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

Save this PDF as:

Size: px
Start display at page:

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

## Transcription

2 An option can either be American or European. American options can be exercised at any time up to the expiration date, whereas European can be exercised only on the expiration date itself. II. METHODOLOGY A. Option Pricing Models The main idea underlying option pricing models is that a credit default will occur when the economic value of the borrower s assets falls below the economic value of its debt. A loan taken out by the company or individual can be associated with the purchase of an option which would allow the equity investors or the individual to satisfy the claims of the debt lenders by handing over the company or the individual s assets instead of repaying the debt in the case of default [3]. The price the company pays for this option corresponds to the risk premium included in the interest on the loan. The probability that the option will be exercised becomes the probability of default. B. Black Scholes Model The Black-Scholes Merton formula [2] for prices of a European call option and put option on a non-dividend paying stock is K is the strike price of the option r is the continuously compounded risk-free rate is the stock price volatility T is the time to maturity of the option. C. Default Probability (Firm s) We link the Black-Scholes Model [2] to the to the debt obligations of a firm in the form of zero coupon bonds at time T years: is the asset value, D is the debt obligation, T is the time to maturity, is the asset volatility, is the riskfree interest rate are respectively the probabilities relative to the exercise of the option and debt repayment From the basic accounting equation Implying that the value of the debt today becomes: And the value of the company s debt at time T is The Black-Scholes Merton (BSM) default probability is simply the probability that the market value of firm s assets is less than the face value of the firm s liabilities, D, at time T (i.e. ) [4]. With risk neutral valuation framework or risk neutral world, the expected value (E) of a call and put respectively at option at maturity is From the BSM model equity can be viewed as a call option on the value of the firm s assets. The expected value of the firm s equity at time T is N(x) is the cumulative probability distribution function for a standardized normal distribution is the stock price at time And the value of the equity today is (9) 14

3 And the asset volatility, can be calculated from is the standard normal distribution. Assume that a loan defaults if the value of the borrower s assets at the loan maturity T falls below the contractual amount B payable. Let s denote to be the value of the i-th borrower s assets and assume that it follows a stochastic process described by the equation the volatility of equity of listed companies can be calculated form historical prices of stock Equity holders (shareholders) of a company have a call option on the assets value. That is equity holders have the right but not the obligation to pay off the debt holders (liabilities of the firm). Debt holders on the other hand have a put option on the firm s assets, which is implied in the limited liability of the company. The put option in this case is the right but not the obligation to sell the firm s assets in case of default. D. Default Probability (Clients ) Default probabilities of clients of banks can be modeled to follow the BSM default probability. We consider a random variable X which follows a normal distribution with mean µ and variance where X is the loss incurred due to customers default on loans then more concisely we can deduce that losses follow a normal distribution where and are the instantaneous mean and variance, respectively and is an increment of the Wiener process. B which is the total contractual amount payable at maturity T, is assumed to be prepaid at time t >0 The logarithm of the total asset value at time normally distributed with the mean And variance is given by The assets value at Time T is is a standard normal random variable. The probability of default denoted p of the i-th loan is then is With its probability distribution function given by Then it follows that A loan will be in default if the value of the borrower s assets at the time of maturity of the loan at time T is less than the contractual amount payable It follows that the probability of default is given by We denote p the probability of default by It follows that 15

4 Therefore N is the cumulative normal distribution function (. III. DATA ANALYSIS AND RESULTS The analysis focused on a bank which is assumed to be a listed member of the Ghana Stock Exchange. The source of data used was from the bank s financial statement. The following assumptions were made to evaluate the default probability. 1. The bank stock is not a dividend paying stock 2. Risk neutral valuation 3. There is no arbitrage opportunity 4. There are no transaction costs. 5. A perfect market is assumed 6. The debt of the firm is considered to be in the form of zero coupon bonds. Assuming from the bank s financial statement, that the current market value of its total asset ( is GHȼ516,632,000 and that it has an impending debt obligation ( ) of GHȼ465,546,000 in five years time, the BSM default model was used to evaluate the default probability of the bank. The bank s asset volatility was evaluated by means of equation (10).The equity volatility of the bank was estimated from the assumed 15 days trading information of the company s stock on the stock exchange. The analysis focused on a bank which is assumed to be a listed member of the Ghana Stock Exchange. The source of data used was from the bank s financial statement. The following assumptions were made to evaluate the default probability. 1. The bank stock is not a dividend paying stock 2. Risk neutral valuation 3. There is no arbitrage opportunity 4. There are no transaction costs A perfect market is assumed 6. The debt of the firm is considered to be in the form of zero coupon bonds. Assuming from the bank s financial statement, the current market value of its total asset ( is GHȼ516,632,000 and has an impending debt obligation ( ) of GHȼ465,546,000 in five years time. The BSM default model was used to evaluate the default probability of the bank [5]. The bank s asset volatility was evaluated by means of equation (10).The equity volatility of the bank was estimated from the assumed 15 days trading information of the company s stock on the stock exchange which is displayed in Table 1. Day Thus, TABLE I DAILY SHARE PRICES Share price(ghȼ)

5 The study assumed the bank to have a total of 302,039,348 shares and therefore with a current market value of the equity to be GHȼ78,530, The bank s volatility of equity was evaluated using equation (11) yielding Expected share price of the company The BSM default probability of the bank is thus given by This implies that the bank has a virtually zero probability of defaulting on its current long term debt obligations. From equation (5) The Bank s equity five years from now is given by And IV. DISCUSSION The discussion focused on a simple situation. If the bank wants to borrow from external sources in the form of debt financing, the lender s purchases a claim on the firm's assets, and thereby becomes a partial owner of the company. The value of bank s assets increases by the amount received. The new total value of the firm's assets is equal to the value of the stock (equity) and the value of the debt. With time, the market value of the bank s assets will change as the market perception of the future earning power of the company changes. These changes obviously involve considerable uncertainty. What concerns the lender is the market value of bank s assets when the term of the debt elapses. Three situations are possible. 1. The market value of the bank s asset is greater than the face value of the debt 2. The market value of the bank s asset is at least that the debt obligation payable 3. The market value of the bank s asset value is less than face value of the debt In the first situation, the equity holders of the bank would have a call option on the asset value. That is the Equity holders would have the right but not the obligation to pay off the company s debt holders. Clearly the (GHȼ ). With an expected equity payoff The expected payoff of the company s equity at time T (5 years) = Dividing the expected payoff at time T=5 years by the total shareholder base of the company yields the expected share price of the company within five years. In such a situation the stockholders will pay off the debt since an equity value of GHȼ386,236,167.5 is an indication of the bank s stock performance. The shareholders can honor the debt obligation since the company s asset value is enough for them to do so.if the Bank however does not have enough cash to settle the debt obligation, the stockholders can raise it by selling a part of the assets at their market value. It will be in the interest of the Stockholders of the bank to pay its lenders, otherwise the lenders would force the bank to bankruptcy. The Lenders on the other hand have a put option on the assets value which is implied in the limited liability of the bank. 17

6 The put option in this sense becomes the right but not the obligation to sell off Bank s assets which causes the stockholders to lose control of the firm. In the situation that the market value of the bank s assets falls below the amount due the lender s, then the bank cannot repay its lenders since the bank would have an expected equity payoff The bank cannot raise additional cash be it by equity finance or debt finance since no other lender would refinance the loan, because that would mean taking over the loss from the original lender. It is also not possible to raise additional equity, since the banks stock would be nonperforming and an expected equity payoff of is an indication of the stock s non-performance. The company has to declare bankruptcy. Because of the limited liability of the shareholder the expected payoff of the shareholders.the stockholders of bank get nothing, while the lenders take over the assets. The lenders will thus realize a loss equal to the difference between the face value of the debt and the market value of assets. V. CONCLUSION In addition to the many useful applications of the Black- Scholes Merton model for evaluating or quantifying credit risk, one of its strongest attributes is its ability to resonate with two key stakeholder groups; shareholders and debt holders. The model predicted effectively the company s probability of default without the use of enormous data which makes it a mathematically elegant model for banks to work with. It is a high time Ghanaian indigenous banks considered the use of option pricing methodologies in its day to day credit risk management. The result above has shown strong and robust means of evaluating the bank s probability of default and hence the corresponding credit risk. Acknowledgment The authors are very grateful to Messieurs Bismark Owusu Tawiah, Collins Knight Arthur and Miss Victoria Naaki Armah for research assistance. REFERENCES [1] Basel Committee on Banking Supervision (1999) Principles for the Management of Credit Risk Published by the Risk Management Group of the Basel Committee on Banking Supervision [2] Hull, J.C (2009) Options, Futures and Other Derivatives, Seventh Edition, Pearson Prentice Hall, New Jersey [3] Kirmße, S. (1996): Die Bepreisung und Steuerung von Ausfallrisiken im Firmenkundengeschäft der Kreditinstitute, Frankfurt am Main 1996 [4] [5] Glasserman, P (2000). Probability Models of Credit Risk, Columbia Business School, pp 2 43 [6] World Bank (2005) Credit Risk and Emergence of Credit Derivatives [7] Alexander J. McNeil, Rüdiger Frey, & Paul Embrechts (2006) Quantitative Risk Management: Concepts, Techniques, and Tools, Princeton Series in Finance Darrell Duffie and Stephen Schaefer, Series Editors 18

### 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

### 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 5 Financial Forwards and Futures

Chapter 5 Financial Forwards and Futures Question 5.1. Four different ways to sell a share of stock that has a price S(0) at time 0. Question 5.2. Description Get Paid at Lose Ownership of Receive Payment

### 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

### Example 1. Consider the following two portfolios: 2. Buy one c(s(t), 20, τ, r) and sell one c(s(t), 10, τ, r).

Chapter 4 Put-Call Parity 1 Bull and Bear Financial analysts use words such as bull and bear to describe the trend in stock markets. Generally speaking, a bull market is characterized by rising prices.

### 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.

### TPPE17 Corporate Finance 1(5) SOLUTIONS RE-EXAMS 2014 II + III

TPPE17 Corporate Finance 1(5) SOLUTIONS RE-EXAMS 2014 II III Instructions 1. Only one problem should be treated on each sheet of paper and only one side of the sheet should be used. 2. The solutions folder

### 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

### Finance 2 for IBA (30J201) F. Feriozzi Re-sit exam June 18 th, 2012. Part One: Multiple-Choice Questions (45 points)

Finance 2 for IBA (30J201) F. Feriozzi Re-sit exam June 18 th, 2012 Part One: Multiple-Choice Questions (45 points) Question 1 Assume that capital markets are perfect. Which of the following statements

### Jorge Cruz Lopez - Bus 316: Derivative Securities. Week 11. The Black-Scholes Model: Hull, Ch. 13.

Week 11 The Black-Scholes Model: Hull, Ch. 13. 1 The Black-Scholes Model Objective: To show how the Black-Scholes formula is derived and how it can be used to value options. 2 The Black-Scholes Model 1.

### Betting on Volatility: A Delta Hedging Approach. Liang Zhong

Betting on Volatility: A Delta Hedging Approach Liang Zhong Department of Mathematics, KTH, Stockholm, Sweden April, 211 Abstract In the financial market, investors prefer to estimate the stock price

### 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

### 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

### 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

### 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

### 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

### 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,

### 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

### 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

### 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

### 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

### 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

### A Genetic Algorithm to Price an European Put Option Using the Geometric Mean Reverting Model

Applied Mathematical Sciences, vol 8, 14, no 143, 715-7135 HIKARI Ltd, wwwm-hikaricom http://dxdoiorg/11988/ams144644 A Genetic Algorithm to Price an European Put Option Using the Geometric Mean Reverting

### Week 12. Options on Stock Indices and Currencies: Hull, Ch. 15. Employee Stock Options: Hull, Ch. 14.

Week 12 Options on Stock Indices and Currencies: Hull, Ch. 15. Employee Stock Options: Hull, Ch. 14. 1 Options on Stock Indices and Currencies Objective: To explain the basic asset pricing techniques used

### LOCKING IN TREASURY RATES WITH TREASURY LOCKS

LOCKING IN TREASURY RATES WITH TREASURY LOCKS Interest-rate sensitive financial decisions often involve a waiting period before they can be implemen-ted. This delay exposes institutions to the risk that

### Hedging with Futures and Options: Supplementary Material. Global Financial Management

Hedging with Futures and Options: Supplementary Material Global Financial Management Fuqua School of Business Duke University 1 Hedging Stock Market Risk: S&P500 Futures Contract A futures contract on

### 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

### 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

### Valuation of the Surrender Option Embedded in Equity-Linked Life Insurance. Brennan Schwartz (1976,1979) Brennan Schwartz

Valuation of the Surrender Option Embedded in Equity-Linked Life Insurance Brennan Schwartz (976,979) Brennan Schwartz 04 2005 6. Introduction Compared to traditional insurance products, one distinguishing

### EC372 Bond and Derivatives Markets Topic #5: Options Markets I: fundamentals

EC372 Bond and Derivatives Markets Topic #5: Options Markets I: fundamentals R. E. Bailey Department of Economics University of Essex Outline Contents 1 Call options and put options 1 2 Payoffs on options

### 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),

### 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

### One Period Binomial Model

FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 One Period Binomial Model These notes consider the one period binomial model to exactly price an option. We will consider three different methods of pricing

### Forwards and Futures

Prof. Alex Shapiro Lecture Notes 16 Forwards and Futures I. Readings and Suggested Practice Problems II. Forward Contracts III. Futures Contracts IV. Forward-Spot Parity V. Stock Index Forward-Spot Parity

### Article from: Risk Management. June 2009 Issue 16

Article from: Risk Management June 2009 Issue 16 CHAIRSPERSON S Risk quantification CORNER Structural Credit Risk Modeling: Merton and Beyond By Yu Wang The past two years have seen global financial markets

### 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

### Pricing Forwards and Swaps

Chapter 7 Pricing Forwards and Swaps 7. Forwards Throughout this chapter, we will repeatedly use the following property of no-arbitrage: P 0 (αx T +βy T ) = αp 0 (x T )+βp 0 (y T ). Here, P 0 (w T ) is

### 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

### 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

### 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

### t = 1 2 3 1. Calculate the implied interest rates and graph the term structure of interest rates. t = 1 2 3 X t = 100 100 100 t = 1 2 3

MØA 155 PROBLEM SET: Summarizing Exercise 1. Present Value [3] You are given the following prices P t today for receiving risk free payments t periods from now. t = 1 2 3 P t = 0.95 0.9 0.85 1. Calculate

### 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:

### 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

### 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

### Chapter 21 Valuing Options

Chapter 21 Valuing Options Multiple Choice Questions 1. Relative to the underlying stock, a call option always has: A) A higher beta and a higher standard deviation of return B) A lower beta and a higher

### 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

### Option Pricing Theory and Applications. Aswath Damodaran

Option Pricing Theory and Applications Aswath Damodaran 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

### 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

### FUNDING INVESTMENTS FINANCE 238/738, Spring 2008, Prof. Musto Class 6 Introduction to Corporate Bonds

FUNDING INVESTMENTS FINANCE 238/738, Spring 2008, Prof. Musto Class 6 Introduction to Corporate Bonds Today: I. Equity is a call on firm value II. Senior Debt III. Junior Debt IV. Convertible Debt V. Variance

### 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

### 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

### 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:

### 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

### Ind AS 102 Share-based Payments

Ind AS 102 Share-based Payments Mayur Ankolekar Consulting Actuary Current Issues in Pension Seminar at Mumbai The Institute of Actuaries of India August 21, 2015 Page 1 Session Objectives 1. To appreciate

### 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

### Session X: Lecturer: Dr. Jose Olmo. Module: Economics of Financial Markets. MSc. Financial Economics. Department of Economics, City University, London

Session X: Options: Hedging, Insurance and Trading Strategies Lecturer: Dr. Jose Olmo Module: Economics of Financial Markets MSc. Financial Economics Department of Economics, City University, London Option

### Likewise, the payoff of the better-of-two note may be decomposed as follows: Price of gold (US\$/oz) 375 400 425 450 475 500 525 550 575 600 Oil price

Exchange Options Consider the Double Index Bull (DIB) note, which is suited to investors who believe that two indices will rally over a given term. The note typically pays no coupons and has a redemption

### OPTIONS MARKETS AND VALUATIONS (CHAPTERS 16 & 17)

OPTIONS MARKETS AND VALUATIONS (CHAPTERS 16 & 17) WHAT ARE OPTIONS? Derivative securities whose values are derived from the values of the underlying securities. Stock options quotations from WSJ. A call

### 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

### CHAPTER 20: OPTIONS MARKETS: INTRODUCTION

CHAPTER 20: OPTIONS MARKETS: INTRODUCTION PROBLEM SETS 1. Options provide numerous opportunities to modify the risk profile of a portfolio. The simplest example of an option strategy that increases risk

### Lecture 3: Put Options and Distribution-Free Results

OPTIONS and FUTURES Lecture 3: Put Options and Distribution-Free Results Philip H. Dybvig Washington University in Saint Louis put options binomial valuation what are distribution-free results? option

### Chapter 7: Capital Structure: An Overview of the Financing Decision

Chapter 7: Capital Structure: An Overview of the Financing Decision 1. Income bonds are similar to preferred stock in several ways. Payment of interest on income bonds depends on the availability of sufficient

### CHAPTER 20: OPTIONS MARKETS: INTRODUCTION

CHAPTER 20: OPTIONS MARKETS: INTRODUCTION 1. Cost Profit Call option, X = 95 12.20 10 2.20 Put option, X = 95 1.65 0 1.65 Call option, X = 105 4.70 0 4.70 Put option, X = 105 4.40 0 4.40 Call option, X

### 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

### VALUE 11.125%. \$100,000 2003 (=MATURITY

NOTES H IX. How to Read Financial Bond Pages Understanding of the previously discussed interest rate measures will permit you to make sense out of the tables found in the financial sections of newspapers

### 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 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

### CHAPTER 20. Hybrid Financing: Preferred Stock, Warrants, and Convertibles

CHAPTER 20 Hybrid Financing: Preferred Stock, Warrants, and Convertibles 1 Topics in Chapter Types of hybrid securities Preferred stock Warrants Convertibles Features and risk Cost of capital to issuers

### Manual for SOA Exam FM/CAS Exam 2.

Manual for SOA Exam FM/CAS Exam 2. Chapter 7. Derivatives markets. c 2009. Miguel A. Arcones. All rights reserved. Extract from: Arcones Manual for the SOA Exam FM/CAS Exam 2, Financial Mathematics. Fall

### 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

1.0 FINANCING PRINCIPLES Module 1: Corporate Finance and the Role of Venture Capital Financing Financing Principles 1.01 Introduction to Financing Principles 1.02 Capitalization of a Business 1.03 Capital

### 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

### 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

### 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.

### University of Essex. Term Paper Financial Instruments and Capital Markets 2010/2011. Konstantin Vasilev Financial Economics Bsc

University of Essex Term Paper Financial Instruments and Capital Markets 2010/2011 Konstantin Vasilev Financial Economics Bsc Explain the role of futures contracts and options on futures as instruments

### What Do Short-Term Liquidity Ratios Measure? What Is Working Capital? How Is the Current Ratio Calculated? How Is the Quick Ratio Calculated?

What Do Short-Term Liquidity Ratios Measure? What Is Working Capital? HOCK international - 2004 1 HOCK international - 2004 2 How Is the Current Ratio Calculated? How Is the Quick Ratio Calculated? HOCK

### RISK DISCLOSURE STATEMENT

RISK DISCLOSURE STATEMENT You should note that there are significant risks inherent in investing in certain financial instruments and in certain markets. Investment in derivatives, futures, options and

### CHAPTER 22: FUTURES MARKETS

CHAPTER 22: FUTURES MARKETS 1. a. The closing price for the spot index was 1329.78. The dollar value of stocks is thus \$250 1329.78 = \$332,445. The closing futures price for the March contract was 1364.00,

### 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

### CFA Level -2 Derivatives - I

CFA Level -2 Derivatives - I EduPristine www.edupristine.com Agenda Forwards Markets and Contracts Future Markets and Contracts Option Markets and Contracts 1 Forwards Markets and Contracts 2 Pricing and

### 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

### CHAPTER 22: FUTURES MARKETS

CHAPTER 22: FUTURES MARKETS PROBLEM SETS 1. There is little hedging or speculative demand for cement futures, since cement prices are fairly stable and predictable. The trading activity necessary to support

### 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

### Option Pricing Applications in Valuation!

Option Pricing Applications in Valuation! Equity Value in Deeply Troubled Firms Value of Undeveloped Reserves for Natural Resource Firm Value of Patent/License 73 Option Pricing Applications in Equity

### Option Pricing Beyond Black-Scholes Dan O Rourke

Option Pricing Beyond Black-Scholes Dan O Rourke January 2005 1 Black-Scholes Formula (Historical Context) Produced a usable model where all inputs were easily observed Coincided with the introduction

### FNCE 301, Financial Management H Guy Williams, 2006

Stock Valuation Stock characteristics Stocks are the other major traded security (stocks & bonds). Options are another traded security but not as big as these two. - Ownership Stockholders are the owner

### Jung-Soon Hyun and Young-Hee Kim

J. Korean Math. Soc. 43 (2006), No. 4, pp. 845 858 TWO APPROACHES FOR STOCHASTIC INTEREST RATE OPTION MODEL Jung-Soon Hyun and Young-Hee Kim Abstract. We present two approaches of the stochastic interest

### 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.

### Note 8: Derivative Instruments

Note 8: Derivative Instruments Derivative instruments are financial contracts that derive their value from underlying changes in interest rates, foreign exchange rates or other financial or commodity prices

### Forwards, Swaps and Futures

IEOR E4706: Financial Engineering: Discrete-Time Models c 2010 by Martin Haugh Forwards, Swaps and Futures These notes 1 introduce forwards, swaps and futures, and the basic mechanics of their associated

### Introduction, Forwards and Futures

Introduction, Forwards and Futures Liuren Wu Zicklin School of Business, Baruch College Fall, 2007 (Hull chapters: 1,2,3,5) Liuren Wu Introduction, Forwards & Futures Option Pricing, Fall, 2007 1 / 35

### 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

### MTH6120 Further Topics in Mathematical Finance Lesson 2

MTH6120 Further Topics in Mathematical Finance Lesson 2 Contents 1.2.3 Non-constant interest rates....................... 15 1.3 Arbitrage and Black-Scholes Theory....................... 16 1.3.1 Informal

### Lecture 2 Bond pricing. Hedging the interest rate risk

Lecture 2 Bond pricing. Hedging the interest rate risk IMQF, Spring Semester 2011/2012 Module: Derivatives and Fixed Income Securities Course: Fixed Income Securities Lecturer: Miloš Bo ović Lecture outline

### EC247 FINANCIAL INSTRUMENTS AND CAPITAL MARKETS TERM PAPER

EC247 FINANCIAL INSTRUMENTS AND CAPITAL MARKETS TERM PAPER NAME: IOANNA KOULLOUROU REG. NUMBER: 1004216 1 Term Paper Title: Explain what is meant by the term structure of interest rates. Critically evaluate

### Rigensis Bank AS Information on the Characteristics of Financial Instruments and the Risks Connected with Financial Instruments

Rigensis Bank AS Information on the Characteristics of Financial Instruments and the Risks Connected with Financial Instruments Contents 1. Risks connected with the type of financial instrument... 2 Credit

### International Financial Reporting Standard 2

International Financial Reporting Standard 2 Share-based Payment OBJECTIVE 1 The objective of this IFRS is to specify the financial reporting by an entity when it undertakes a share-based payment transaction.

Chapter 6 Interest rates and Bond Valuation 2012 Pearson Prentice Hall. All rights reserved. 4-1 Interest Rates and Required Returns: Interest Rate Fundamentals The interest rate is usually applied to