# Empirical Study of effect of using Weibull. NIFTY index options

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Empirical Study of effect of using Weibull distribution in Black Scholes Formula on NIFTY index options 11 th Global Conference of Actuaries Harsh Narang Vatsala Deora

2 Overview NIFTY NSE index options Black Scholes model (lognormal distribution) Replacing the lognormal distribution with Wiblldi Weibull distribution ib ti Comparison on the basis of difference in prices Comparison on the basis of ease of use and implementation

3 NSE Nifty options S&P CNX Nifty (Nifty) is a 50 stock index comprising the largest and the most liquid companies in India Option trading began in June 2001 Uses Black Scholes model (lognormal) A call option gives its holder the right to buy whereas put option gives its holder the right to sell Analysis based on call prices of Nifty Options (European type)

4 Black Scholes model Cornerstone of option pricing since its inception Used in both financial literature as well as practice Arbitrary assumption of lognormal distribution Deficient in pricing deep in the money and deep out of the money options using statistical estimates of volatility. Still widely prevalent due to simplicity of the model and ease of use

5 Option pricing using lognormal distribution Under general asset pricing theory with the noarbitrage condition, the current price of an asset is equal to the present value of its expectedpayoffs payoffs discounted at an appropriate rate. W th t di id d id We assume that dividends are paid continuously, and that the dividend amount is proportional to the level of the index.

6 Black Scholes Formula where is the modified forward price that occurs in the terms d1 and d2: S 0 : the price of the underlying security at time 0. 0 p y g y X : exercise price r : the risk free rate per annum with continuous compounding σ 2 : variance rate of return for the underlying security T : time to expiration c : market value of call option at time 0 p : market value of put option at time 0 q: the dividend yield N(d): cumulative probability distribution function for a standardized normal distribution at d.

7 Limitations & Modifications Concept of implied volatility Value of implied volatility for different strike prices should theoretically be identical, but is usually seen to vary. The third central moment of the price distribution is significantly negative. Two solutions o s are aepossible: Adjusting the lognormal distribution by a skewness correction term Using a distribution with natural negative skewness

8 Weibull Distribution It is widely used in survival analysis & in extreme value theory Its skewness is negative for most parameter values which makes it more appropriate for valuing Nifty options

9 Option pricing using weibull distribution Derived using the method of equivalent martingale measures If it is assumed that the stock price at expiration, S T, is distributed Weibull: F d (a) denotes the cumulative distribution function at point a of χ 2 distributed random variable with degrees of freedom d and where

10 Data used for analysis The NSE nifty daily dil returns from 1 st January 2001 to 1 st January 2008 have been used. Then comparison between the two distributions was based on call prices of Nifty Options (European type) for the period of about three months from 3 rd October 2007 to 27th December The daily MIBOR (Mumbai Inter Bank Offer Rate) rate has been taken as the risk free interest rate. Data for thinly traded options (less than 100 contracts on a given day) was excluded from the study.

11 NSE nifty Frequency Plot of Closing Values 01/01/2001 / 01/01/2008/ / Frequency

12 NSE nifty returns data Daily Returns Weekly Returns Mean Variance Std Error Skewness Kurtosis Nifty closing values do not follow a lognormal distribution and the returns do not conform to the normal distribution and hence pricing the returns with the lognormal assumption is inaccurate.

13 NSE Nifty Frequency plot of daily returns ( ) Normal Distribution Frequency Plot

14 Methodology O i i i d b i i i i i O 2 i E Optimization done by minimizing i1 i O i is the market price and E is the theoretical price usinga given setof parameters Hypothesis: The total error using Weibull distribution ib i is less than that by using Lognormal assumption. For comparison, we compute the Error Sum of Squares (ESS) for the two approaches by summing the square of the difference between the predicted and the actual prices. These two ESS are then compared for statistically significant difference using the paired t test. n

15 Results t Test: Paired Two Sample for Means: Combined Black Scholes Weibull Mean Variance Observations Pearson Correlation Hypothesized Mean Difference Df 48 t Stat P(T<=t) one tail t Critical one tail P(T<=t) two tail t Critical two tailtail

16 Results t Test: Paired Two Sample for 25 Oct Nov Dec 2007 Means Black Scholes Weibull Black Scholes Weibull Black Scholes Weibull Mean Variance E E Observations Pearson Correlation Hypothesized ed Mean Difference Df t Stat P(T<=t) one tail t Critical one tail P(T<=t) two tail t Critical twotail

17 Volatility Smile es Pa arameter Valu Comparison of volatility smile Oct 12 Weibull Param meter Values Lognormal Comparsion of volatility smile december Lognomal Weibull Comparison of volatility smile Oct 5 Comparison of Volatility smile Nov r Value Paramete Lognormal Weibull r Value Paramete Lognormal Weibull 0 0

18 Conclusion The main advantages of the proposed distribution model over the pre existing models are its simple algebraic form, comparable to that of the Black Scholes model with lognormal distribution, the ease of the model s implementation, and the absenceof pricing biases suchas those generated by the standard Black Scholes formula. As a result, the proposed model has a potential for a wide range of practical applications.

19 Future Work In this study, we have calculated implied volatility based on today s data to predict tomorrow s prices. This can be extended to explore whether the modified approach gives significantly ifi better prices for longer durations or not. A related study can be conducted regarding comparisonofdifferent of distributional assumptions ( g and h, GB2, Burr) with the Weibull distribution in pricing of the Nifty options.

20 Thank you

21 References Savickas,R., A Simple Option Pricing Formula, Department of Finance, George Washington University (2001) Black, F. and M. Scholes, The pricing of options and corporate liabilities, Journal of Political Economy 81, McDonald, J. B. and R. M. Bookstaber, Option Pricing for Generalized Distributions. Communications in Statistics Theory Meth. Vol. 20, Hull, J. C., Options, Futures, & Other Derivatives. Upper Saddle River, NJ: Prentice Hall, Inc., Data on option prices, Retrieved February 23, 2008 from Saurabha, Rritu and Tiwari, Manvendra, Empirical Study of the effect of including Skewness and Kurtosis in Black Scholes option pricing gformula on S&P CNX Nifty fyindex Options, IIM Lucknow (2007) Merrill, C. and D. F. Babbel, Interest Rate Option Pricing Revisited. The Journal of Futures Markets, Vol. 16, No. 8, Misra, D., Kannan,R. and Misra, Sangeeta D. (2006): Implied Volatility Surfaces: A study of NSE Nifty Options, International Research Journal of Finance & Economics. Dutta and Babel, Extracting Probabilistic Information from the Prices of Interest Rate Options: Tests of Distributional Assumptions, Wharton financial institutions center(2002)

### 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

### Options Pricing with Skewness and Kurtosis Adjustments

Options Pricing with Skewness and Kurtosis Adjustments N. S. Nilakantan, Faculty, KJSIMSR, Mumbai, India. Email: Nilakantan@somaiya.edu Achal Jain, PGFS 2012-14, KJSIMSR, Mumbai, India. Email: achaljain.simsr@gmail.com

### Study on the Volatility Smile of EUR/USD Currency Options and Trading Strategies

Prof. Joseph Fung, FDS Study on the Volatility Smile of EUR/USD Currency Options and Trading Strategies BY CHEN Duyi 11050098 Finance Concentration LI Ronggang 11050527 Finance Concentration An Honors

### 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

About Volatility Index Volatility Index is a measure of market s expectation of volatility over the near term. Volatility is often described as the rate and magnitude of changes in prices and in finance

### Black Scholes Merton Approach To Modelling Financial Derivatives Prices Tomas Sinkariovas 0802869. Words: 3441

Black Scholes Merton Approach To Modelling Financial Derivatives Prices Tomas Sinkariovas 0802869 Words: 3441 1 1. Introduction In this paper I present Black, Scholes (1973) and Merton (1973) (BSM) general

### Oscillatory Reduction in Option Pricing Formula Using Shifted Poisson and Linear Approximation

EPJ Web of Conferences 68, 0 00 06 (2014) DOI: 10.1051/ epjconf/ 20146800006 C Owned by the authors, published by EDP Sciences, 2014 Oscillatory Reduction in Option Pricing Formula Using Shifted Poisson

### 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

### Chapter 1: Financial Markets and Financial Derivatives

Chapter 1: Financial Markets and Financial Derivatives 1.1 Financial Markets Financial markets are markets for financial instruments, in which buyers and sellers find each other and create or exchange

### Bond Options, Caps and the Black Model

Bond Options, Caps and the Black Model Black formula Recall the Black formula for pricing options on futures: C(F, K, σ, r, T, r) = Fe rt N(d 1 ) Ke rt N(d 2 ) where d 1 = 1 [ σ ln( F T K ) + 1 ] 2 σ2

### 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

### How to Value Employee Stock Options

John Hull and Alan White One of the arguments often used against expensing employee stock options is that calculating their fair value at the time they are granted is very difficult. This article presents

### IMPLIED VOLATILITY SKEWS AND STOCK INDEX SKEWNESS AND KURTOSIS IMPLIED BY S&P 500 INDEX OPTION PRICES

IMPLIED VOLATILITY SKEWS AND STOCK INDEX SKEWNESS AND KURTOSIS IMPLIED BY S&P 500 INDEX OPTION PRICES Charles J. Corrado Department of Finance University of Missouri - Columbia Tie Su Department of Finance

### VALUING REAL OPTIONS USING IMPLIED BINOMIAL TREES AND COMMODITY FUTURES OPTIONS

VALUING REAL OPTIONS USING IMPLIED BINOMIAL TREES AND COMMODITY FUTURES OPTIONS TOM ARNOLD TIMOTHY FALCON CRACK* ADAM SCHWARTZ A real option on a commodity is valued using an implied binomial tree (IBT)

### 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

### A Study of the Relation Between Market Index, Index Futures and Index ETFs: A Case Study of India ABSTRACT

Rev. Integr. Bus. Econ. Res. Vol 2(1) 223 A Study of the Relation Between Market Index, Index Futures and Index ETFs: A Case Study of India S. Kevin Director, TKM Institute of Management, Kollam, India

### Quantitative Methods for Finance

Quantitative Methods for Finance Module 1: The Time Value of Money 1 Learning how to interpret interest rates as required rates of return, discount rates, or opportunity costs. 2 Learning how to explain

### IMPLIED VOLATILITY SKEWS AND STOCK INDEX SKEWNESS AND KURTOSIS IMPLIED BY S&P 500 INDEX OPTION PRICES

IMPLIED VOLATILITY SKEWS AND STOCK INDEX SKEWNESS AND KURTOSIS IMPLIED BY S&P 500 INDEX OPTION PRICES Charles J. Corrado Department of Finance 14 Middlebush Hall University of Missouri Columbia, MO 6511

### NOTES ON THE BANK OF ENGLAND OPTION-IMPLIED PROBABILITY DENSITY FUNCTIONS

1 NOTES ON THE BANK OF ENGLAND OPTION-IMPLIED PROBABILITY DENSITY FUNCTIONS Options are contracts used to insure against or speculate/take a view on uncertainty about the future prices of a wide range

### 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.

### DETERMINATION OF INTEREST RATE ARBITRAGE ACROSS MARKETS IN INDIA

SRJIS/Sameer Anvekar & Prakash Salvi. (170179) DETERMINATION OF INTEREST RATE ARBITRAGE ACROSS MARKETS IN INDIA Samir Anvekar Research Student, Department of Economics, University of Mumbai. Prakash Salvi.

### 1 The Black-Scholes Formula

1 The Black-Scholes Formula In 1973 Fischer Black and Myron Scholes published a formula - the Black-Scholes formula - for computing the theoretical price of a European call option on a stock. Their paper,

### 24. Pricing Fixed Income Derivatives. through Black s Formula. MA6622, Ernesto Mordecki, CityU, HK, 2006. References for this Lecture:

24. Pricing Fixed Income Derivatives through Black s Formula MA6622, Ernesto Mordecki, CityU, HK, 2006. References for this Lecture: John C. Hull, Options, Futures & other Derivatives (Fourth Edition),

### No-Arbitrage Condition of Option Implied Volatility and Bandwidth Selection

Kamla-Raj 2014 Anthropologist, 17(3): 751-755 (2014) No-Arbitrage Condition of Option Implied Volatility and Bandwidth Selection Milos Kopa 1 and Tomas Tichy 2 1 Institute of Information Theory and Automation

### Hedging Illiquid FX Options: An Empirical Analysis of Alternative Hedging Strategies

Hedging Illiquid FX Options: An Empirical Analysis of Alternative Hedging Strategies Drazen Pesjak Supervised by A.A. Tsvetkov 1, D. Posthuma 2 and S.A. Borovkova 3 MSc. Thesis Finance HONOURS TRACK Quantitative

### 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

### 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

### Journal of Financial and Economic Practice

Analyzing Investment Data Using Conditional Probabilities: The Implications for Investment Forecasts, Stock Option Pricing, Risk Premia, and CAPM Beta Calculations By Richard R. Joss, Ph.D. Resource Actuary,

### Journal Of Financial And Strategic Decisions Volume 10 Number 2 Summer 1997

Journal Of Financial And Strategic Decisions Volume 10 Number 2 Summer 1997 AN EMPIRICAL INVESTIGATION OF PUT OPTION PRICING: A SPECIFICATION TEST OF AT-THE-MONEY OPTION IMPLIED VOLATILITY Hongshik Kim,

### 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

### Beyond Black- Scholes: Smile & Exo6c op6ons. Professor André Farber Solvay Brussels School of Economics and Management Université Libre de Bruxelles

Beyond Black- Scholes: Smile & Exo6c op6ons Professor André Farber Solvay Brussels School of Economics and Management Université Libre de Bruxelles 1 What is a Vola6lity Smile? Rela6onship between implied

### Information Content of Right Option Tails: Evidence from S&P 500 Index Options

Information Content of Right Option Tails: Evidence from S&P 500 Index Options Greg Orosi* October 17, 2015 Abstract In this study, we investigate how useful the information content of out-of-the-money

### 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

### EPS 625 INTERMEDIATE STATISTICS FRIEDMAN TEST

EPS 625 INTERMEDIATE STATISTICS The Friedman test is an extension of the Wilcoxon test. The Wilcoxon test can be applied to repeated-measures data if participants are assessed on two occasions or conditions

### 10. Fixed-Income Securities. Basic Concepts

0. Fixed-Income Securities Fixed-income securities (FIS) are bonds that have no default risk and their payments are fully determined in advance. Sometimes corporate bonds that do not necessarily have certain

### UNDERSTANDING THE DEPENDENT-SAMPLES t TEST

UNDERSTANDING THE DEPENDENT-SAMPLES t TEST A dependent-samples t test (a.k.a. matched or paired-samples, matched-pairs, samples, or subjects, simple repeated-measures or within-groups, or correlated groups)

### An Empirical Analysis of Option Valuation Techniques. Using Stock Index Options

An Empirical Analysis of Option Valuation Techniques Using Stock Index Options Mohammad Yamin Yakoob 1 Duke University Durham, NC April 2002 1 Mohammad Yamin Yakoob graduated cum laude from Duke University

### call option put option strike price/exercise price expiration date/maturity

OPTIONS Objective This chapter introduces the readers to the concept of options which include calls and puts. All basic concepts like option buyer and seller, European and American options, and payoff

### 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

### 1 The Black-Scholes model: extensions and hedging

1 The Black-Scholes model: extensions and hedging 1.1 Dividends Since we are now in a continuous time framework the dividend paid out at time t (or t ) is given by dd t = D t D t, where as before D denotes

### 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

### A comparison between different volatility models. Daniel Amsköld

A comparison between different volatility models Daniel Amsköld 211 6 14 I II Abstract The main purpose of this master thesis is to evaluate and compare different volatility models. The evaluation is based

### Implied Volatility Surface

Implied Volatility Surface Liuren Wu Zicklin School of Business, Baruch College Options Markets (Hull chapter: 16) Liuren Wu Implied Volatility Surface Options Markets 1 / 18 Implied volatility Recall

### Charles University, Faculty of Mathematics and Physics, Prague, Czech Republic.

WDS'09 Proceedings of Contributed Papers, Part I, 148 153, 2009. ISBN 978-80-7378-101-9 MATFYZPRESS Volatility Modelling L. Jarešová Charles University, Faculty of Mathematics and Physics, Prague, Czech

### Tutorial: Structural Models of the Firm

Tutorial: Structural Models of the Firm Peter Ritchken Case Western Reserve University February 16, 2015 Peter Ritchken, Case Western Reserve University Tutorial: Structural Models of the Firm 1/61 Tutorial:

### A Comparison of ITM and OTM Protective-Puts and Covered-Calls

A Comparison of ITM and OTM Protective-Puts and Covered-Calls Mihir Dash Department of Quantitative Methods, School of Business, Alliance University Chikkahagade Cross, Chandapura-Anekal Road, Anekal,

### Institutional Finance 08: Dynamic Arbitrage to Replicate Non-linear Payoffs. Binomial Option Pricing: Basics (Chapter 10 of McDonald)

Copyright 2003 Pearson Education, Inc. Slide 08-1 Institutional Finance 08: Dynamic Arbitrage to Replicate Non-linear Payoffs Binomial Option Pricing: Basics (Chapter 10 of McDonald) Originally prepared

### Black-Scholes Equation for Option Pricing

Black-Scholes Equation for Option Pricing By Ivan Karmazin, Jiacong Li 1. Introduction In early 1970s, Black, Scholes and Merton achieved a major breakthrough in pricing of European stock options and there

### Naïve Bayes Classifier And Profitability of Options Gamma Trading

Naïve Bayes Classifier And Profitability of Options Gamma Trading HYUNG SUP LIM hlim2@stanford.edu Abstract At any given time during the lifetime of financial options, one can set up a variety of delta-neutral

### 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

### UNDERSTANDING THE INDEPENDENT-SAMPLES t TEST

UNDERSTANDING The independent-samples t test evaluates the difference between the means of two independent or unrelated groups. That is, we evaluate whether the means for two independent groups are significantly

### 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

### Implied Volatility Skews in the Foreign Exchange Market. Empirical Evidence from JPY and GBP: 1997-2002

Implied Volatility Skews in the Foreign Exchange Market Empirical Evidence from JPY and GBP: 1997-2002 The Leonard N. Stern School of Business Glucksman Institute for Research in Securities Markets Faculty

### 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

### 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,

### The Performance of Option Trading Software Agents: Initial Results

The Performance of Option Trading Software Agents: Initial Results Omar Baqueiro, Wiebe van der Hoek, and Peter McBurney Department of Computer Science, University of Liverpool, Liverpool, UK {omar, wiebe,

### 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.

### 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

### THE RELATIONSHIP BETWEEN WORKING CAPITAL MANAGEMENT AND DIVIDEND PAYOUT RATIO OF FIRMS LISTED IN NAIROBI SECURITIES EXCHANGE

International Journal of Economics, Commerce and Management United Kingdom Vol. III, Issue 11, November 2015 http://ijecm.co.uk/ ISSN 2348 0386 THE RELATIONSHIP BETWEEN WORKING CAPITAL MANAGEMENT AND DIVIDEND

### 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

### Application of options in hedging of crude oil price risk

AMERICAN JOURNAL OF SOCIAL AND MANAGEMEN SCIENCES ISSN rint: 156-154, ISSN Online: 151-1559 doi:1.551/ajsms.1.1.1.67.74 1, ScienceHuβ, http://www.scihub.org/ajsms Application of options in hedging of crude

### 1. Volatility Index. 2. India VIX* 3. India VIX :: computation methodology

1. Volatility Index Volatility Index is a measure of market s expectation of volatility over the near term. Usually, during periods of market volatility, market moves steeply up or down and the volatility

### ACCOUNTING AND FINANCE DIVISION (www.accountingandfinance.stir.ac.uk)

ACCOUNTING AND FINANCE DIVISION (www.accountingandfinance.stir.ac.uk) MSc in Finance MSc in Investment Analysis MSc in Banking and Finance MSc in Computing for Financial Markets MSc in International Accounting

### Credit volatility : Options and beyond

Credit volatility : Options and beyond Jerome.Brun@sgcib.com joint work with F. Rhazal, L. Luciani, T. Bounhar Société Générale Corporate and Investment Banking 25 June 2008, Credit Risk Workshop EVRY

### Option Pricing Basics

Option Pricing Basics Aswath Damodaran Aswath Damodaran 1 What is an option? An option provides the holder with the right to buy or sell a specified quantity of an underlying asset at a fixed price (called

### DETERMINING THE VALUE OF EMPLOYEE STOCK OPTIONS. Report Produced for the Ontario Teachers Pension Plan John Hull and Alan White August 2002

DETERMINING THE VALUE OF EMPLOYEE STOCK OPTIONS 1. Background Report Produced for the Ontario Teachers Pension Plan John Hull and Alan White August 2002 It is now becoming increasingly accepted that companies

### QUANTIZED INTEREST RATE AT THE MONEY FOR AMERICAN OPTIONS

QUANTIZED INTEREST RATE AT THE MONEY FOR AMERICAN OPTIONS L. M. Dieng ( Department of Physics, CUNY/BCC, New York, New York) Abstract: In this work, we expand the idea of Samuelson[3] and Shepp[,5,6] for

### Determination of Forward and Futures Prices. Chapter 5

Determination of Forward and Futures Prices Chapter 5 Fundamentals of Futures and Options Markets, 8th Ed, Ch 5, Copyright John C. Hull 2013 1 Consumption vs Investment Assets Investment assets are assets

### Lecture 7: Bounds on Options Prices Steven Skiena. http://www.cs.sunysb.edu/ skiena

Lecture 7: Bounds on Options Prices Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena Option Price Quotes Reading the

### 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

### Understanding Volatility

Options Trading Forum October 2 nd, 2002 Understanding Volatility Sheldon Natenberg Chicago Trading Co. 440 S. LaSalle St. Chicago, IL 60605 (312) 863-8004 shellynat@aol.com exercise price time to expiration

### 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

### RISKY LOSS DISTRIBUTIONS AND MODELING THE LOSS RESERVE PAY-OUT TAIL

Review of Applied Economics, Vol. 3, No. 1-2, (2007) : 1-23 RISKY LOSS DISTRIBUTIONS AND MODELING THE LOSS RESERVE PAY-OUT TAIL J. David Cummins *, James B. McDonald ** & Craig Merrill *** Although an

Frequently Asked Questions on Derivatives Trading At NSE NATIONAL STOCK EXCHANGE OF INDIA LIMITED QUESTIONS & ANSWERS 1. What are derivatives? Derivatives, such as futures or options, are financial contracts

### 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

### Session IX: Lecturer: Dr. Jose Olmo. Module: Economics of Financial Markets. MSc. Financial Economics

Session IX: Stock Options: Properties, Mechanics and Valuation Lecturer: Dr. Jose Olmo Module: Economics of Financial Markets MSc. Financial Economics Department of Economics, City University, London Stock

### Stock Market Volatility during the 2008 Financial Crisis

Stock Market Volatility during the 2008 Financial Crisis Kiran Manda * The Leonard N. Stern School of Business Glucksman Institute for Research in Securities Markets Faculty Advisor: Menachem Brenner April

### TABLE OF CONTENTS. A. Put-Call Parity 1 B. Comparing Options with Respect to Style, Maturity, and Strike 13

TABLE OF CONTENTS 1. McDonald 9: "Parity and Other Option Relationships" A. Put-Call Parity 1 B. Comparing Options with Respect to Style, Maturity, and Strike 13 2. McDonald 10: "Binomial Option Pricing:

### The Valuation of Currency Options

The Valuation of Currency Options Nahum Biger and John Hull Both Nahum Biger and John Hull are Associate Professors of Finance in the Faculty of Administrative Studies, York University, Canada. Introduction

### How to Calculate Present Values

How to Calculate Present Values Michael Frantz, 2010-09-22 Present Value What is the Present Value The Present Value is the value today of tomorrow s cash flows. It is based on the fact that a Euro tomorrow

### ON DETERMINANTS AND SENSITIVITIES OF OPTION PRICES IN DELAYED BLACK-SCHOLES MODEL

ON DETERMINANTS AND SENSITIVITIES OF OPTION PRICES IN DELAYED BLACK-SCHOLES MODEL A. B. M. Shahadat Hossain, Sharif Mozumder ABSTRACT This paper investigates determinant-wise effect of option prices when

### Chapter 13 The Black-Scholes-Merton Model

Chapter 13 The Black-Scholes-Merton Model March 3, 009 13.1. The Black-Scholes option pricing model assumes that the probability distribution of the stock price in one year(or at any other future time)

### Institute of Actuaries of India Subject CT3 Probability and Mathematical Statistics

Institute of Actuaries of India Subject CT3 Probability and Mathematical Statistics For 2015 Examinations Aim The aim of the Probability and Mathematical Statistics subject is to provide a grounding in

### 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.

### Equity derivative strategy 2012 Q1 update and Trading Volatility

Equity derivative strategy 212 Q1 update and Trading Volatility Colin Bennett (+34) 91 28 9356 cdbennett@gruposantander.com 1 Contents VOLATILITY TRADING FOR DIRECTIONAL INVESTORS Call overwriting Protection

### Figure S9.1 Profit from long position in Problem 9.9

Problem 9.9 Suppose that a European call option to buy a share for \$100.00 costs \$5.00 and is held until maturity. Under what circumstances will the holder of the option make a profit? Under what circumstances

### 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

### A Day in the Life of a Trader

Siena, April 2014 Introduction 1 Examples of Market Payoffs 2 3 4 Sticky Smile e Floating Smile 5 Examples of Market Payoffs Understanding risk profiles of a payoff is conditio sine qua non for a mathematical

### Assessing Credit Risk for a Ghanaian Bank Using the Black- Scholes Model

Assessing Credit Risk for a Ghanaian Bank Using the Black- Scholes Model VK Dedu 1, FT Oduro 2 1,2 Department of Mathematics, Kwame Nkrumah University of Science and Technology, Kumasi, Ghana. Abstract

### A Study of information asymmetry using Bid-Ask spread on firm value: evidence from Tehran Stock Exchange

International Research Journal of Applied and Basic Sciences 2013 Available online at www.irjabs.com ISSN 2251-838X / Vol, 4 (9): 2872-2876 Science Explorer Publications A Study of information asymmetry

### An Introduction to Exotic Options

An Introduction to Exotic Options Jeff Casey Jeff Casey is entering his final semester of undergraduate studies at Ball State University. He is majoring in Financial Mathematics and has been a math tutor

### 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:

### I.e., the return per dollar from investing in the shares from time 0 to time 1,

XVII. SECURITY PRICING AND SECURITY ANALYSIS IN AN EFFICIENT MARKET Consider the following somewhat simplified description of a typical analyst-investor's actions in making an investment decision. First,

### IP Valuation. WIPO Workshop on Innovation, Intellectual Asset Management and Successful Technology Licensing: Wealth Creation in the Arab Region

IP Valuation WIPO Workshop on Innovation, Intellectual Asset Management and Successful Technology Licensing: Wealth Creation in the Arab Region Muscat, Oman, December 12 and 13, 2011 Topics Intangibles

### Percent, Sales Tax, & Discounts

Percent, Sales Tax, & Discounts Many applications involving percent are based on the following formula: Note that of implies multiplication. Suppose that the local sales tax rate is 7.5% and you purchase

### What s Wrong with Multiplying by the Square Root of Twelve

What s Wrong with Multiplying by the Square Root of Twelve Paul D. Kaplan, Ph.D., CFA Director of Research Morningstar Canada January 2013 2013 Morningstar, Inc. All rights reserved. The information in

### CHAPTER 13 SIMPLE LINEAR REGRESSION. Opening Example. Simple Regression. Linear Regression

Opening Example CHAPTER 13 SIMPLE LINEAR REGREION SIMPLE LINEAR REGREION! Simple Regression! Linear Regression Simple Regression Definition A regression model is a mathematical equation that descries the