1 Black-Scholes-Merton approach merits and shortcomings Emilia Matei EC372 Term Paper. Topic 3
2 1. Introduction The Black-Scholes and Merton method of modelling derivatives prices was first introduced in 1973, by the Nobel Prize winners Black, Scholes (1973) and Merton (1973), after which the model is named. Essentially, the Black-Scholes-Merton (BSM) approach shows how the price of an option contract can be determined by using a simple formula of the underlying asset s price and its volatility, the exercise price price of the underlying asset that the contract stipulates time to maturity of the contract and the risk-free interest rate prevailed in the market. Its discovery has highly influenced the pricing and hedging of options since the 1970 s, offering also the possibility of extending the approach to other derivative instruments that have similar characteristics to options. The present paper is intended to offer an assessment of this approach, by discussing its economic and empirical validity. An option is a financial contract, the value of which is derived from the price of an underlying asset, hence the title of derivatives attributable to these types of contracts. The holder of such a contract purchases the right, but not the obligation, to buy or sell the asset specified in the option contract the underlying. The classification of options varies according to their scope and flexibility in the terms of their fulfilment. Hence, a call/ put option owner has the possibility of buying/ selling the underlying asset at the specified contract price and date, for a specified fee the option premium. This is the price modelled using the BSM approach, which is straightforward in implementation for European-style options. American options can be exercised any time prior to or at maturity, whereas European options can be executed only at the maturity date, hence the simplicity in modelling European option prices. The second
3 participating party to the option contract is the option writer, who must satisfy the terms of the agreement to buy or sell the asset, should the option holder decide to exercise his right. The paper is organised as follows. Section 2 will explain the role of the assumptions underlying the BSM model. Section 3 will provide an overview of the meaning of volatility and how it is modelled and will discuss the main critiques of the BSM approach. The following section will summarize other general, yet important applications of this model. Section 5 concludes on the main aspects of the assessment of the BSM method. 2. Main assumptions of BSM approach The BSM approach offers a model of general equilibrium of options prices, which is valid only under a certain set of assumptions. Some of the assumptions are necessary for every model of option price determination, whereas the BSM model requires two additional assumptions regarding the trading of options and the underlying asset s price. The BSM model is based on the following assumptions: 1. Frictionless markets, which implies that the cost of trading is zero, and there are no legal restrictions on trading in the options and in the underlying asset, or on short-selling the asset. Moreover, perfect divisibility of the asset ensures that one unit can be readily fragmented into tradable parts.
4 2. The underlying asset offers no additional payments during the life of the option (for example, stock that pays no dividends and is protected against stock splits) Unlimited borrowing and lending is possible at a constant risk-free interest rate. 4. The arbitrage principle is at work no profits can be made from an investment with zero initial outlay and zero risk. 5. Investor preference assumption given a constant variance of the underlying asset s price, investors will always choose more wealth to less. 6. Continuous trading in the asset markets. 7. The logarithm of the price of the underlying asset has a normal distribution and the changes in this price are described by a geometric Brownian motion (a samplecontinuous process). Given these assumptions, Black and Scholes (1973) arrived at the following formulas 2 for a European call option premium, c, and a European put option premium, p: (1) (2) with (3) and (4) 1 Black and Scholes (1973) use this assumption because option prices change when there is a change in the payout policy of a company; however, Merton (1973) proposes an adjustment in the reasoning of the model to account for dividend payments during the life of the option. 2 Source: Hull (Introduction to Futures and Options Markets, 2011, Seventh Edition, Pearson Education).
5 The determinants of the prices for the European call and put options are: the stock price -, the strike price - X, time until option expires (time to maturity) - T, the risk-free interest rate - r and the volatility of the stock price (measured by its standard deviation) -. The N(x) function represents the cumulative probability function for a variable with zero mean and constant variance of one, following a standard normal distribution N(0,1). N(x) essentially gives the probability that a variable with these mean and variance properties will be less than x. The first five assumptions are also required for the Capital Asset Pricing Model of Sharpe (1964) and Lintner (1965), being viewed as standard assumptions of asset price derivation. The innovation of the BSM comes from assumptions 6 and 7, which are essential to the model s predictions. Moreover, it is important to note that the price of a European option as derived in the BSM model does not depend on the expected return and total supply of the underlying asset, and investors attitudes to risk. Hence, investors may have different expectations on the rate of return on the asset and the model will still holds. However, investors must agree that the geometric Brownian motion process best describes price fluctuations in the asset. In order for the predictions of the model to hold, the volatility term is valued the same by each investor. The factors that determine the option price, as they appear in the formulas (1), (2), (3) and (4), are computed with ease and a high degree of accuracy, except for the standard deviation term. The measurement difficulties of this variable stem from the fact that volatility is not an observed value, rather it is necessary to estimate it using information on the asset price movements.
6 The arbitrage principle is a core element in the economic validity of the model. As Merton (1998) describes the reasoning, in order to minimise the risk associated with issuing an option, it is possible to construct a replicating portfolio consisting of both risk-free and risky assets and whose payoffs perfectly reproduce the payoffs of the option. This is obtained through dynamic hedging which means that the weights in this portfolio are constantly amended until it consists of only the underlying asset of the option contract. Such perfect hedging is possible in the BSM model through the continuous trading and frictionless markets assumptions. Continuous trading implies that hedging is possible by trading in the option and in the underlying asset simultaneously. Otherwise, arbitrage opportunities arise and the option price is no longer determined by the formula derived in the model. The unrealistic nature of the model s assumptions would make dynamic hedging strategies impossible in practice. This implies that hedging options payoffs is risky. The capital asset pricing model shows that firmspecific risk can be eliminated through a well-diversified portfolio. Hence, by diversification, the return on the replicating portfolio yields market risk only and the option payoffs are successfully hedged. 3. Main critiques Whereas the main merit of the BSM approach is its simplicity in application, the BSM model was developed purely theoretical, leaving most of the literature on option pricing theory to question its assumptions: their unrealistic nature may prevent the model from producing accurate predictions when tested empirically. Another approach used in testing the theory is to compare the model s predictions on the option prices with the observed market prices. If the
7 predictions are constantly accurate, the BSM model would be the best model, regardless of its assumptions. The literature on these two approaches shows mixed results, of which the most prominent ones are discussed in the following. One of the most noted shortcomings of the model is related to the way the volatility term is computed. Measurement errors in the option price determinants, such as volatility, cause misleading predictions. In order to clearly show how this affects the model s predictions, a distinction between explicit and implicit volatility 3 is necessary. Explicit volatility is measured by the standard deviation of realised asset prices. In contrast, implicit volatility is that measure of volatility which satisfies the Black-Scholes formula. To be obtained, one would substitute the observed option price in the relevant formula (either for a call or for a put option), and compute the value of σ. The estimated volatility is forward-looking, based on expectations on the future and is used in evaluating the risk of holding an asset. More importantly to the analysis of the BSM model is the equality of the estimated volatilities from different option contracts written on the same underlying asset. If these equalities do not hold, there is evidence that volatility of the asset s price is not constant. It is found that implicit volatility patterns take many graphical forms, most observed being volatility smiles, smirks and skews. A study by Macbeth and Merville (1979) shows that the implied volatility varies whether the option considered is out or in the money and that the longer the time to maturity the less volatile the underlying asset price is. Black and Scholes (1972) themselves admit the drawbacks of their formula, more precisely the inaccurate measure of explicit volatility which overestimates the value of an option written on an asset with a high volatility of its return. 3 Source: Bailey, R. E. (The Economics of Financial Markets, 2005, Cambridge University Press)
8 Given that volatility varies over time, whether the asset price follows a geometric Brownian motion is a good approximation of reality is treated with scepticism in the literature. Bollerslev and Zhou (2007) use a model-free approach to account for time-varying volatility and find better predictive power than for the Black-Scholes model. Their argument is built on the difference between explicit and implicit volatilities of asset prices, which would be accounted by a risk premium. Asset indivisibility assumption is violated because very often option contracts are traded in blocks on organised exchanges, hence the impossibility of trading in a single option from that block. Furthermore, in practice, dynamic hedging is not feasible due to market frictions such as excessive transaction costs, and discontinuous trading. Hence the resulting portfolio is only an approximation of a replicating portfolio. Haug and Taleb (2010) argue the weakness of the Black-Scholes formula to jumps in the underlying asset s price and also to tail events. These cause the variance of the portfolio constructed using dynamic hedging to increase abruptly, thus experiencing enormous losses on this portfolio. In the light of these findings, the Black- Scholes approach can have high associated risks, which need to be accounted for by a modification in the analysis. The above mentioned difficulties in applying the model can result in discrepancies between the observed market prices of options and the prices predicted by the Black-Scholes formula. As a result, the literature on derivatives pricing has found additional relevant factors that determine the prices of option contracts. Garleanu, Pedersen and Poteshman (2009) study the implications of option demand on option prices and find that demand is a relevant predictor.
9 The BSM model has its shortcomings when it comes to accurately predicting option prices. However, the intuition used to construct it is relevant for the theory of option pricing as it provides a benchmark for assessing competing model. Moreover, it can be easily modified to account for different relaxations of assumptions, while in other aspects it can prove difficult to amend 4. Black and Scholes (1973) restricted their analysis to European style options, but the assumptions imposed can be relaxed so as to extend this theory, for example to the Americanstyle options. Merton (1973) shows that the Black-Scholes approach can be applied to American call options, provided that the underlying asset pays no dividends during the life of the option. He demonstrates the it is not profitable for an investor to exercise a call option on such assets prior to maturity, hence his argument also allows for varying interest rates over time; however, the crucial assumption of homogeneous beliefs of investors must still hold in order to derive the Black-Scholes formula. Moreover, Merton (1973) also demonstrates the Cox and Ross (1976) allow for discontinuous asset price movements, hence eliminating the geometric Brownian motion unrealistic assumption. 4. Alternative applications of the BSM approach The Black-Scholes-Merton analysis of option prices can be generally applied to contingent claims securities that have returns dependent on the returns of other assets. The contingent claims analysis is thus concerned with finding such a relationship and using it to determine the price of the contingent claim. This is the same reasoning applied in the BSM model. 4 This is the case of time-varying volatility, as argued by Bollerslev and Zhou (2011).
10 Merton (1998) outlines the numerous applications of the Black-Scholes formula that have evolved together with financial innovations. One of the significant applications of the BSM approach to option-like securities involved the pricing of insurance contracts such as loan guarantees and deposit insurance policies and is attributable to Merton (1977). In essence, the strategy of acquiring both a put option on an asset and the asset as well can be viewed as an insurance policy against losses that would result from a decline in the asset price. Similarly, loan guarantees and deposit insurance policies protect the owner from losses incurred in the event of default. Loan guarantees insure banks from losing the payments on the loans they issue should a borrower default. Deposit insurance, in a similar fashion, are guarantees that depositors will receive the full amount or some percentage of it in the case of a bank run. To make the similarity of deposit insurance and loan guarantees with a put option evident, the insurance company (or the government), writes a put option that gives its holder the right but not the obligation to sell the underlying asset: the deposit made with a bank where the holder is a depositor who wishes to protect himself should the bank default and the loan issued where a bank holds the option on the loan and will exercise it in the case that the borrower defaults on his payments. Merton (1977) uses the Black-Scholes formula to price the insurance contracts described above. Another application of the option pricing theory of BSM is to revolving credit agreements (or line of credit agreements, as they represent one of the ways in which banks offer credit), which are contracts similar to options in the sense that a company in need of finance for its projects signs an agreement with another company who is obliged to lend to it the amount needed should this be requested. The company which owns this call option makes a decision according
11 to the cost of borrowing that other banks impose, the aim of the company being to borrow at the lowest rate possible. Hawkins (1982) uses the theoretical set-up of BSM model to calculate the prices of revolving credit agreements. 5. Conclusion The option-pricing model developed by Black, Scholes and Merton in 1973 provides a straightforward way of computing the prices of option contracts, being widely used by traders since its publication. The main strengths discussed in are the following. By using the information provided in the option contract (which are observed values) and the estimated volatility of the underlying asset price (unobserved value) one can easily determine the option price through the Black-Scholes formulas. The study of the model s assumptions and predictions have enabled Merton (1977) along with other researchers to broaden its applicability to other contingent claims such as line-credit agreements and insurance contracts. As with any model of asset pricing, the weaknesses emerge from an unrealistic set of assumptions, which cause difficulties in estimation and evaluating the model s predictions. Given these shortcomings, the BSM approach is still one of the most accessible and relied-upon methods. Researchers have worked on amending the model to incorporate assumptions closer to reality and have discovered that by knowing the model s weaknesses, its application can still prove useful in analysing option prices.
12 Bibliography Bailey, R. E. (2005). The Economics of Financial Markets. United Kingdom: Cambridge University Press. Black, F., & Scholes, M. (1973). The Pricing of Options and Corporate Liabilities. The Journal of Political Economy, 81 (3), Black, F., & Scholes, M. (1972). The Valuation of Option Contracts and a Test of Market Efficiency. The Journal of Finance, 27 (2), Bollerslev, T., & Zhou, H. ( ). Expected Stock Returns and Variance Risk Premia. (unpublished) Retrieved April 2013, from Cox, J. C., & Ross, S. A. (1976). The Valuation of Options For Alternative Stochastic Processes. Journal of Financial Economics, 3 (1), Garleanu, N., Pedersen, L. H., & Poteshman, A. M. (2009). Demand-Based Option Pricing. The Review of Financial Studies, 22 (10), Haug, E. G., & Taleb, N. N. (2010). Option traders use (very) sophisticated heuristics, never the Black Scholes formula. (unpublished) Retrieved April 2013, from Hawkins, G. D. (1982). An Analysis of Revolving Credit Agreements. Journal of Financial Economics, 10 (1), Lintner, J. (1965). The Valuation of Risk Assets and the Selection of Risky Investments in Stock Portfolios and Capital Budgets. Review of Economics and Statistics, 47 (1), Macbeth, J. D., & Merville, L. J. (1979). An Empirical Examination of the Black-Scholes Call Option Pricing Model. Journal of Finance, 34 (5), Merton, R. C. (1977). An Analytical Derivation of the Cost of Deposit Insurance and Loan Guarantees. Journal of Banking and Finance, 1 (1), Merton, R. C. (1998). Applications of Option-Pricing Theory: Twenty-Five Years Later. The American Economic Review, 88 (3), Merton, R. C. (1973). Theory of Rational Option Pricing. The Bell Journal of Economics and Management Science, 4 (1), Sharpe, W. F. (1964). Capital Asset Prices: A Theory of Market Equilibrium under Conditions of Risk. Journal of Financial Economicsv, 10 (3), Word count: 2995
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
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
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.
European Options Pricing Using Monte Carlo Simulation Alexandros Kyrtsos Division of Materials Science and Engineering, Boston University email@example.com European options can be priced using the analytical
Chap 3 CAPM, Arbitrage, and Linear Factor Models 1 Asset Pricing Model a logical extension of portfolio selection theory is to consider the equilibrium asset pricing consequences of investors individually
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
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
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:
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
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
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,
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.
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
Put-Call Parity chris bemis May 22, 2006 Recall that a replicating portfolio of a contingent claim determines the claim s price. This was justified by the no arbitrage principle. Using this idea, we obtain
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
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
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
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
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
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 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
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
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
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
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
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
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
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
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
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
Rendleman and Bartter  present a simple two-state model of option pricing. The states of the world evolve like the branches of a tree. Given the current state, there are two possible states next period.
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
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.
Finance Theory MIT Sloan MBA Program Andrew W. Lo Harris & Harris Group Professor, MIT Sloan School Lectures 10 11 11: : Options Critical Concepts Motivation Payoff Diagrams Payoff Tables Option Strategies
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
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
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
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
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
Lecture 10: Multi-period Model Options Black-Scholes-Merton model Prof. Markus K. Brunnermeier 1 Binomial Option Pricing Consider a European call option maturing at time T with ihstrike K: C T =max(s T
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
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
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
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.
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
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 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
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
OPTIONS and FUTURES Lecture 2: Binomial Option Pricing and Call Options Philip H. Dybvig Washington University in Saint Louis binomial model replicating portfolio single period artificial (risk-neutral)
780 w Interest Rate Models The Behavior of Bonds and Interest Rates Before discussing how a bond market-maker would delta-hedge, we first need to specify how bonds behave. Suppose we try to model a zero-coupon
June 2008 Supplement to Characteristics and Risks of Standardized Options This supplement supersedes and replaces the April 2008 Supplement to the booklet entitled Characteristics and Risks of Standardized
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
CHAPTER 21: OPTION VALUATION 1. Put values also must increase as the volatility of the underlying stock increases. We see this from the parity relation as follows: P = C + PV(X) S 0 + PV(Dividends). Given
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
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,
Schonbucher Chapter 9: Firm alue and Share Priced-Based Models Updated 07-30-2007 (References sited are listed in the book s bibliography, except Miller 1988) For Intensity and spread-based models of default
Discussions of Monte Carlo Simulation in Option Pricing TIANYI SHI, Y LAURENT LIU PROF. RENATO FERES MATH 350 RESEARCH PAPER INTRODUCTION Having been exposed to a variety of applications of Monte Carlo
The Black-Scholes-Merton formula (Hull 13.5 13.8) Assume S t is a geometric Brownian motion w/drift. Want market value at t = 0 of call option. European call option with expiration at time T. Payout at
Caput Derivatives: October 30, 2003 Exam + Answers Total time: 2 hours and 30 minutes. Note 1: You are allowed to use books, course notes, and a calculator. Question 1. [20 points] Consider an investor
VALUATION IN DERIVATIVES MARKETS September 2005 Rawle Parris ABN AMRO Property Derivatives What is a Derivative? A contract that specifies the rights and obligations between two parties to receive or deliver
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
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
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
Understanding N(d 1 ) and N(d 2 ): Risk-Adjusted Probabilities in the Black-Scholes Model 1 Lars Tyge Nielsen INSEAD Boulevard de Constance 77305 Fontainebleau Cedex France E-mail: nielsen@freiba51 October
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
Two-State Options John Norstad firstname.lastname@example.org http://www.norstad.org January 12, 1999 Updated: November 3, 2011 Abstract How options are priced when the underlying asset has only two possible
Mathematical Finance: Stochastic Modeling of Asset Prices José E. Figueroa-López 1 1 Department of Statistics Purdue University Worcester Polytechnic Institute Department of Mathematical Sciences December
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
Risk/Arbitrage Strategies: An Application to Stock Option Portfolio Management Vincenzo Bochicchio, Niklaus Bühlmann, Stephane Junod and Hans-Fredo List Swiss Reinsurance Company Mythenquai 50/60, CH-8022
LECTURE 10.1 Default risk in Merton s model Adriana Breccia March 12, 2012 1 1 MERTON S MODEL 1.1 Introduction Credit risk is the risk of suffering a financial loss due to the decline in the creditworthiness
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,
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
1 CHAPTER 5 OPTION PRICING THEORY AND MODELS In general, the value of any asset is the present value of the expected cash flows on that asset. In this section, we will consider an exception to that rule
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
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
Four Derivations of the Black Scholes PDE by Fabrice Douglas Rouah www.frouah.com www.volopta.com In this Note we derive the Black Scholes PDE for an option V, given by @t + 1 + rs @S2 @S We derive the
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
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
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
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
MØA 155 PROBLEM SET: Summarizing Exercise 1. Present Value  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
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,
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)
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:
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,
Your consent to our cookies if you continue to use this website.