An Introduction to the Mathematics of Finance. Basu, Goodman, Stampfli
|
|
- Shanon Daniel
- 5 years ago
- Views:
Transcription
1 An Introduction to the Mathematics of Finance Basu, Goodman, Stampfli 1998
2 Click here to see Chapter One.
3 Chapter 2 Binomial Trees, Replicating Portfolios, and Arbitrage 2.1 Pricing an Option A Special Case A Call option on a stock is the right, but not the obligation, to purchase the stock on a certain day. This day is termed the strike date or date of expiration. The purchase price is agreed on in advance, it is termed the strike price or exercise price. A Put option on a stock is a similar right, but not the obligation to sell the stock on a certain date. It is understood that one does not need to own a share of stock to exercise a put. Example A Call Option. Ford 27 1/2 Dec / /8 27 1/2 Mar /8 30 Nov / /8 30 Dec 84 3/4 28 5/8 30 Mar /8 28 5/8 32 1/2 Dec 59 1/4 28 5/8 32 1/2 Mar / /8 35 Mar /16 The Wall Street Journal We have a stock presently priced at $100. In exactly one year the stock price will be either $90 or $120. We are not given probabilities. The current interest rate is 5% (a dollar invested today is worth $1.05 in one year). What is a fair price for the option on the stock with a strike of $105 and expiring in one year? 3
4 4 CHAPTER 2. REPLICATION AND ARBITRAGE We will present two methods which answer this question. One method could be called the probabilistic (expected value) approach and the other the financial (game theory, arbitrage) approach. Some readers may feel we do not have enough information to answer the question. They are right in that we will have to make some assumptions. The assumptions are all part of the approach or solution to the problem and we feel no concern at adding them as we go along Financial Method Let V = price of the option, and let S = price of the stock. We will construct a portfolio as follows: We buy a shares of the option and b shares of the stock. The numbers a or b may be negative. If b, for example, is negative, it indicates that we are short selling the stock. Let Π 0 = the value of the portfolio at time t =0. Π 0 = av + bs. At this point we do not know a or b. We will next value the portfolio at time t =1. Since there are two possible states (scenarios) at t = 1, we take them separately. (Up State, S = $120) Π 1 = a( ) + b120. Why does the term appear? We convert the option into stock and sell the stock. The cost of the conversion is $105 per share and the sale price is $120 per share. (Down State, S = $90) Π 1 = a 0+b90. We seem to have made little progress but now for a brilliant stroke. We cannot control the stock performance. So we use a trick from game theory. Eliminate Uncertainty. We will choose a and b so that Π 1 does not depend on the outcome. Thus we set a( ) + b120 = a 0+b90.
5 2.1. PRICING AN OPTION A SPECIAL CASE 5 This gives the equation 15a = 30b and we make the choice b = +1 and a = 2. This strategy tells us we should sell two options and buy one share of stock. If we do so, the outcome is deterministic; we can ignore the real world. Valuing the Option. Note that Π 0 = 2V and Π 1 = (+1)120 = 90. We could have taken the funds Π 0 and invested them at 5%. So the value of that investment should equal the value Π 1 of the stock-option trade. In other words 1.05Π 0 =1.05(100 2V )=Π 1 =90. Thus, 100 2V = and so V =7.14. Arbitrage. Suppose V > Then we could make a deterministic, risk free profit by borrowing money at 5% to buy the stock and sell the options. If, on the other hand, V < 7.14, then we could make a deterministic, risk free profit by buying two options and selling one share of stock and investing the net proceeds at 5%. Note that arbitrage forces or drives the price to $7.14. Let s see this arbitrage in action. We will put concrete numbers into the scenario. Assume the price of the option is V = $8. Our analysis tells us the options are overpriced. So, we will buy one share of stock and sell two shares of the option. We will borrow the money necessary to fund this position. The cost at time 0 is = 84. We borrow the $84 (none of our money is at stake). After one year our position is worth $90 (independent of the stock s up down movement). We repay the loan, handing over This gives us a profit of = = $1.80. Note that our profit was a certainty, independent of events. And, we might have scaled up our purchases, perhaps, purchased a thousand shares of stock and sold two thousand options. The reader may wish to work out the details when the options are underpriced at $6 per share (exercise 5).
6 6 CHAPTER 2. REPLICATION AND ARBITRAGE The Probabilistic Approach. Stock Price Tree 120 p p 90 Let s begin by thinking about this situation using one characteristic of real markets. We know the stock price is $100, the up price is $120, and the down price is $90. Suppose we were viewing real market behavior over a one year period. A reasonable choice of p is one where the expected return of the stock is on the order of 15%. This return is much larger than if we invested the $100 in a secure bank account, and the following calculations suggest why this is so. A p value that roughly matches this expected return is p =.90. This seems to produce an attractive situation. The expected payoff is given by E(P )=.9( ) +.1(90 100) = $17. Notice that the expected return each year is 17%. But there is still some uncertainty. Since only a probability of success is involved, the situation here is that 90% of the time you make $20 and 10% of the time you lose $10. Many investors would purchase stock under these circumstances. The healthy price increase of the stock would offset the occasional loss so that this investment choice is attractive to those who can afford the risk of some loss. However, each investor is different. How do we decide what is a reasonable amount of risk and reward for this stock? To circumvent this impossible task we introduce a hypothetical investor, hereafter designated by HI, who has the following characteristics: I. The HI is risk neutral; this is the oposite of a bird in the hand outlook. A risk neutral investor is risk indifferent; a certain dollar is no
7 2.1. PRICING AN OPTION A SPECIAL CASE 7 more preferable than an expected dollar. Most people are not risk neutral. The insurance industry is based on this fact. II. Our HI thus has no preference between the stock introduced above and a risk free investment. Given these assumptions, what value of p in our stock model will make the stock and the risk free return (.05) equally attractive to the investor? p 1-p 90 If we form a portfolio, Π, consisting of one share of stock then Π 0 = $100 and E[Π 1 ]=p120 + (1 p)90 =30p +90 after one year. If we had simply invested the $100 at the risk free rate, then the value of that selection would be $105 at the end of one year. Being risk neutral, our HI sees these investments as equal. That is, 30p + 90 = 105 which implies p =0.5. Important Caveat. The value we have just found does not necessarily correspond to investor sentiment, nor to some real but unknown probability associated with the stock. It is simply the probability which produces a stock return equivalent (in the hypothetical value sense) to the risk free return. Now, take this p value and use it to compute the expected value for a call option on the stock. Let C be our call price. Then E[C] =p( ) + +(1 p)(90 105) +
8 8 CHAPTER 2. REPLICATION AND ARBITRAGE = =$7.50. However, we pay for the call today and receive our payoff in one year, so we should discount the expected return on the call. We then arrive at the value E[C]/1.05 = $7.50/1.05 = $7.14. Astonishingly, this is exactly the price we calculated using the financial (arbitrage, replicating portfolio) method. 2.2 Replicating Portfolios We have encountered several powerful techniques in Section 2.1. We wish to try them out in a slightly more general environment. The Context. As before, we have a stock where S 0 is the stock price at t = 0 and the stock can achieve one of two possible values at time t = τ. S 0 S u S d V U D We also have a derivative security, V, whose value at t = τ will depend on the performance of S. IfS goes up, V will be worth U. IfS goes down, V will be worth D. What is a fair price for V today? At first glance this may seem to be an impossible question. We know almost nothing about V. It could be an option, but it could be some complicated combination of options, futures, swaps, or derived values. To price V, we will employ an ingenious device known as a replicating portfolio A Portfolio Match We need to fix one more item. The risk-free interest rate will be denoted by r and we make use of the short-term lending and borrowing possible at this rate. Let s record such amounts of money in terms of a bond by assuming
9 2.2. REPLICATING PORTFOLIOS 9 the bond is intially worth $1. Then the value of such a bond at time t is e rt. Our portfolio, Π, will consist of a units of the stock and b units of the bond. The value, Π 0, of the portfolio at time t = 0 is just Π 0 = as 0 + b. Let s compute the value of Π at t = τ. Our stock model gives two future values for the portfolio: Up State Down State Π τ = as u + be rτ Π τ = as d + be rτ We now set as u + be rτ = U as d + be rτ = D (2.1) so that the value of our portfolio, Π, is identical to the value of the derivative security. The portfolio replicates V. Since the portfolio and the derivative have the same value at t = τ, they should have the same value today. After all, they are indistinguishable at the next future date. We conclude that V 0 = as 0 + b. This expression for V 0 is rather remarkable when we solve for a and b using the equations (??). We have two linear equations whose solution is a = U D (2.2) and b = [U U D ] S u e rτ. Although these expressions look complicated, they produce simpler expressions for portfolio values. We focus on the expression for V 0 (which is just as 0 + b). V 0 = U D S 0 + [U U D ] S u e rτ, and we separate the U terms and the D terms to get [ S 0 V 0 = U + e rτ S u e rτ] [ S 0 + D + S u e rτ]
10 10 CHAPTER 2. REPLICATION AND ARBITRAGE [ e = e rτ rτ S 0 U S ] [ d + e rτ S u D erτ S ] 0. There is something very special here. The U coefficient, ignoring the exponential term, is q = erτ S 0 S d and the D coefficient, S u e rτ S 0, is just 1 q. So, our portfolio value does simplify to V 0 = e rτ [qu +(1 q)d]. (2.3) Expected Value Pricing Approach. It appears from equation (??) that the present value of a portfolio is obtained by discounting (e rτ is referred to as a discount factor) an expected value of the future portfolio values. Indeed, the formula for q would serve as some sort of probability if we could check that the condition 0 q 1 holds. Let s look at the value for q once more: q = erτ S 0 S d. (2.4) If q were negative, this stock would be a great buy; its worst value, S d,at the future date would exceed the return we would get by initially investing $ S 0 in the bond. Note that the bond return would be $ e rτ S 0. This is an example of a sure money-making scheme known as arbitrage, and we believe this is too good to be true in the real world. Equally unrealistic is the case that 1 q is negative. We see from the expression just before equation (??) that 1 q = S u e rτ S 0, and, if this were negative, then the stock would be a dud. In this case, the best future value of the stock, S u, is not as large as the return on an initial bond investment of $ S 0. There would be no reason to buy such a stock. Again, a real market would not support such stock behavior.
11 2.2. REPLICATING PORTFOLIOS 11 as This motivates us to rewrite our simple portfolio value in equation (??) V 0 = e rτ E q [V 1 ]. (2.5) The subscript in equation (??) will indicate that we are using the specially computed no-arbitrage pricing probability, q, given by (??) The No-arbitrage Pricing Probability We wish to emphasize that a good way to remember the q-value in the context of this section is to consider the most simple, one period portfolio to own: a single share of the stock. This corresponds to the tree S 0 S u q 1-q S d and thus (after discounting) e rτ S 0 = qs u +(1 q)s d. (2.6) When the U and D values are the (two) future stock values, equation (??) includes, as a special case, equation (??). The reader should take a few moments to solve this equation for q in terms of S 0, S u, and S d.heor she will appreciate the fact that equation (??) determines the no-arbitrage pricing probability, q. It will be important to remember one other quantity. In this section we have focussed on a portfolio that matches or replicates the results of some other equity. The key idea is to hold the correct number of shares of the stock. Our formula for this number of shares, equation (??), is just # of shares = U D (2.7) Notice that this ratio compares the change in the equity being replicated to the change in the stock price. This ratio is termed the delta quantity when it determines an investment process. You will see the pricing probability, q,
12 12 CHAPTER 2. REPLICATION AND ARBITRAGE and the delta quantity,, appear in many calculations of portfolio behavior later on. The techniques and results of this section are so elegant and powerful, one naturally wonders whether they can be extended to a more general situation. Suppose our stock took on just three values: up, middle, and down (a Goldilock s stock). Alas, there is no way to apply the foregoing approach without making truely ridiculous assumptions. We present the case in the next section. 2.3 Limits of the Binomial Arbitrage Method We have just seen that the replicating portfolio method, combined with the absence of arbitrage, exactly determines an option price. It converts what appears to be a stochastic situation into a deterministic one, in the binomial context. However, a stock that takes only one of two values isn t very realistic. Can we extend this approach to a stock that takes on three values? The answer is, no (unless we make unreasonable assumptions). Let s see why. S u S 0 S m S d V U M D Stock Tree Our portfolio will (as before) consist of a units of the stock S Option Tree b units of a bond (price is $1 at an interest rate r). the value of the portfolio at t =0is Π 0 = a S + b We now set the portfolio value at t = 1 equal to the option value for the three scenarios:
13 2.3. LIMITS OF THE BINOMIAL ARBITRAGE METHOD 13 Up Case Middle Case Down Case as u + be r = U as m + be r = M as d + be r = D We wish to solve for a and b. But, in general that is impossible since we have more equations to satisfy (three) than unknowns (two). To put it another way, we need a three dimensional solution space, and yet, what we have is two dimensional. However, all is not lost. Our observation suggests we should add another financial instrument. Let s begin by adding a bond with a different interest rate, R. Thus, our new portfolio consists of a units of the stock S b units of Bond One (price is $1 at an interest rate r). c units of Bond Two (price is $1 at an interest rate R). Again, we set up the portfolio value at t = 1 in the three scenarios: Up Case Middle Case Down Case as u + be r + ce R = U as m + be r + ce R = M as d + be r + ce R = D Now we have three equations and three unknowns; but, look at the second and third column on the left hand side. They are identical. So, again we cannot solve for a, b, c in general. Again, our solution space is only two dimensional (or the space of the column vector is two dimensional). And, we need three dimensions. We still have one more strategy. Let s add another stock, P, which also takes exactly three values, V u, V, and V d at t = 1. So, our new portfolio consists of a units of the stock b units of the bond (price is $1 at an interest rate r) c units of the stock P. We set up the portfolio value at t = 1 and we make the following:
14 14 CHAPTER 2. REPLICATION AND ARBITRAGE Enormous Assumption 1. When S S u ; P P u 2. when S S m ; P P m 3. When S S d ; P P d. Thus, the portfolio values in the three cases become as u + be r + cp u = U as m + be r + cp m = M as d + be r + cp d = D We now have three independent equations and three unknowns so we can solve for a, b, c. Having done so, we can determine the initial portfolio value. Indeed, Π 0 = a S + b + cp But look back at our Enormous Assumption. We have won a hollow victory. First, it doesn t seem reasonable that if we wish to price an option on Ford, we should be using the additional stock International Paper. But, second, it is even more unreasonable to assume that Ford and International Paper move up, down, or sideways in tandem. We don t have to assume that P u, P m, and P d occur in any particular order, but the linked, coupled, or tandem movement is just too much to swallow. 2.4 Repeated Binomial Trees and Arbitrage In the next chapter we look at repeated binomial trees in detail. But, let s look at the two step case before moving on. We take our binomial tree when S 1 = XS 0 and X is a random variable and takes either the values u or d. So, our tree becomes
15 2.4. REPEATED BINOMIAL TREES AND ARBITRAGE 15 S 0 us 0 ds 0 u 2 S 0 uds 0 dus 0 d 2 S 0 Since u d = d u, we can simplify our tree by representing it as: u 2 S 0 us 0 S 0 uds 0 ds 0 Time d 2 S 0 Can we use a replicating portfolio to price an option by arbitrage in this two step process? More precisely, can we make the outcome deterministic? Can we protect the seller of the option from any loss? The answer is no, but since this limitation is so important, let s look at this in detail. If we disregard the intermediate time step we have a trinomial (three outcome) situation, and the comments from the previous discussion apply. But, here we have a lot more information via the intermediate step. Can that work to our advantage?
16 16 CHAPTER 2. REPLICATION AND ARBITRAGE Step 1 We first use the replicating portfolio method (RPM) to obtain a sure-thing deterministic price for the portfolio at time t = 1. Step 2 We would like to use the RPM again to price the option at t =2. But there is a problem. Are we now in the up state or the down state? Since the state at t = 1 is the result of a random process, we don t know the answer at t = 1. Moreover, the price we should assign to the option depends on the state at t = 1. let s illustrate this with a concrete example based on our earlier example. Let The tree for the process is: S 0 = 100 u =1.2 d =.9 r =.05 (discrete) Strike Price for Option = 110 Option Payoff at t = Time Using RPM or the Expectation Method, we arrive at the deterministic price of $4.76 for the portfolio at t =1. Step 2 We wish to price the option (deterministically if possible) from t =1 to t = 2. But look what happens. If we are in the Down state the option value or price is 0. it can never rise above the strike price.
17 2.4. REPEATED BINOMIAL TREES AND ARBITRAGE 17 If we are in the Up state, the option clearly has some value, and if we work through the numbers, we find that is $ The $16.19 is unimportant. What is important is the fact that we don t know in advance just which state we will be in at t = 1. There is no possible way to determine our state at t = 1 ahead of the event. We can take some sort of a weighted average over the up state and the down state, and that is precisely what we will do in the next chapter. However, the main point of this section declares that we can not deterministically price even a binomial option in the two step case.
Option Pricing. Chapter Discrete Time
Chapter 7 Option Pricing 7.1 Discrete Time In the next section we will discuss the Black Scholes formula. To prepare for that, we will consider the much simpler problem of pricing options when there are
More informationHelp Session 2. David Sovich. Washington University in St. Louis
Help Session 2 David Sovich Washington University in St. Louis TODAY S AGENDA 1. Refresh the concept of no arbitrage and how to bound option prices using just the principle of no arbitrage 2. Work on applying
More informationMartingale Pricing Theory in Discrete-Time and Discrete-Space Models
IEOR E4707: Foundations of Financial Engineering c 206 by Martin Haugh Martingale Pricing Theory in Discrete-Time and Discrete-Space Models These notes develop the theory of martingale pricing in a discrete-time,
More informationCorporate Finance, Module 21: Option Valuation. Practice Problems. (The attached PDF file has better formatting.) Updated: July 7, 2005
Corporate Finance, Module 21: Option Valuation Practice Problems (The attached PDF file has better formatting.) Updated: July 7, 2005 {This posting has more information than is needed for the corporate
More informationLecture 6: Option Pricing Using a One-step Binomial Tree. Thursday, September 12, 13
Lecture 6: Option Pricing Using a One-step Binomial Tree An over-simplified model with surprisingly general extensions a single time step from 0 to T two types of traded securities: stock S and a bond
More informationSYSM 6304: Risk and Decision Analysis Lecture 6: Pricing and Hedging Financial Derivatives
SYSM 6304: Risk and Decision Analysis Lecture 6: Pricing and Hedging Financial Derivatives M. Vidyasagar Cecil & Ida Green Chair The University of Texas at Dallas Email: M.Vidyasagar@utdallas.edu October
More informationLecture 1 Definitions from finance
Lecture 1 s from finance Financial market instruments can be divided into two types. There are the underlying stocks shares, bonds, commodities, foreign currencies; and their derivatives, claims that promise
More informationThe Binomial Model. Chapter 3
Chapter 3 The Binomial Model In Chapter 1 the linear derivatives were considered. They were priced with static replication and payo tables. For the non-linear derivatives in Chapter 2 this will not work
More informationThe Multistep Binomial Model
Lecture 10 The Multistep Binomial Model Reminder: Mid Term Test Friday 9th March - 12pm Examples Sheet 1 4 (not qu 3 or qu 5 on sheet 4) Lectures 1-9 10.1 A Discrete Model for Stock Price Reminder: The
More informationIncome Taxation and Stochastic Interest Rates
Income Taxation and Stochastic Interest Rates Preliminary and Incomplete: Please Do Not Quote or Circulate Thomas J. Brennan This Draft: May, 07 Abstract Note to NTA conference organizers: This is a very
More informationAppendix to Supplement: What Determines Prices in the Futures and Options Markets?
Appendix to Supplement: What Determines Prices in the Futures and Options Markets? 0 ne probably does need to be a rocket scientist to figure out the latest wrinkles in the pricing formulas used by professionals
More informationExercise 14 Interest Rates in Binomial Grids
Exercise 4 Interest Rates in Binomial Grids Financial Models in Excel, F65/F65D Peter Raahauge December 5, 2003 The objective with this exercise is to introduce the methodology needed to price callable
More informationMathematics in Finance
Mathematics in Finance Robert Almgren University of Chicago Program on Financial Mathematics MAA Short Course San Antonio, Texas January 11-12, 1999 1 Robert Almgren 1/99 Mathematics in Finance 2 1. Pricing
More informationFinance 402: Problem Set 7 Solutions
Finance 402: Problem Set 7 Solutions Note: Where appropriate, the final answer for each problem is given in bold italics for those not interested in the discussion of the solution. 1. Consider the forward
More informationFINANCIAL OPTION ANALYSIS HANDOUTS
FINANCIAL OPTION ANALYSIS HANDOUTS 1 2 FAIR PRICING There is a market for an object called S. The prevailing price today is S 0 = 100. At this price the object S can be bought or sold by anyone for any
More informationLecture 16: Delta Hedging
Lecture 16: Delta Hedging We are now going to look at the construction of binomial trees as a first technique for pricing options in an approximative way. These techniques were first proposed in: J.C.
More informationECON4510 Finance Theory Lecture 10
ECON4510 Finance Theory Lecture 10 Diderik Lund Department of Economics University of Oslo 11 April 2016 Diderik Lund, Dept. of Economics, UiO ECON4510 Lecture 10 11 April 2016 1 / 24 Valuation of options
More informationArbitrage Enforced Valuation of Financial Options. Outline
Arbitrage Enforced Valuation of Financial Options Richard de Neufville Professor of Engineering Systems and of Civil and Environmental Engineering MIT Arbitrage Enforced Valuation Slide 1 of 40 Outline
More informationMixed Strategies. Samuel Alizon and Daniel Cownden February 4, 2009
Mixed Strategies Samuel Alizon and Daniel Cownden February 4, 009 1 What are Mixed Strategies In the previous sections we have looked at games where players face uncertainty, and concluded that they choose
More informationLECTURE 2: MULTIPERIOD MODELS AND TREES
LECTURE 2: MULTIPERIOD MODELS AND TREES 1. Introduction One-period models, which were the subject of Lecture 1, are of limited usefulness in the pricing and hedging of derivative securities. In real-world
More informationCHAPTER 17 OPTIONS AND CORPORATE FINANCE
CHAPTER 17 OPTIONS AND CORPORATE FINANCE Answers to Concept Questions 1. A call option confers the right, without the obligation, to buy an asset at a given price on or before a given date. A put option
More informationChapter 2. An Introduction to Forwards and Options. Question 2.1
Chapter 2 An Introduction to Forwards and Options Question 2.1 The payoff diagram of the stock is just a graph of the stock price as a function of the stock price: In order to obtain the profit diagram
More informationECON 459 Game Theory. Lecture Notes Auctions. Luca Anderlini Spring 2017
ECON 459 Game Theory Lecture Notes Auctions Luca Anderlini Spring 2017 These notes have been used and commented on before. If you can still spot any errors or have any suggestions for improvement, please
More informationLecture 16. Options and option pricing. Lecture 16 1 / 22
Lecture 16 Options and option pricing Lecture 16 1 / 22 Introduction One of the most, perhaps the most, important family of derivatives are the options. Lecture 16 2 / 22 Introduction One of the most,
More informationB6302 Sample Placement Exam Academic Year
Revised June 011 B630 Sample Placement Exam Academic Year 011-01 Part 1: Multiple Choice Question 1 Consider the following information on three mutual funds (all information is in annualized units). Fund
More informationOption Pricing Models for European Options
Chapter 2 Option Pricing Models for European Options 2.1 Continuous-time Model: Black-Scholes Model 2.1.1 Black-Scholes Assumptions We list the assumptions that we make for most of this notes. 1. The underlying
More informationAdvanced Corporate Finance. 5. Options (a refresher)
Advanced Corporate Finance 5. Options (a refresher) Objectives of the session 1. Define options (calls and puts) 2. Analyze terminal payoff 3. Define basic strategies 4. Binomial option pricing model 5.
More informationBINARY OPTIONS: A SMARTER WAY TO TRADE THE WORLD'S MARKETS NADEX.COM
BINARY OPTIONS: A SMARTER WAY TO TRADE THE WORLD'S MARKETS NADEX.COM CONTENTS To Be or Not To Be? That s a Binary Question Who Sets a Binary Option's Price? And How? Price Reflects Probability Actually,
More informationNotes: This is a closed book and closed notes exam. The maximal score on this exam is 100 points. Time: 75 minutes
M375T/M396C Introduction to Financial Mathematics for Actuarial Applications Spring 2013 University of Texas at Austin Sample In-Term Exam II - Solutions This problem set is aimed at making up the lost
More informationCash Flows on Options strike or exercise price
1 APPENDIX 4 OPTION PRICING 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 when we will look
More informationIntroduction Random Walk One-Period Option Pricing Binomial Option Pricing Nice Math. Binomial Models. Christopher Ting.
Binomial Models Christopher Ting Christopher Ting http://www.mysmu.edu/faculty/christophert/ : christopherting@smu.edu.sg : 6828 0364 : LKCSB 5036 October 14, 2016 Christopher Ting QF 101 Week 9 October
More informationMathematics of Finance Final Preparation December 19. To be thoroughly prepared for the final exam, you should
Mathematics of Finance Final Preparation December 19 To be thoroughly prepared for the final exam, you should 1. know how to do the homework problems. 2. be able to provide (correct and complete!) definitions
More information2 The binomial pricing model
2 The binomial pricing model 2. Options and other derivatives A derivative security is a financial contract whose value depends on some underlying asset like stock, commodity (gold, oil) or currency. The
More informationProblem Set. Solutions to the problems appear at the end of this document.
Problem Set Solutions to the problems appear at the end of this document. Unless otherwise stated, any coupon payments, cash dividends, or other cash payouts delivered by a security in the following problems
More informationDevelopmental Math An Open Program Unit 12 Factoring First Edition
Developmental Math An Open Program Unit 12 Factoring First Edition Lesson 1 Introduction to Factoring TOPICS 12.1.1 Greatest Common Factor 1 Find the greatest common factor (GCF) of monomials. 2 Factor
More informationFinance 100 Problem Set 6 Futures (Alternative Solutions)
Finance 100 Problem Set 6 Futures (Alternative Solutions) Note: Where appropriate, the final answer for each problem is given in bold italics for those not interested in the discussion of the solution.
More informationPractice of Finance: Advanced Corporate Risk Management
MIT OpenCourseWare http://ocw.mit.edu 15.997 Practice of Finance: Advanced Corporate Risk Management Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
More informationChapter 6: Supply and Demand with Income in the Form of Endowments
Chapter 6: Supply and Demand with Income in the Form of Endowments 6.1: Introduction This chapter and the next contain almost identical analyses concerning the supply and demand implied by different kinds
More information1.1 Basic Financial Derivatives: Forward Contracts and Options
Chapter 1 Preliminaries 1.1 Basic Financial Derivatives: Forward Contracts and Options A derivative is a financial instrument whose value depends on the values of other, more basic underlying variables
More informationIntroduction to Financial Mathematics and Engineering. A guide, based on lecture notes by Professor Chjan Lim. Julienne LaChance
Introduction to Financial Mathematics and Engineering A guide, based on lecture notes by Professor Chjan Lim Julienne LaChance Lecture 1. The Basics risk- involves an unknown outcome, but a known probability
More informationThe internal rate of return (IRR) is a venerable technique for evaluating deterministic cash flow streams.
MANAGEMENT SCIENCE Vol. 55, No. 6, June 2009, pp. 1030 1034 issn 0025-1909 eissn 1526-5501 09 5506 1030 informs doi 10.1287/mnsc.1080.0989 2009 INFORMS An Extension of the Internal Rate of Return to Stochastic
More informationMATH 425: BINOMIAL TREES
MATH 425: BINOMIAL TREES G. BERKOLAIKO Summary. These notes will discuss: 1-level binomial tree for a call, fair price and the hedging procedure 1-level binomial tree for a general derivative, fair price
More informationLecture 17 Option pricing in the one-period binomial model.
Lecture: 17 Course: M339D/M389D - Intro to Financial Math Page: 1 of 9 University of Texas at Austin Lecture 17 Option pricing in the one-period binomial model. 17.1. Introduction. Recall the one-period
More information******************************* The multi-period binomial model generalizes the single-period binomial model we considered in Section 2.
Derivative Securities Multiperiod Binomial Trees. We turn to the valuation of derivative securities in a time-dependent setting. We focus for now on multi-period binomial models, i.e. binomial trees. This
More informationnon linear Payoffs Markus K. Brunnermeier
Institutional Finance Lecture 10: Dynamic Arbitrage to Replicate non linear Payoffs Markus K. Brunnermeier Preceptor: Dong Beom Choi Princeton University 1 BINOMIAL OPTION PRICING Consider a European call
More informationsample-bookchapter 2015/7/7 9:44 page 1 #1 THE BINOMIAL MODEL
sample-bookchapter 2015/7/7 9:44 page 1 #1 1 THE BINOMIAL MODEL In this chapter we will study, in some detail, the simplest possible nontrivial model of a financial market the binomial model. This is a
More informationChapter 15: Jump Processes and Incomplete Markets. 1 Jumps as One Explanation of Incomplete Markets
Chapter 5: Jump Processes and Incomplete Markets Jumps as One Explanation of Incomplete Markets It is easy to argue that Brownian motion paths cannot model actual stock price movements properly in reality,
More informationNo-arbitrage Pricing Approach and Fundamental Theorem of Asset Pricing
No-arbitrage Pricing Approach and Fundamental Theorem of Asset Pricing presented by Yue Kuen KWOK Department of Mathematics Hong Kong University of Science and Technology 1 Parable of the bookmaker Taking
More informationOption Pricing. Simple Arbitrage Relations. Payoffs to Call and Put Options. Black-Scholes Model. Put-Call Parity. Implied Volatility
Simple Arbitrage Relations Payoffs to Call and Put Options Black-Scholes Model Put-Call Parity Implied Volatility Option Pricing Options: Definitions A call option gives the buyer the right, but not the
More informationGame Theory Notes: Examples of Games with Dominant Strategy Equilibrium or Nash Equilibrium
Game Theory Notes: Examples of Games with Dominant Strategy Equilibrium or Nash Equilibrium Below are two different games. The first game has a dominant strategy equilibrium. The second game has two Nash
More informationStochastic Models. Introduction to Derivatives. Walt Pohl. April 10, Department of Business Administration
Stochastic Models Introduction to Derivatives Walt Pohl Universität Zürich Department of Business Administration April 10, 2013 Decision Making, The Easy Case There is one case where deciding between two
More information( 0) ,...,S N ,S 2 ( 0)... S N S 2. N and a portfolio is created that way, the value of the portfolio at time 0 is: (0) N S N ( 1, ) +...
No-Arbitrage Pricing Theory Single-Period odel There are N securities denoted ( S,S,...,S N ), they can be stocks, bonds, or any securities, we assume they are all traded, and have prices available. Ω
More informationFixed-Income Securities Lecture 5: Tools from Option Pricing
Fixed-Income Securities Lecture 5: Tools from Option Pricing Philip H. Dybvig Washington University in Saint Louis Review of binomial option pricing Interest rates and option pricing Effective duration
More informationStochastic Finance - A Numeraire Approach
Stochastic Finance - A Numeraire Approach Stochastické modelování v ekonomii a financích 28th November and 5th December 2011 1 Motivation for Numeraire Approach 1 Motivation for Numeraire Approach 2 1
More informationBUSM 411: Derivatives and Fixed Income
BUSM 411: Derivatives and Fixed Income 12. Binomial Option Pricing Binomial option pricing enables us to determine the price of an option, given the characteristics of the stock other underlying asset
More informationThe Black-Scholes PDE from Scratch
The Black-Scholes PDE from Scratch chris bemis November 27, 2006 0-0 Goal: Derive the Black-Scholes PDE To do this, we will need to: Come up with some dynamics for the stock returns Discuss Brownian motion
More informationProbability. An intro for calculus students P= Figure 1: A normal integral
Probability An intro for calculus students.8.6.4.2 P=.87 2 3 4 Figure : A normal integral Suppose we flip a coin 2 times; what is the probability that we get more than 2 heads? Suppose we roll a six-sided
More informationA Scholar s Introduction to Stocks, Bonds and Derivatives
A Scholar s Introduction to Stocks, Bonds and Derivatives Martin V. Day June 8, 2004 1 Introduction This course concerns mathematical models of some basic financial assets: stocks, bonds and derivative
More informationELEMENTS OF MATRIX MATHEMATICS
QRMC07 9/7/0 4:45 PM Page 5 CHAPTER SEVEN ELEMENTS OF MATRIX MATHEMATICS 7. AN INTRODUCTION TO MATRICES Investors frequently encounter situations involving numerous potential outcomes, many discrete periods
More informationThe Binomial Lattice Model for Stocks: Introduction to Option Pricing
1/33 The Binomial Lattice Model for Stocks: Introduction to Option Pricing Professor Karl Sigman Columbia University Dept. IEOR New York City USA 2/33 Outline The Binomial Lattice Model (BLM) as a Model
More informationHedging Credit Derivatives in Intensity Based Models
Hedging Credit Derivatives in Intensity Based Models PETER CARR Head of Quantitative Financial Research, Bloomberg LP, New York Director of the Masters Program in Math Finance, Courant Institute, NYU Stanford
More informationCertainty and Uncertainty in the Taxation of Risky Returns
Certainty and Uncertainty in the Taxation of Risky Returns Thomas J. Brennan This Draft: October 21, 2009 Preliminary and Incomplete Please Do Not Quote Abstract I extend the general equilibrium techniques
More informationPractical Hedging: From Theory to Practice. OSU Financial Mathematics Seminar May 5, 2008
Practical Hedging: From Theory to Practice OSU Financial Mathematics Seminar May 5, 008 Background Dynamic replication is a risk management technique used to mitigate market risk We hope to spend a certain
More informationlast problem outlines how the Black Scholes PDE (and its derivation) may be modified to account for the payment of stock dividends.
224 10 Arbitrage and SDEs last problem outlines how the Black Scholes PDE (and its derivation) may be modified to account for the payment of stock dividends. 10.1 (Calculation of Delta First and Finest
More informationBinomial Option Pricing
Binomial Option Pricing The wonderful Cox Ross Rubinstein model Nico van der Wijst 1 D. van der Wijst Finance for science and technology students 1 Introduction 2 3 4 2 D. van der Wijst Finance for science
More informationFinance 197. Simple One-time Interest
Finance 197 Finance We have to work with money every day. While balancing your checkbook or calculating your monthly expenditures on espresso requires only arithmetic, when we start saving, planning for
More informationHedging and Pricing in the Binomial Model
Hedging and Pricing in the Binomial Model Peter Carr Bloomberg LP and Courant Institute, NYU Continuous Time Finance Lecture 2 Wednesday, January 26th, 2005 One Period Model Initial Setup: 0 risk-free
More informationNotes for Lecture 5 (February 28)
Midterm 7:40 9:00 on March 14 Ground rules: Closed book. You should bring a calculator. You may bring one 8 1/2 x 11 sheet of paper with whatever you want written on the two sides. Suggested study questions
More informationMS-E2114 Investment Science Exercise 10/2016, Solutions
A simple and versatile model of asset dynamics is the binomial lattice. In this model, the asset price is multiplied by either factor u (up) or d (down) in each period, according to probabilities p and
More informationDERIVATIVE SECURITIES Lecture 5: Fixed-income securities
DERIVATIVE SECURITIES Lecture 5: Fixed-income securities Philip H. Dybvig Washington University in Saint Louis Interest rates Interest rate derivative pricing: general issues Bond and bond option pricing
More informationPricing Options with Binomial Trees
Pricing Options with Binomial Trees MATH 472 Financial Mathematics J. Robert Buchanan 2018 Objectives In this lesson we will learn: a simple discrete framework for pricing options, how to calculate risk-neutral
More informationLecture 8: Introduction to asset pricing
THE UNIVERSITY OF SOUTHAMPTON Paul Klein Office: Murray Building, 3005 Email: p.klein@soton.ac.uk URL: http://paulklein.se Economics 3010 Topics in Macroeconomics 3 Autumn 2010 Lecture 8: Introduction
More informationFINANCIAL OPTION ANALYSIS HANDOUTS
FINANCIAL OPTION ANALYSIS HANDOUTS 1 2 FAIR PRICING There is a market for an object called S. The prevailing price today is S 0 = 100. At this price the object S can be bought or sold by anyone for any
More informationThe parable of the bookmaker
The parable of the bookmaker Consider a race between two horses ( red and green ). Assume that the bookmaker estimates the chances of red to win as 5% (and hence the chances of green to win are 75%). This
More informationIntroduction to Real Options
IEOR E4706: Foundations of Financial Engineering c 2016 by Martin Haugh Introduction to Real Options We introduce real options and discuss some of the issues and solution methods that arise when tackling
More informationPage 1. Real Options for Engineering Systems. Financial Options. Leverage. Session 4: Valuation of financial options
Real Options for Engineering Systems Session 4: Valuation of financial options Stefan Scholtes Judge Institute of Management, CU Slide 1 Financial Options Option: Right (but not obligation) to buy ( call
More informationMonte Carlo Methods in Structuring and Derivatives Pricing
Monte Carlo Methods in Structuring and Derivatives Pricing Prof. Manuela Pedio (guest) 20263 Advanced Tools for Risk Management and Pricing Spring 2017 Outline and objectives The basic Monte Carlo algorithm
More information1 The continuous time limit
Derivative Securities, Courant Institute, Fall 2008 http://www.math.nyu.edu/faculty/goodman/teaching/derivsec08/index.html Jonathan Goodman and Keith Lewis Supplementary notes and comments, Section 3 1
More informationPricing theory of financial derivatives
Pricing theory of financial derivatives One-period securities model S denotes the price process {S(t) : t = 0, 1}, where S(t) = (S 1 (t) S 2 (t) S M (t)). Here, M is the number of securities. At t = 1,
More informationMSC FINANCIAL ENGINEERING PRICING I, AUTUMN LECTURE 6: EXTENSIONS OF BLACK AND SCHOLES RAYMOND BRUMMELHUIS DEPARTMENT EMS BIRKBECK
MSC FINANCIAL ENGINEERING PRICING I, AUTUMN 2010-2011 LECTURE 6: EXTENSIONS OF BLACK AND SCHOLES RAYMOND BRUMMELHUIS DEPARTMENT EMS BIRKBECK In this section we look at some easy extensions of the Black
More informationEconomic Risk and Decision Analysis for Oil and Gas Industry CE School of Engineering and Technology Asian Institute of Technology
Economic Risk and Decision Analysis for Oil and Gas Industry CE81.98 School of Engineering and Technology Asian Institute of Technology January Semester Presented by Dr. Thitisak Boonpramote Department
More informationThe Pennsylvania State University. The Graduate School. Department of Industrial Engineering AMERICAN-ASIAN OPTION PRICING BASED ON MONTE CARLO
The Pennsylvania State University The Graduate School Department of Industrial Engineering AMERICAN-ASIAN OPTION PRICING BASED ON MONTE CARLO SIMULATION METHOD A Thesis in Industrial Engineering and Operations
More informationSection 5.6 Factoring Strategies
Section 5.6 Factoring Strategies INTRODUCTION Let s review what you should know about factoring. (1) Factors imply multiplication Whenever we refer to factors, we are either directly or indirectly referring
More informationLecture Notes for Chapter 6. 1 Prototype model: a one-step binomial tree
Lecture Notes for Chapter 6 This is the chapter that brings together the mathematical tools (Brownian motion, Itô calculus) and the financial justifications (no-arbitrage pricing) to produce the derivative
More informationFixed-Income Options
Fixed-Income Options Consider a two-year 99 European call on the three-year, 5% Treasury. Assume the Treasury pays annual interest. From p. 852 the three-year Treasury s price minus the $5 interest could
More informationMATH 5510 Mathematical Models of Financial Derivatives. Topic 1 Risk neutral pricing principles under single-period securities models
MATH 5510 Mathematical Models of Financial Derivatives Topic 1 Risk neutral pricing principles under single-period securities models 1.1 Law of one price and Arrow securities 1.2 No-arbitrage theory and
More informationIn chapter 5, we approximated the Black-Scholes model
Chapter 7 The Black-Scholes Equation In chapter 5, we approximated the Black-Scholes model ds t /S t = µ dt + σ dx t 7.1) with a suitable Binomial model and were able to derive a pricing formula for option
More informationECON Microeconomics II IRYNA DUDNYK. Auctions.
Auctions. What is an auction? When and whhy do we need auctions? Auction is a mechanism of allocating a particular object at a certain price. Allocating part concerns who will get the object and the price
More informationComputational Finance Least Squares Monte Carlo
Computational Finance Least Squares Monte Carlo School of Mathematics 2019 Monte Carlo and Binomial Methods In the last two lectures we discussed the binomial tree method and convergence problems. One
More informationOPTION VALUATION Fall 2000
OPTION VALUATION Fall 2000 2 Essentially there are two models for pricing options a. Black Scholes Model b. Binomial option Pricing Model For equities, usual model is Black Scholes. For most bond options
More informationECO OPTIONS AND FUTURES SPRING Options
ECO-30004 OPTIONS AND FUTURES SPRING 2008 Options These notes describe the payoffs to European and American put and call options the so-called plain vanilla options. We consider the payoffs to these options
More informationLecture Quantitative Finance Spring Term 2015
and Lecture Quantitative Finance Spring Term 2015 Prof. Dr. Erich Walter Farkas Lecture 06: March 26, 2015 1 / 47 Remember and Previous chapters: introduction to the theory of options put-call parity fundamentals
More informationArbitrage Pricing. What is an Equivalent Martingale Measure, and why should a bookie care? Department of Mathematics University of Texas at Austin
Arbitrage Pricing What is an Equivalent Martingale Measure, and why should a bookie care? Department of Mathematics University of Texas at Austin March 27, 2010 Introduction What is Mathematical Finance?
More informationBest Reply Behavior. Michael Peters. December 27, 2013
Best Reply Behavior Michael Peters December 27, 2013 1 Introduction So far, we have concentrated on individual optimization. This unified way of thinking about individual behavior makes it possible to
More informationIntroduction to Binomial Trees. Chapter 12
Introduction to Binomial Trees Chapter 12 Fundamentals of Futures and Options Markets, 8th Ed, Ch 12, Copyright John C. Hull 2013 1 A Simple Binomial Model A stock price is currently $20. In three months
More information1.1 Interest rates Time value of money
Lecture 1 Pre- Derivatives Basics Stocks and bonds are referred to as underlying basic assets in financial markets. Nowadays, more and more derivatives are constructed and traded whose payoffs depend on
More informationAppendix A Financial Calculations
Derivatives Demystified: A Step-by-Step Guide to Forwards, Futures, Swaps and Options, Second Edition By Andrew M. Chisholm 010 John Wiley & Sons, Ltd. Appendix A Financial Calculations TIME VALUE OF MONEY
More informationChristiano 362, Winter 2006 Lecture #3: More on Exchange Rates More on the idea that exchange rates move around a lot.
Christiano 362, Winter 2006 Lecture #3: More on Exchange Rates More on the idea that exchange rates move around a lot. 1.Theexampleattheendoflecture#2discussedalargemovementin the US-Japanese exchange
More informationForwards, Futures, Options and Swaps
Forwards, Futures, Options and Swaps A derivative asset is any asset whose payoff, price or value depends on the payoff, price or value of another asset. The underlying or primitive asset may be almost
More informationB6302 B7302 Sample Placement Exam Answer Sheet (answers are indicated in bold)
B6302 B7302 Sample Placement Exam Answer Sheet (answers are indicated in bold) Part 1: Multiple Choice Question 1 Consider the following information on three mutual funds (all information is in annualized
More information