIONS and FUTURES Lecture 4: The Black-Scholes model Philip H. Dybvig Washington University in Saint Louis Black-Scholes option pricing model Lognormal price process Call price Put price Using Black-Scholes

### CHAPTER 20. Financial Options. Chapter Synopsis

CHAPTER 20 Financial Options Chapter Synopsis 20.1 Option Basics A financial option gives its owner the right, but not the obligation, to buy or sell a financial asset at a fixed price on or until a specified

### Option Values. Option Valuation. Call Option Value before Expiration. Determinants of Call Option Values

Option Values Option Valuation Intrinsic value profit that could be made if the option was immediately exercised Call: stock price exercise price : S T X i i k i X S Put: exercise price stock price : X

### Model-Free Boundaries of Option Time Value and Early Exercise Premium

Model-Free Boundaries of Option Time Value and Early Exercise Premium Tie Su* Department of Finance University of Miami P.O. Box 248094 Coral Gables, FL 33124-6552 Phone: 305-284-1885 Fax: 305-284-4800

### Put-Call Parity and Synthetics

Courtesy of Market Taker Mentoring LLC TM Excerpt from Trading Option Greeks, by Dan Passarelli Chapter 6 Put-Call Parity and Synthetics In order to understand more-complex spread strategies involving

### 6. Foreign Currency Options

6. Foreign Currency Options So far, we have studied contracts whose payoffs are contingent on the spot rate (foreign currency forward and foreign currency futures). he payoffs from these instruments are

### w w w.c a t l e y l a k e m a n.c o m 0 2 0 7 0 4 3 0 1 0 0

A ADR-Style: for a derivative on an underlying denominated in one currency, where the derivative is denominated in a different currency, payments are exchanged using a floating foreign-exchange rate. The

### Review of Basic Options Concepts and Terminology

Review of Basic Options Concepts and Terminology March 24, 2005 1 Introduction The purchase of an options contract gives the buyer the right to buy call options contract or sell put options contract some

### UNDERSTANDING EQUITY OPTIONS

UNDERSTANDING EQUITY OPTIONS The Options Industry Council (OIC) is a non-profit association created to educate the investing public and brokers about the benefits and risks of exchange-traded options.

### Valuing Coca-Cola and Pepsi Options Using the Black-Scholes Option Pricing Model

Valuing Coca-Cola and Pepsi Options Using the Black-Scholes Option Pricing Model John C. Gardner, University of New Orleans Carl B. McGowan, Jr., CFA, Norfolk State University ABSTRACT In this paper, we

### An Introduction to Modeling Stock Price Returns With a View Towards Option Pricing

An Introduction to Modeling Stock Price Returns With a View Towards Option Pricing Kyle Chauvin August 21, 2006 This work is the product of a summer research project at the University of Kansas, conducted

### 2. Exercising the option - buying or selling asset by using option. 3. Strike (or exercise) price - price at which asset may be bought or sold

Chapter 21 : Options-1 CHAPTER 21. OPTIONS Contents I. INTRODUCTION BASIC TERMS II. VALUATION OF OPTIONS A. Minimum Values of Options B. Maximum Values of Options C. Determinants of Call Value D. Black-Scholes

### CHAPTER 7 FUTURES AND OPTIONS ON FOREIGN EXCHANGE SUGGESTED ANSWERS AND SOLUTIONS TO END-OF-CHAPTER QUESTIONS AND PROBLEMS

CHAPTER 7 FUTURES AND OPTIONS ON FOREIGN EXCHANGE SUGGESTED ANSWERS AND SOLUTIONS TO END-OF-CHAPTER QUESTIONS AND PROBLEMS QUESTIONS 1. Explain the basic differences between the operation of a currency

### Valuing Stock Options: The Black-Scholes-Merton Model. Chapter 13

Valuing Stock Options: The Black-Scholes-Merton Model Chapter 13 Fundamentals of Futures and Options Markets, 8th Ed, Ch 13, Copyright John C. Hull 2013 1 The Black-Scholes-Merton Random Walk Assumption