Geometric Brownian motion makes sense for the call price because call prices cannot be negative. Now Using Ito's lemma we can nd an expression for dc,


 Edmund Dalton
 2 years ago
 Views:
Transcription
1 12 Option Pricing We are now going to apply our continuoustime methods to the pricing of nonstandard securities. In particular, we will consider the class of derivative securities known as options in this section of the notes. We will derive the famous BlackScholes option pricing model in this section, for which Scholes recently won the Nobel prize (Black died a few years ago). The original paper by Black and Scholes is the supplementary reading that goes with this lecture note. The paper is a model of how all good papers should be written. There is not a lot of extraneous information in there  it's pretty short. However, its inuence has been enormous. They seem to have maximized the ratio of economics to math in their paper. If you do not know much about options, you may want to talk to me about other readings. We will use a set of powerpoint notes to review some preliminary denitions and concepts The BlackScholes Model We will assume that stock prices follow geometric Brownian motion, ds S = dt + dw (236) We also know that a particular call option's price is a function of time and its underlying stock price, C = C(S; t): (237) Since the call option's value is a function of time and the stock price, we know that it will follow a diusion process. We posit (and later verify) that the call option's value follows geometric Brownian motion, dc(s; t) C = ^dt +^dw: (238) 95
2 Geometric Brownian motion makes sense for the call price because call prices cannot be negative. Now Using Ito's lemma we can nd an expression for dc, dc = C S ds + C t dt C SSdS 2 ; (239) and we can take the expectation of dc, E[dC] =C S Sdt + C t dt c SS 2 dt: (240) In order to verify that the call option's price follows Geometric Brownian motion we need to solve for ^ and ^. If we can nd expressions for both of these terms then the call price really does follow GBM. We solve for the terms using our expression for dc and E[dC]. Beginning with ^, ^ = E[dC] Cdt = C SS + C t + 1 C 2 SSS 2 2 : (241) C The term for ^ is ^ = dc, E[dC] CdW = C S(dS, E[dS]) + 1 C 2 SS(dS 2, E[dS 2 ]) ; (242) CdW which simplies to ^ = C SS C : (243) So we have shown that the call option price follows GBM with particular values for ^ and ^. Now we want to think about a portfolio, or a position that involves putting w dollars in the call option and putting (1, w) in the stock. The stochastic proces that will describe the value of our position, V, through time is another GBM 96
3 process, dv V = w dc C +(1,w)dS S : (244) Using the processes for dc and ds, we can be more specic about the process driving the value of our position, dv V =[w^+(1,w)]dt +[w^+(1,w)]dw: (245) We choose a value for w that eliminates all the risk in our position in the next instant by setting the coecient on the dw term above to zero, w =, ^ : (246) Choosing this particular w gives us the riskfree process dv V w =[w ^+(1,w )]dt = rdt (247) We have apparently done a very curious thing. By carefully choosing w, we can make the coecient on the dw term go to zero. If we set up our position just right, we can eliminate all of the risk in the portfolio. This is called a dynamic hedge. The riskfree position that we create will only be riskfree over the next instant  to maintain a riskfree position, we will have to constantly adjust w as both S and C move around through time. This is the continuoustime analog to one node of the binomial tree that we saw in the previous set of notes. The continuoustime hedge is the basis for most arbitrage pricing in continuous time. To eectively hedge continuously you must have very small transactions costs. Some researchers derive noarbitrage results with transactions costs  they nd that rather than arriving at a unique noarbitrage price, they get a range of possible option values that don't admit 97
4 arbitrage. This is a highly techincal research area that is dominated by statisticians and mathematicians. We have shown how to convert our position to a riskfree security through a dynamic hedge. Since this position is riskless over the next instant, dt, its payo must be equal to the payo of a riskfree security over the next instant (by the law of one price),, ^ ^, ^ = r: (248), ^ Doing a little bit of algebra, we can rearrange this to be, ^, r = ^ (, r): (249) Substituting the expressions for ^ and ^ into this expression, C S S + C t C SS 2 S 2 C, r = CSS C (, r); (250) which further simplies to C S S + C t C SS 2 S 2, rc = C S S(, r): (251) Dropping C S S from both sides, C S Sr + C t C SS 2 S 2, rc =0; (252) which gives us the equation that we have been solving for, (C S, C)r + C t C SS 2 S 2 =0; (253) which is a famous dierential equation. This is known as the heat transfer equation, 98
5 and its solution was derived many years ago. For the BlackScholes formula, we solve the heat transfer equation subject to the following boundary conditions, Boundary conditions : C(0;t)= 0 C(S; T ) = max[s, X; 0] (254) C(S; t) S: Unfortunately, you will just have to take it on faith (for now) that the solution to the heat transfer equation is actually the BlackScholes formula. For reference purposes, the equation can be written as C(S; t) =SN(d 1 ),Xe,r N (d 2 ); (255) where d 1 = ln(s=x)+(r+2 =2) p ; d 2 = d 1, p ; (256) and is the option's time to maturity, N () is the cumulative normal distribution, and X is the strike price. This is not the last time that we will see the BlackScholes formula. We will derive it again with \change of measure" methods when we talk about the absence of arbitrage in continuoustime. Let's talk about the model a little more right now to get a little more understanding of how it works Solving for Delta An important quantity for the BlackScholes model is what is commonly called, or the sensitivity of the option's call price to changes in the value of the option's underlying security. As we stated previously, in the BlackScholes framework, a call 99
6 option's is just equal to N (d 1 ). We should prove this rigorously. We start by taking the derivative of the BlackScholes call option = N(d 1), : (257) Remembering that we can express d 1 as, d 1 = ln(s=x) p + r p p ; (258) we can take the derivative ofd 1 with respect to = 1 S p : (259) Now using the chain (d 1 = f(d 1) S p : (260) Since d 2 is just d 1 minus p,asimilar result holds for N (d 2 (d 2 = f(d 1, p ) S p : (261) In both of these expressions, f() is just the standard normal distribution, 2 =2 f(x) = e,x p : (262) 2 Returning to our expression for, we can now state = N(d 1)+ f(d 1) p,xe,r S f(d 1, p ) p : (263) 100
7 Therefore, our result ( = N (d 1 )) will hold as long as f(d 1 )= Xe,r S f(d 1, p ): (264) To see that this condition is true, we express the ratio of f(d 1 ) to f(d 2 )as e,d2 1 =2 e,(d 1, p ) 2 =2 =Xe,r S : (265) Next, we express the denominator as e,(d 1, p )=2 = e,(d2 1,2d 1 p + 2 )=2 ; (266) which yields the condition e,d 1 p + 2 =2 = Xe,r S : (267) Finally, substituting the denition of d 1 into our condition, e, ln(s=x),(r+2 =2) + 2 =2 = Xe,r S ; (268) which we can verify as true. Thus, we have shown = N(d 1): (269) What do we use this term for? For one thing, tells us how to perform our continuoustime hedge. As we stated in the notes on the binomial tree pricing model, we always want to hold shares of stock for each written call option in a riskfree portfolio. As the components of d 1 change, will change. We can update our position by constantly checking if we have shares of stock per short call. 101
8 A second reason for caring about is that it tells us about the expected return on an option. To see this, we will dene another quantity known as omega, C = SN(d 1 ) SN(d 1 ), Xe,r N (d 2 ) > 1: (270) Omega is the elasticity ofthe call option's price to changes in the value of the call's underlying security. Thus, If the underlying asset's value changes by 3% because the market moves 3%, the call option's price should move by times 3%. In other words, it can be shown that C = S : (271) Since is always greater than one (as demonstrated above), the expected return of a call option written on an asset with a positive beta is always higher than the expected return of the underlying asset in a BlackScholes/CAPM world. There are, of course, much more general things that can be said about option returns. However, the intuition behind options returns arguments is the BlackScholes/CAPM case. We have spent a great deal of time discussing option pricing. This may seem unwarranted because options are just one type of nancial instrument available to market participants. However, Black and Scholes point out at the end of their article that lots of securities actually have optionlike characteristics. Consider, for example, the stock of a company that has some level of debt outstanding that must be paid in a lump sum. If the company plans to liquidate immediately after paying o its debt then the company's stock can be thought of as an option on the value of the rm. If the rm is suciently protable then the option will expire in the money and stockholders will be paid. Otherwise, all the rm's value will go to bondholders and the option (equity) will expire worthless. 102
9 12.3 Homework Problems 1. Assume that returns are driven by a kfactor model, so that ds i =S i = i dt + i1 df 1t + :::+ ik df kt + i dw i (272) and assume that each factor follows geometric Brownian motion with drift fj and volatility fj for the jth factor. Let the Wiener process of each factor (dw fj ) be independent of the corresponding process of each other factor and let it be independent ofdw i 8i. Assume also all of the assumptions that make the BlackScholes formula valid. Show that the jth factor beta of a call option on stock i is equal to i times the stock's factor beta, ij. 2. Suppose you buy one share of stock, write a oneyear call option on the stock with a strike price (X) of10, and buy a oneyear put with X = 10. Your total outlay for this position is $9.50. what is the riskfree rate? 3. Show that an atthemoney call option must cost more than an atthemoney put on the same stock and with the same maturity. 4. Show that the dierence between the prices of two European call options with strike prices X 1 and X 2 (where X 1 < X 2 ) is less than or equal to the present value of (X 2, X 1 ). 5. In the numerical example of a twostage binomial tree in the powerpoint notes, solve for the value of a call option with a strike price of
On BlackScholes Equation, Black Scholes Formula and Binary Option Price
On BlackScholes Equation, Black Scholes Formula and Binary Option Price Abstract: Chi Gao 12/15/2013 I. BlackScholes Equation is derived using two methods: (1) riskneutral measure; (2)  hedge. II.
More information第 9 讲 : 股 票 期 权 定 价 : BS 模 型 Valuing Stock Options: The BlackScholes Model
1 第 9 讲 : 股 票 期 权 定 价 : BS 模 型 Valuing Stock Options: The BlackScholes Model Outline 有 关 股 价 的 假 设 The BS 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
More informationFour Derivations of the Black Scholes PDE by Fabrice Douglas Rouah www.frouah.com www.volopta.com
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
More informationThe BlackScholes Formula
FIN40008 FINANCIAL INSTRUMENTS SPRING 2008 The BlackScholes Formula These notes examine the BlackScholes formula for European options. The BlackScholes formula are complex as they are based on the
More informationMertonBlackScholes model for option pricing. Peter Denteneer. 22 oktober 2009
MertonBlackScholes model for option pricing Instituut{Lorentz voor Theoretische Natuurkunde, LION, Universiteit Leiden 22 oktober 2009 With inspiration from: J. Tinbergen, T.C. Koopmans, E. Majorana,
More informationEuropean Options Pricing Using Monte Carlo Simulation
European Options Pricing Using Monte Carlo Simulation Alexandros Kyrtsos Division of Materials Science and Engineering, Boston University akyrtsos@bu.edu European options can be priced using the analytical
More informationCS 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
More informationCall 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
More informationOne Period Binomial Model
FIN40008 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
More informationCaput Derivatives: October 30, 2003
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
More informationLecture 12: The BlackScholes Model Steven Skiena. http://www.cs.sunysb.edu/ skiena
Lecture 12: The BlackScholes Model Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena The BlackScholesMerton Model
More informationIntroduction to Binomial Trees
11 C H A P T E R Introduction to Binomial Trees A useful and very popular technique for pricing an option involves constructing a binomial tree. This is a diagram that represents di erent possible paths
More informationA 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
More informationOption 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
More informationMathematical Finance
Mathematical Finance Option Pricing under the RiskNeutral Measure Cory Barnes Department of Mathematics University of Washington June 11, 2013 Outline 1 Probability Background 2 Black Scholes for European
More informationTwoState Option Pricing
Rendleman and Bartter [1] present a simple twostate 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.
More informationBinomial lattice model for stock prices
Copyright c 2007 by Karl Sigman Binomial lattice model for stock prices Here we model the price of a stock in discrete time by a Markov chain of the recursive form S n+ S n Y n+, n 0, where the {Y i }
More informationDoes BlackScholes framework for Option Pricing use Constant Volatilities and Interest Rates? New Solution for a New Problem
Does BlackScholes framework for Option Pricing use Constant Volatilities and Interest Rates? New Solution for a New Problem Gagan Deep Singh Assistant Vice President Genpact Smart Decision Services Financial
More informationValuing Options / Volatility
Chapter 5 Valuing Options / Volatility Measures Now that the foundation regarding the basics of futures and options contracts has been set, we now move to discuss the role of volatility in futures and
More informationMoreover, under the risk neutral measure, it must be the case that (5) r t = µ t.
LECTURE 7: BLACK SCHOLES THEORY 1. Introduction: The Black Scholes Model In 1973 Fisher Black and Myron Scholes ushered in the modern era of derivative securities with a seminal paper 1 on the pricing
More informationThe BlackScholes Model
The BlackScholes Model Liuren Wu Zicklin School of Business, Baruch College Options Markets (Hull chapter: 12, 13, 14) Liuren Wu The BlackScholes Model Options Markets 1 / 19 The BlackScholesMerton
More informationFUNDING INVESTMENTS FINANCE 238/738, Spring 2008, Prof. Musto Class 5 Review of Option Pricing
FUNDING INVESTMENTS FINANCE 238/738, Spring 2008, Prof. Musto Class 5 Review of Option Pricing I. PutCall Parity II. OnePeriod Binomial Option Pricing III. Adding Periods to the Binomial Model IV. BlackScholes
More informationOptions: 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
More informationInstitutional Finance 08: Dynamic Arbitrage to Replicate Nonlinear Payoffs. Binomial Option Pricing: Basics (Chapter 10 of McDonald)
Copyright 2003 Pearson Education, Inc. Slide 081 Institutional Finance 08: Dynamic Arbitrage to Replicate Nonlinear Payoffs Binomial Option Pricing: Basics (Chapter 10 of McDonald) Originally prepared
More informationSchonbucher Chapter 9: Firm Value and Share PricedBased Models Updated 07302007
Schonbucher Chapter 9: Firm alue and Share PricedBased Models Updated 07302007 (References sited are listed in the book s bibliography, except Miller 1988) For Intensity and spreadbased models of default
More informationOption Valuation. Chapter 21
Option Valuation Chapter 21 Intrinsic and Time Value intrinsic value of inthemoney options = the payoff that could be obtained from the immediate exercise of the option for a call option: stock price
More informationTwoState Model of Option Pricing
Rendleman and Bartter [1] put forward a simple twostate model of option pricing. As in the BlackScholes model, to buy the stock and to sell the call in the hedge ratio obtains a riskfree portfolio.
More informationBINOMIAL OPTION PRICING
Darden Graduate School of Business Administration University of Virginia BINOMIAL OPTION PRICING Binomial option pricing is a simple but powerful technique that can be used to solve many complex optionpricing
More informationJorge Cruz Lopez  Bus 316: Derivative Securities. Week 11. The BlackScholes Model: Hull, Ch. 13.
Week 11 The BlackScholes Model: Hull, Ch. 13. 1 The BlackScholes Model Objective: To show how the BlackScholes formula is derived and how it can be used to value options. 2 The BlackScholes Model 1.
More information2. Exercising the option  buying or selling asset by using option. 3. Strike (or exercise) price  price at which asset may be bought or sold
Chapter 21 : Options1 CHAPTER 21. OPTIONS Contents I. INTRODUCTION BASIC TERMS II. VALUATION OF OPTIONS A. Minimum Values of Options B. Maximum Values of Options C. Determinants of Call Value D. BlackScholes
More informationJorge Cruz Lopez  Bus 316: Derivative Securities. Week 9. Binomial Trees : Hull, Ch. 12.
Week 9 Binomial Trees : Hull, Ch. 12. 1 Binomial Trees Objective: To explain how the binomial model can be used to price options. 2 Binomial Trees 1. Introduction. 2. One Step Binomial Model. 3. Risk Neutral
More informationFinancial Modeling. Class #06B. Financial Modeling MSS 2012 1
Financial Modeling Class #06B Financial Modeling MSS 2012 1 Class Overview Equity options We will cover three methods of determining an option s price 1. BlackScholesMerton formula 2. Binomial trees
More informationContents. iii. MFE/3F Study Manual 9 th edition 10 th printing Copyright 2015 ASM
Contents 1 PutCall Parity 1 1.1 Review of derivative instruments................................. 1 1.1.1 Forwards........................................... 1 1.1.2 Call and put options....................................
More informationExam MFE Spring 2007 FINAL ANSWER KEY 1 B 2 A 3 C 4 E 5 D 6 C 7 E 8 C 9 A 10 B 11 D 12 A 13 E 14 E 15 C 16 D 17 B 18 A 19 D
Exam MFE Spring 2007 FINAL ANSWER KEY Question # Answer 1 B 2 A 3 C 4 E 5 D 6 C 7 E 8 C 9 A 10 B 11 D 12 A 13 E 14 E 15 C 16 D 17 B 18 A 19 D **BEGINNING OF EXAMINATION** ACTUARIAL MODELS FINANCIAL ECONOMICS
More informationThe BlackScholes pricing formulas
The BlackScholes pricing formulas Moty Katzman September 19, 2014 The BlackScholes differential equation Aim: Find a formula for the price of European options on stock. Lemma 6.1: Assume that a stock
More informationTABLE OF CONTENTS. A. PutCall 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. PutCall Parity 1 B. Comparing Options with Respect to Style, Maturity, and Strike 13 2. McDonald 10: "Binomial Option Pricing:
More information1. Assume that a (European) call option exists on this stock having on exercise price of $155.
MØA 155 PROBLEM SET: Binomial Option Pricing Exercise 1. Call option [4] A stock s current price is $16, and there are two possible prices that may occur next period: $15 or $175. The interest rate on
More information10 Binomial Trees. 10.1 Onestep model. 1. Model structure. ECG590I Asset Pricing. Lecture 10: Binomial Trees 1
ECG590I Asset Pricing. Lecture 10: Binomial Trees 1 10 Binomial Trees 10.1 Onestep model 1. Model structure ECG590I Asset Pricing. Lecture 10: Binomial Trees 2 There is only one time interval (t 0, t
More informationPutCall Parity. chris bemis
PutCall 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
More informationChapter 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 riskadjusted discount rate. Part D Introduction to derivatives. Forwards
More informationLecture Note of Bus 41202, Spring 2012: Stochastic Diffusion & Option Pricing
Lecture Note of Bus 41202, Spring 2012: Stochastic Diffusion & Option Pricing Key concept: Ito s lemma Stock Options: A contract giving its holder the right, but not obligation, to trade shares of a common
More informationIntroduction to ArbitrageFree Pricing: Fundamental Theorems
Introduction to ArbitrageFree Pricing: Fundamental Theorems Dmitry Kramkov Carnegie Mellon University Workshop on Interdisciplinary Mathematics, Penn State, May 810, 2015 1 / 24 Outline Financial market
More informationUnderstanding Options and Their Role in Hedging via the Greeks
Understanding Options and Their Role in Hedging via the Greeks Bradley J. Wogsland Department of Physics and Astronomy, University of Tennessee, Knoxville, TN 379961200 Options are priced assuming that
More informationChapter 9 Parity and Other Option Relationships
Chapter 9 Parity and Other Option Relationships Question 9.1. This problem requires the application of putcallparity. We have: Question 9.2. P (35, 0.5) C (35, 0.5) e δt S 0 + e rt 35 P (35, 0.5) $2.27
More informationLectures. Sergei Fedotov. 20912  Introduction to Financial Mathematics. No tutorials in the first week
Lectures Sergei Fedotov 20912  Introduction to Financial Mathematics No tutorials in the first week Sergei Fedotov (University of Manchester) 20912 2010 1 / 1 Lecture 1 1 Introduction Elementary economics
More information15 Solution 1.4: The dividend growth model says that! DIV1 = $6.00! k = 12.0%! g = 4.0% The expected stock price = P0 = $6 / (12% 4%) = $75.
1 The present value of the exercise price does not change in one hour if the riskfree rate does not change, so the change in call put is the change in the stock price. The change in call put is $4, so
More information1 The BlackScholes model: extensions and hedging
1 The BlackScholes 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
More informationτ θ What is the proper price at time t =0of this option?
Now by Itô s formula But Mu f and u g in Ū. Hence τ θ u(x) =E( Mu(X) ds + u(x(τ θ))) 0 τ θ u(x) E( f(x) ds + g(x(τ θ))) = J x (θ). 0 But since u(x) =J x (θ ), we consequently have u(x) =J x (θ ) = min
More informationBetting 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
More informationFinancial 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 equitylinked securities requires an understanding of financial
More informationEconomics 1723: Capital Markets Lecture 20
Economics 1723: Capital Markets Lecture 20 John Y. Campbell Ec1723 November 14, 2013 John Y. Campbell (Ec1723) Lecture 20 November 14, 2013 1 / 43 Key questions What is a CDS? What information do we get
More informationTHE BLACKSCHOLES MODEL AND EXTENSIONS
THE BLACSCHOLES MODEL AND EXTENSIONS EVAN TURNER Abstract. This paper will derive the BlackScholes pricing model of a European option by calculating the expected value of the option. We will assume that
More informationDerivation of Local Volatility by Fabrice Douglas Rouah www.frouah.com www.volopta.com
Derivation of Local Volatility by Fabrice Douglas Rouah www.frouah.com www.volopta.com The derivation of local volatility is outlined in many papers and textbooks (such as the one by Jim Gatheral []),
More informationLecture 11: RiskNeutral Valuation Steven Skiena. skiena
Lecture 11: RiskNeutral Valuation Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena RiskNeutral Probabilities We can
More informationOptions 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 PutCall Parity Relationship. I. Preliminary Material Recall the payoff
More informationOverview. 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 Noarbitrage bounds on option prices Binomial option pricing BlackScholesMerton
More informationContents. iii. MFE/3F Study Manual 9th edition Copyright 2011 ASM
Contents 1 PutCall Parity 1 1.1 Review of derivative instruments................................................ 1 1.1.1 Forwards............................................................ 1 1.1.2 Call
More informationLecture 3.1: Option Pricing Models: The Binomial Model
Important Concepts Lecture 3.1: Option Pricing Models: The Binomial Model The concept of an option pricing model The one and two period binomial option pricing models Explanation of the establishment and
More informationModelFree Boundaries of Option Time Value and Early Exercise Premium
ModelFree Boundaries of Option Time Value and Early Exercise Premium Tie Su* Department of Finance University of Miami P.O. Box 248094 Coral Gables, FL 331246552 Phone: 3052841885 Fax: 3052844800
More informationCHAPTER 21: OPTION VALUATION
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
More informationCA Final Strategic Financial Management, Paper 2, Chapter 5. CA.Tarun Mahajan,
CA Final Strategic Financial Management, Paper 2, Chapter 5 CA.Tarun Mahajan, Options Types of options Speculation using options Valuation of options In futures both parties have right as well as duty
More informationDiusion processes. Olivier Scaillet. University of Geneva and Swiss Finance Institute
Diusion processes Olivier Scaillet University of Geneva and Swiss Finance Institute Outline 1 Brownian motion 2 Itô integral 3 Diusion processes 4 BlackScholes 5 Equity linked life insurance 6 Merton
More informationBlackScholes Equation for Option Pricing
BlackScholes 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
More informationThe BlackScholes Formula
ECO30004 OPTIONS AND FUTURES SPRING 2008 The BlackScholes Formula The BlackScholes Formula We next examine the BlackScholes formula for European options. The BlackScholes formula are complex as they
More informationLecture 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
More informationFinite Differences Schemes for Pricing of European and American Options
Finite Differences Schemes for Pricing of European and American Options Margarida Mirador Fernandes IST Technical University of Lisbon Lisbon, Portugal November 009 Abstract Starting with the BlackScholes
More informationChapter 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
More information4. Options Markets. 4.6. Applications
4. Options Markets 4.6. Applications 4.6.1. Options on Stock Indices e.g.: Dow Jones Industrial (European) S&P 500 (European) S&P 100 (American) LEAPS S becomes to Se qt Basic Properties with Se qt c Se
More informationLECTURE 9: A MODEL FOR FOREIGN EXCHANGE
LECTURE 9: A MODEL FOR FOREIGN EXCHANGE 1. Foreign Exchange Contracts There was a time, not so long ago, when a U. S. dollar would buy you precisely.4 British pounds sterling 1, and a British pound sterling
More informationThe BlackScholesMerton Approach to Pricing Options
he BlackScholesMerton Approach to Pricing Options Paul J Atzberger Comments should be sent to: atzberg@mathucsbedu Introduction In this article we shall discuss the BlackScholesMerton approach to determining
More informationValuing Stock Options: The BlackScholesMerton Model. Chapter 13
Valuing Stock Options: The BlackScholesMerton Model Chapter 13 Fundamentals of Futures and Options Markets, 8th Ed, Ch 13, Copyright John C. Hull 2013 1 The BlackScholesMerton Random Walk Assumption
More informationBUS 316 NOTES AND ANSWERS BINOMIAL OPTION PRICING
BUS 316 NOTES AND ANSWERS BINOMIAL OPTION PRICING 3. Suppose there are only two possible future states of the world. In state 1 the stock price rises by 50%. In state 2, the stock price drops by 25%. The
More informationVannaVolga Method for Foreign Exchange Implied Volatility Smile. Copyright Changwei Xiong 2011. January 2011. last update: Nov 27, 2013
VannaVolga Method for Foreign Exchange Implied Volatility Smile Copyright Changwei Xiong 011 January 011 last update: Nov 7, 01 TABLE OF CONTENTS TABLE OF CONTENTS...1 1. Trading Strategies of Vanilla
More informationIntroduction 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
More informationFinancial Modeling. An introduction to financial modelling and financial options. Conall O Sullivan
Financial Modeling An introduction to financial modelling and financial options Conall O Sullivan Banking and Finance UCD Smurfit School of Business 31 May / UCD Maths Summer School Outline Introduction
More informationArticle 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
More informationLecture 21 Options Pricing
Lecture 21 Options Pricing Readings BM, chapter 20 Reader, Lecture 21 M. Spiegel and R. Stanton, 2000 1 Outline Last lecture: Examples of options Derivatives and risk (mis)management Replication and Putcall
More informationConsider a European call option maturing at time T
Lecture 10: Multiperiod Model Options BlackScholesMerton 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
More informationBlackScholes Option Pricing Model
BlackScholes Option Pricing Model Nathan Coelen June 6, 22 1 Introduction Finance is one of the most rapidly changing and fastest growing areas in the corporate business world. Because of this rapid change,
More informationEC3070 FINANCIAL DERIVATIVES
BINOMIAL OPTION PRICING MODEL A OneStep Binomial Model The Binomial Option Pricing Model is a simple device that is used for determining the price c τ 0 that should be attributed initially to a call option
More informationOn MarketMaking and DeltaHedging
On MarketMaking and DeltaHedging 1 Market Makers 2 MarketMaking and BondPricing On MarketMaking and DeltaHedging 1 Market Makers 2 MarketMaking and BondPricing What to market makers do? Provide
More informationLecture 15. Sergei Fedotov. 20912  Introduction to Financial Mathematics. Sergei Fedotov (University of Manchester) 20912 2010 1 / 6
Lecture 15 Sergei Fedotov 20912  Introduction to Financial Mathematics Sergei Fedotov (University of Manchester) 20912 2010 1 / 6 Lecture 15 1 BlackScholes Equation and Replicating Portfolio 2 Static
More informationLecture 6: Option Pricing Using a Onestep Binomial Tree. Friday, September 14, 12
Lecture 6: Option Pricing Using a Onestep Binomial Tree An oversimplified model with surprisingly general extensions a single time step from 0 to T two types of traded securities: stock S and a bond
More informationLecture 4: The BlackScholes model
OPTIONS and FUTURES Lecture 4: The BlackScholes model Philip H. Dybvig Washington University in Saint Louis BlackScholes option pricing model Lognormal price process Call price Put price Using BlackScholes
More informationLecture 17/18/19 Options II
1 Lecture 17/18/19 Options II Alexander K. Koch Department of Economics, Royal Holloway, University of London February 25, February 29, and March 10 2008 In addition to learning the material covered in
More informationProtective Put Strategy Profits
Chapter Part Options and Corporate Finance: Basic Concepts Combinations of Options Options Call Options Put Options Selling Options Reading The Wall Street Journal Combinations of Options Valuing Options
More informationOPTION VALUATION. Topics in Corporate Finance P A R T 8. ON JULY 7, 2008, the closing stock prices for LEARNING OBJECTIVES
LEARNING OBJECTIVES After studying this chapter, you should understand: LO1 The relationship between stock prices, call prices, and put prices using put call parity. LO2 The famous Black Scholes option
More informationMore Exotic Options. 1 Barrier Options. 2 Compound Options. 3 Gap Options
More Exotic Options 1 Barrier Options 2 Compound Options 3 Gap Options More Exotic Options 1 Barrier Options 2 Compound Options 3 Gap Options Definition; Some types The payoff of a Barrier option is path
More information7: The CRR Market Model
Ben Goldys and Marek Rutkowski School of Mathematics and Statistics University of Sydney MATH3075/3975 Financial Mathematics Semester 2, 2015 Outline We will examine the following issues: 1 The CoxRossRubinstein
More informationNotes on BlackScholes Option Pricing Formula
. Notes on BlackScholes Option Pricing Formula by DeXing Guan March 2006 These notes are a brief introduction to the BlackScholes formula, which prices the European call options. The essential reading
More informationPricing Options: Pricing Options: The Binomial Way FINC 456. The important slide. Pricing options really boils down to three key concepts
Pricing Options: The Binomial Way FINC 456 Pricing Options: The important slide Pricing options really boils down to three key concepts Two portfolios that have the same payoff cost the same. Why? A perfectly
More informationMATH3075/3975 Financial Mathematics
MATH3075/3975 Financial Mathematics Week 11: Solutions Exercise 1 We consider the BlackScholes model M = B, S with the initial stock price S 0 = 9, the continuously compounded interest rate r = 0.01 per
More informationChapter 2 Introduction to Option Management
Chapter 2 Introduction to Option Management The prize must be worth the toil when one stakes one s life on fortune s dice. Dolon to Hector, Euripides (Rhesus, 182) In this chapter we discuss basic concepts
More informationJungSoon Hyun and YoungHee Kim
J. Korean Math. Soc. 43 (2006), No. 4, pp. 845 858 TWO APPROACHES FOR STOCHASTIC INTEREST RATE OPTION MODEL JungSoon Hyun and YoungHee Kim Abstract. We present two approaches of the stochastic interest
More informationLECTURE 10.1 Default risk in Merton s model
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
More informationOPTIONS and FUTURES Lecture 2: Binomial Option Pricing and Call Options
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 (riskneutral)
More informationThe 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
More informationHOW TO VALUE EMPLOYEE STOCK OPTIONS. John Hull and Alan White September, 2002
HOW TO VALUE EMPLOYEE STOCK OPTIONS John Hull and Alan White September, 2002 Joseph L. Rotman School of Management University of Toronto 105 St George Street Toronto, ON M5S 3E6 Canada Telephone Hull:
More information1 Pricing options using the Black Scholes formula
Lecture 9 Pricing options using the Black Scholes formula Exercise. Consider month options with exercise prices of K = 45. The variance of the underlying security is σ 2 = 0.20. The risk free interest
More informationLecture 14 Option pricing in the oneperiod binomial model.
Lecture: 14 Course: M339D/M389D  Intro to Financial Math Page: 1 of 9 University of Texas at Austin Lecture 14 Option pricing in the oneperiod binomial model. 14.1. Introduction. Recall the oneperiod
More information