Pricing of Financial Derivatives: Integral Form and Solution Vector for American Options, Notas de estudo de Economia

Baixe Pricing of Financial Derivatives: Integral Form and Solution Vector for American Options e outras Notas de estudo em PDF para Economia, somente na Docsity! FINANCE AND CAPITAL MARKETS SERIES DERIVATIVES AND INTERNAL MODELS Fourth Edition Hans-Peter Deutsch DERIVATIVES AND INTERNAL MODELS © Hans-Peter Deutsch 2009 All rights reserved. No reproduction, copy or transmission of this publication may be made without written permission. No portion of this publication may be reproduced, copied or transmitted save with written permission or in accordance with the provisions of the Copyright, Designs and Patents Act 1988, or under the terms of any licence permitting limited copying issued by the Copyright Licensing Agency, Saffron House, 6-10 Kirby Street, London EC1N 8TS. Any person who does any unauthorized act in relation to this publication may be liable to criminal prosecution and civil claims for damages. The author has asserted his right to be identified as the author of this work in accordance with the Copyright, Designs and Patents Act 1988. First published in 2009 by PALGRAVE MACMILLAN Palgrave Macmillan in the UK is an imprint of Macmillan Publishers Limited, registered in England, company number 785998, of Houndmills, Basingstoke, Hampshire RG21 6XS. Palgrave Macmillan in the US is a division of St Martin’s Press LLC, 175 Fifth Avenue, New York, NY 10010. Palgrave Macmillan is the global academic imprint of the above companies and has companies and representatives throughout the world. Palgrave™ and Macmillan™ are registered trademarks in the United States, the United Kingdom, Europe and other countries. ISBN-13: 978–0–230–22215–1 ISBN-10: 0–230–22215–3 This book is printed on paper suitable for recycling and made from fully managed and sustained forest sources. Logging, pulping and manufacturing processes are expected to conform to the environmental regulations of the country of origin. A catalogue record for this book is available from the British Library. A catalog record for this book is available from the Library of Congress. 10 9 8 7 6 5 4 3 2 1 18 17 16 15 14 13 12 11 10 09 Printed and bound in Great Britain by CPI Antony Rowe, Chippenham and Eastbourne Contents List of Figures xi List of Tables xv Preface xvii Part I Fundamentals 1 Introduction 3 2 Fundamental Risk Factors of Financial Markets 7 2.1 Interest rates 7 2.2 Market prices 21 2.3 An intuitive model for financial risk factors 22 2.4 Ito processes and stochastic analysis 32 3 Financial Instruments: A System of Derivatives and Underlyings 48 3.1 Spot transactions 49 3.2 Forward transactions 56 3.3 Options 59 Part II Methods 4 Overview of the Assumptions 65 5 Present Value Methods, Yields, and Traditional Risk Measures 68 5.1 Present value and yield to maturity 68 5.2 Internal rate of return and net present value 70 5.3 Accrued interest, residual debt, and par rates 73 5.4 Traditional sensitivities of interest rate instruments 76 6 Arbitrage 84 6.1 Forward contracts 84 6.2 Options 89 v vi CONTENTS 7 The Black-Scholes Differential Equation 94 7.1 The Black-Scholes equation from arbitrage arguments 95 7.2 The Black-Scholes equation and the backward equation 101 7.3 The relationship to the heat equation 105 8 Integral Forms and Analytic Solutions in the Black-Scholes World 109 8.1 Option prices as solutions of the heat equation 109 8.2 Option prices and transition probabilities 111 8.3 Compilation of Black-Scholes option prices for different underlyings 114 9 Numerical Solutions Using Finite Differences 121 9.1 Discretizing the Black-Scholes equation 122 9.2 Difference schemes 129 9.3 Convergence criteria 151 9.4 Discrete dividends 156 9.5 Example 157 10 Binomial and Trinomial Trees 161 10.1 General trees 161 10.2 Recombinant trees 165 10.3 The relationship between random walk and binomial parameters 173 10.4 The binomial model with infinitesimal steps 176 10.5 Trinomial trees 178 11 Monte Carlo Simulations 184 11.1 A simple example: The area of a disk 186 11.2 The general approach to Monte Carlo simulations 190 11.3 Monte Carlo simulation of risk factors 191 11.4 Pricing 197 12 Hedging 199 12.1 Replicating portfolios as synthetic derivatives 199 12.2 Hedging derivatives with spot transactions 200 12.3 Hedging derivatives with forward contracts 203 12.4 Hedge-ratios for arbitrary combinations of financial instruments 209 12.5 “Greek” risk management with sensitivities 211 12.6 Computation of the greek risk variables 218 13 Martingales and Numeraires 224 13.1 The martingale property 224 13.2 The numeraire 226 13.3 Self-financing portfolio strategies 230 CONTENTS ix 26.4 The optimal portfolio 544 26.5 The efficient frontier revisited 547 27 Active Management and Benchmarking 552 27.1 The capital asset pricing model 552 27.2 Benchmarking against an index 554 27.3 Benchmark and characteristic portfolios 559 27.4 Relations between Sharpe ratio and information ratio 567 Part VI Market Data 28 Interest Rate Term Structures 575 28.1 Bootstrapping 576 28.2 Interpolations 588 29 Volatility 590 29.1 Implied volatilities 590 29.2 Local volatility surfaces 592 29.3 Volatility transformations 599 30 Market Parameter from Historical Time Series 615 30.1 Historical yields, volatility, and correlation 615 30.2 Error estimates 617 30.3 Return and covariance estimates 627 31 Time Series Modeling 634 31.1 Stationary time series and autoregressive models 637 31.2 Calibration of time series models 648 32 Forecasting with Time Series Models 656 32.1 Forecasting with autoregressive models 657 32.2 Volatility forecasts with GARCH(p, q) processes 660 32.3 Volatility forecasts with GARCH(1,1) processes 665 32.4 Volatility forecasts with moving averages 668 33 Principal Component Analysis 671 33.1 The general procedure 671 33.2 Principal component analysis of the German term structure 678 34 Pre-Treatment of Time Series and Assessment of Models 682 34.1 Pre-treatment of time series 682 34.2 Measuring the goodness of time series models 686 A Probability and Statistics 699 A.1 Probability, expectation, and variance 699 A.2 Multivariate distributions, covariance, correlation, and beta 701 x CONTENTS A.3 Moments and characteristic functions 705 A.4 A Collection of important distributions 710 A.5 Transformations between distributions 734 Bibliography 739 Index 745 Figures 2.1 Determining the length of a time period using different day count conventions 10 2.2 The general discount factor for the time span from t to T 12 2.3 Interest rates for the same discount factor based on different day count and compounding conventions 18 2.4 The sequence of the times t, T and T ′ 19 2.5 The general forward discount factor for the time between T and T ′ as seen at time t 19 2.6 Forward rates for periods starting in T = 1, 2, . . . , 15 years for terms T ′ − T = 1, 2, . . . , 15 years 21 2.7 A random walk in two dimensions 23 2.8 End-of-day values of a stock price over a period of 500 trading days 27 2.9 The distribution from Table 2.5 with μdt = 0 and σ√dt = 1 32 2.10 The trinomial tree used to derive the forward equation 42 2.11 The trinomial tree used to derive the backward equation 45 3.1 Examples of common money market instruments 50 3.2 Examples of common capital market securities 54 5.1 Cash flow table of a portfolio 72 5.2 Summary of the different yields 75 9.1 Part of the solution surface of an American plain vanilla call on an underlying with a discrete dividend payment 155 10.1 The first steps in a recombining binomial tree 165 10.2 A simple trinomial tree 179 11.1 A disk with diameter 2m in a square with sides of length 2m 187 11.3 Simulation of ln(S(t)/S(0)) over 2000 steps. 1 step corresponds to 1 day 192 xi xiv FIGURES 34.5 Daily standard deviations of the S&P500 returns from GARCH (solid line), MA (dotted line), and EWMA (dash-dotted line) 697 A.1 2000 iid random numbers uniformly distributed between −1/2 and +1/2 735 A.2 The random numbers from Figure A.1 after Transformation A.97 with n= 12 735 Tables 2.1 The commonly used day count conventions 8 2.2 Business day conventions 11 2.3 Effects of business day conventions 11 2.4 Interest rate factors in general notation and their specific form for the four most commonly used compounding methods 13 2.5 Statistical properties of the logarithm of a risk factor and of the risk factor itself 31 3.1 Some examples of international money market instruments 53 3.2 Interest payment modes in international bond markets 56 6.1 Boundaries for the values of plain vanilla options 89 9.1 Properties of the three most commonly used finite difference methods 152 10.1 Interpretation of the various components in the Black-Scholes option pricing formulae for plain vanilla calls and puts 178 12.1 Sensitivities for options and forward contracts with respect to the spot price S(t) 210 12.2 Definitions of the “Greeks” 212 12.3 Examples of Omega and Delta 215 15.1 Present value and sensitivities of a zero bond in the four most commonly used compounding methods 317 15.2 Long/short conventions for swaps and swaptions 327 17.1 Caplets and floorlets as options on zero bonds 370 17.2 Swaptions as options on coupon bonds 376 18.1 Parameters for the valuation of knock-out options using Equation 18.6 389 xv xvi TABLES 18.2 Classification and “brute force” valuation methods for exotic options 394 21.1 The pros and cons of the most common value at risk methods 469 34.1 Confidence level α = 99% 694 34.2 Confidence level α = 95% 695 CE +41 o Fundamentals This page intentionally left blank CHAPTER 1 Introduction The explosive development of derivative financial instruments continues to provide new possibilities and increasing flexibility to manage finance and risk in a way specifically tailored to the needs of individual investors or firms. This holds in particular for banks and financial services companies who deal primarilywith financial products, but is also becoming increasingly important in other sectors as well. Active financial and risk management in corporate treasury can make a significant contribution to the stability and profitability of a company. For example, the terms (price, interest rate, etc.) of transactions to be concluded at a future date can be fixed today, if desired, even giving the company the option of declining to go aheadwith the transaction later on. These types of transactions obviously have some very attractive uses such as arranging a long-term fixed-rate credit agreement at a specified interest rate a year in advance of the actual transaction with the option to forgo the agreement if the anticipated need for money proves to have been unwarranted (this scenario is realized using what is known as a “payer swaption”) or providing a safeguard against fluctuations in a foreign currency exchange rate by establishing a minimum rate of exchange today for changing foreign currency into euros at a future date (using a foreign currency option). However, the complexity of today’s financial instruments and markets and the growing pace of technological progress have led to an almost uncontrollable increase in the risks involved in trading and treasury while simultaneously reducing drastically the time available for decision making. Thus the inappropriate use of financial instruments may quickly result in losses wiping out gains achieved over years in a company’s primary business (for example, the production and sale of automobiles or computer chips). In recent years we have seen an increase in the number of sizable losses incurred in consequence of derivative transactions which, in some cases, have resulted in bankruptcy. This phenomenon has not been restricted to banks but has involved companies in various other sectors as well. Spectac- ular examples include Metallgesellschaft (oil futures), Procter & Gamble 3 6 DERIVATIVES AND INTERNAL MODELS theoretical and technical. The concepts elaborated upon in Part II will then be applied to specific financial instruments in Part III. This separate treatment of general methods and specific financial instruments will contribute to a clearer understanding of this rather complex material. Once the pricing of themost common financial instruments has been dealt with in Part III, the determination of the risks associated with these instru- mentswill be presented in Part IV. The information about the prices and risks of financial instruments can then be used for decision making, specifically of course for trading decisions and the management of investment portfo- lios. This is demonstrated in Part V. Finally, methods for determining and analyzing the market data and historical time series of market risk factors will be the topic of Part VI. CHAPTER 2 Fundamental Risk Factors of Financial Markets The fundamental risk factors in financial markets are the market parame- ters which determine the price of the financial instruments being traded. They include foreign currency exchange rates and the price of commodities and stocks and, of course, interest rates. Fluctuations in these fundamen- tal risks induce fluctuations in the prices of the financial instruments which they underlie. They constitute an inherent market risk in the financial instru- ments and are therefore referred to as risk factors. The risk factors of a financial instrument are the market parameters (interest rates, foreign cur- rency exchange rates, commodity and stock prices), which, through their fluctuation, produce a change in the price of the financial instrument. The above mentioned risk factors do not exhaust the list of the possible risk fac- tors associated with a financial instrument nor do all risk factors affect the price of each instrument; for example, the value of a 5-year coupon bond in Swiss Francs is not determined by the current market price of gold. The first step in risk management is thus to identify the relevant risk factors of a specified financial instrument. 2 .1 I N T ERE ST R A TE S Various conventions are used in the markets to calculate interest payments. For example, interest rates on securities sold in the US money markets (T-bills, T-bill futures) are computed using linear compounding, whereas in the European money market, simple compounding is used. Interest rates in the capital markets are calculated using discrete or annual compounding 7 8 DERIVATIVES AND INTERNAL MODELS while option prices are determined using the continuous compounding convention. While these conventions are not essential for a fundamental understanding of financial instruments and risk management they are of central importance for the implementation of any pricing, trading, or risk management system. Before entering into a general discussion of spot rates in Section 2.1.5, we will therefore introduce the most important compound- ing conventions here. 2.1.1 Day count conventions Before one of the many compounding conventions are applied to calculate the interest on a certain amount over a period between the time (date) t and a later time T , the number of days between t and T over which interest is accrued must first be counted. The beginning, the end, and the length of this time period (usually measured in years) must be precisely specified. To do this there are again different conventions used in different markets, known as day count conventions, DCC for short. These market usances are usually specified by making use of a forward slash notation: the method for counting the days of the month are specified in front of the slash, the number of days of the year after the slash. A list of the most commonly used conventions is presented in Table 2.1. These conventions compute the length of an interest rate period as follows: Act/Act: The actual number of calendar days are counted and divided by the actual number of days in the year. Actual/365f : The actual number of calendar days between t and T are counted and divided by 365 to obtain the interest period in years regard- less of whether the year concerned is a leap year. This distinguishes the Actual/365f convention from the Act/Act in years. Actual/360: The actual number of calendar days between t and T are counted and divided by 360 to obtain the interest period in years. Table 2.1 The commonly used day count conventions Common designation Alternative designation Act/Act Act, Act/365(l), Act/365(leap), Act/365leap Act/365f Act/365(f), Act/365(fixed), Act/365fixed Act/360 30/360 BondBasis, Bond, 30 30E/360 EuroBondBasis, EuroBond, 30E FUNDAMENTAL RISK FACTORS OF FINANCIAL MARKETS 11 Table 2.2 Business day conventions Convention Adjusted to: Following the following business day Modified Following the following business day as long as this falls within the same month; otherwise the preceding business day Preceding the preceding business day Modified Preceding the preceding banking business day as long as it falls within the same month; otherwise the following business day Table 2.3 Effects of business day conventions Convention Adjustment March 28, 1997 Adjustment April 01, 1997 Following April 01, 1997 April 02, 1997 Modified Following March 27, 1997 April 02, 1997 Preceding March 27, 1997 March 27, 1997 Modified Preceding March 27, 1997 April 02, 1997 Table 2.3 shows as an example the adjustment of March 28, 1996 and April 1, 1997. These examples are interesting in that the days March 29, 1997 and March 30, 1997 fall on a weekend and the days March 28, 1997, March 30, 1997 and March 31, 1997 represent bank holidays (Easter holidays) according to the holiday calendar valid for banks in Frankfurt, Germany. For interest rate instruments, a further distinction is made between whether the adjustment convention holds solely for the payment date of an interest period or for its maturity date as well. The maturity date deter- mines the length of the interest rate period and thus affects the amount of the interest payment (if the length of an interest period is longer, the amount of the interest payment to be made is naturally higher). The payment date determineswhen the interest payments are actuallymade (usually one or two business days after the maturity date) and therefore affects how strongly a payment is discounted, in other words, today’s value of the payment (a later payment is obviously worth less than an earlier one); it is thus relevant when the payment is actually made and not when it was due. If the maturity date of a financial instrument is specified as fixed, i.e., nonmoveable, it is not adjusted. However, the payment date is still adjusted in accordancewith the business day convention for the instrument concerned. The rollover day of an interest rate instrument determines on which day 12 DERIVATIVES AND INTERNAL MODELS and month of each year the rollover from one interest period into the next is to take place, i.e., when the maturity and payment dates of individual interest payments occur. Depending on the selected business day convention, a decision ismade as to how thematurity and payment dates, derived from the rollover date, are to be adjusted. For example, federal bonds are common- ly agreed to be fixed. This means that only the payment date is adjusted to the next valid business day while the maturity date is always the same as the rollover day. For swaps, on the other hand, both the payment and maturity dates are adjusted to the next valid business day subsequent to the rollover day. All these conventions make trading substantially more complicated with- out causing a fundamental change in the properties of the instruments being traded. They are actually unnecessary but the markets are inconceivable without them because of strong historical ties. This is particularly true of holiday calendars, some of which even have religious roots. 2.1.3 Discount factors In order to concentrate on the essentials, a general notation will be observed in this book which holds for all compounding, day count, business day, and other market conventions. To accomplish this, discount factors rather than interest rates will be employed throughout. The discount factor is the value by which a cash flow to be paid at a time T is multiplied in order to obtain the value of the cash flow at an earlier time t (for example, today). Since we (usually) would prefer to have a cash flow today rather than at some future date T , the cash flow paid at time T is worth less today than the same cash flow paid at time t. Thus, the discount factor is generally less than 1 (but greater than 0). A discount factor for discounting from a time T back to an earlier time t at the interest rate R will be referred to using the notation indicated in Figure 2.2. The reader is urged to become familiar with this notation as it will be used throughout the book. Since interest rates are different for different term lengths, the interest rate R is generally a function of both t and T , BR (t,T )Discount factor Interest rate Start of interest period End of interest period Figure 2.2 The general discount factor for the time span from t to T FUNDAMENTAL RISK FACTORS OF FINANCIAL MARKETS 13 i.e., R = R(t, T). In order to keep the notation simple, we will use the notation indicated in Figure 2.2 rather than BR(t,T)(t, T) on the understand- ing that the interest rate R refers to the rate corresponding to the times specified in the argument of the discount factor. If, as is occasionally the case, the interest rate is not dependent on the times (as in some option pric- ing models), we sometimes adopt the convention of denoting the constant interest rate by r and the corresponding discount factor by Br(t, T). The letter B has been used to denote the discount factor because a discount fac- tor is nothing other than the price at time t of a zero-coupon Bond with a maturity T and nominal principal equal to one monetary unit (for example, 1 euro). The discount factor yields the value of a future payment today (discount- ing). Conversely, the future value of a payment today (compounding) is obtained by multiplying the payment by BR(t, T)−1. The interest accrued between times t and T is thus the difference between the compounded value and the original amount, i.e., the original amount multiplied by the factor (BR(t, T)−1 − 1). 2.1.4 Compounding methods The explicit form of the discounting and compounding factors and of the interest accrued are shown for the four most common compounding methods in Table 2.4. At each stage in this book, the results expressed in the general notation given in the first line of Table 2.4 can be converted directly into the explicit expressions of the desired compounding method by replacing the general expression with the appropriate entries in Table 2.4. The factors in Table 2.4 are obtained from intuitive considerations which are now described in detail for each compounding method. Note that interest rates are always quoted per time, for example 6% per annum, 1% per month, or 0.02% per day, etc. Table 2.4 Interest rate factors in general notation and their specific form for the four most commonly used compounding methods Discount factor Compounding factor Interest accrued General BR(t, T) BR(t, T)−1 BR(t, T)−1 − 1 Continuous e−R(T−t) eR(T−t) eR(T−t) − 1 Discrete (1+ R)−(T−t) (1+ R)(T−t) (1+ R)(T−t) − 1 Simple [1+ R(T − t)]−1 1+ R(T − t) R(T − t) Linear 1− R(T − t) 1+ R(T − t) R(T − t) 16 DERIVATIVES AND INTERNAL MODELS for the capital earned by the end of the second year K2 = K1 ( 1+ 1 m R )m = K0 ( 1+ 1 m R )2m and in general after n years Kn = K0 ( 1+ 1 m R )nm . If the time period is measured in the same time unit used to quote the interest rate then T − t = n (even for non-integer valued n) and we thus obtain K(T−t) = K0 ( 1+ 1 m R )(T−t)m . Continuous compounding In the case of continuous compounding, the calculation is performed as if interest payments were made after each infinitesimal small time increment (each payment calculated using simple compounding) with the accumulated interest being immediately reinvested at the same rate. The total capital thus accumulated on an investment over a period of T − t time units is then K(T−t) = lim m→∞K0 ( 1+ 1 m R )(T−t)m = K0eR(T−t). (2.2) The compounding factor is thus given by eR(T−t) as indicated in Table 2.4. Here, Euler’s number e, also called the natural number, arises. Its value is approximately e = 2.718281828459. Euler’s number to the power of some number x is called the exponential function, which is defined by the limit exp(x) := ex := lim m→∞ ( 1+ x m )m . Linear compounding Linear compounding is justified for very short periods of time T−t. For such times, the product R(T − t) is also very small. For example, if R = 3% per annum and the time tomaturityT−t is onemonth= 0.083 years, the product R(T − t) = 0.0025. The square of this product is considerably smaller, namely R(T − t)2 = 0.00000625. Thus, in the case of linear compounding, FUNDAMENTAL RISK FACTORS OF FINANCIAL MARKETS 17 only terms of order R(T − t) are of any importance, i.e., all nonlinear terms are simply neglected. If we represent the discount factor, which is always the inverse of the compounding factor, as a geometric series1 neglecting all terms of higher order, we obtain the discount factor given in Table 2.4: [1+ R(T − t)]−1 ≈ 1− R(T − t)︸ ︷︷ ︸ linear terms + (R(T − t))2 ± · · ·︸ ︷︷ ︸ higher order terms are neglected! . Convention-dependent interest rates As different as the formulas for the discount factors in Table 2.4 may look, we emphasize that it is not the numerical values of the discount factors that are dependent on the compounding convention, but rather the interest rates themselves!After all, today’s value of amonetary unit paid in the futuremust be independent of the convention used for discounting.All convention effects are “absorbed” in the interest rate. Requiring the discount factors for the different compoundingmethods as given inTable 2.4 to be numerically equal enables the conversion of interest rates from one convention into another: e−Rcontinuous(T−t) = 1 (1+ Rdiscrete)T−t = 1 1+ Rsimple(T − t) = 1− Rlinear(T − t). For example, the interest rate necessary to generate a discount factor in discrete discountingwith the same numerical value as a given discount factor in continuous compounding is Rdiscrete = eRcontinuous − 1 =⇒ Rcontinuous = ln(1+ Rdiscrete). However, not only the effects of the compounding methods but also the effects of the day count convention are absorbed in the interest rates. This is demonstrated in the accompanying Excel-sheet Usance.xls and in Figure 2.3. The interest period is the same as in Figure 2.1 but the different day count conventions generate different time lengths (measured in years). The discount factor must be the same for all conventions. The interest rates associated with this single discount factor, however, are strongly influenced by both the day count convention as well as the compounding convention. They vary between 4.82% and 5.14%. As already mentioned, all of the conventions introduced here are actu- ally unnecessary for understanding financial instruments. In order to 1 The expansion used here is (1+ x)−1 = 1− x + x2 − x3 + x4 − x5 ± · · · . Such series expansions can be found in any book of mathematical formulas. The result is now obtained by substituting R(T − t) for x. 18 DERIVATIVES AND INTERNAL MODELS Start date Feb. 15, 00 End date Dec. 31, 00 Discount factor 0.9571 Time period Zero rate (%) Days Years Linear Simple Discrete Contin. Act/act 320 0.87431694 4.91 5.13 5.14 5.01 Act/365f 320 0.87671233 4.89 5.11 5.13 5.00 Act/360 320 0.88888889 4.82 5.04 5.06 4.93 30/360 316 0.87777778 4.89 5.11 5.12 4.99 30/E360 315 0.875 4.90 5.12 5.14 5.01 Figure 2.3 Interest rates for the same discount factor based on different day count and compounding conventions concentrate on the essentials, discount factors rather than interest rates will be predominantly used in this book. For any concrete calculation, the reader should be able to write down explicitly the required discounting factor using Table 2.4 and to calculate the time length T − t using Equation 2.1 (along with the Excel-sheet Usance.xls) after the exact dates t and T in the appropriate business day convention as specified in Table 2.2 have been determined. In what follows, we will therefore work only with the general discount factor given in Figure 2.2. 2.1.5 Spot rates Spot rates are the current yields on securities which generate only one single payment (cash flow) upon maturity. A zero coupon bond is an example of such a security as are coupon bonds whose last coupon payment prior to maturity has already been made. The spot rates as a function of time to maturity T is called the spot rate curve or the term structure. These spot rate curves can be represented by the discount factors BR(t, T). 2.1.6 Forward rates Forward rates are the future spot rates from today’s point of view which are consistent with the current spot rates in the sense of the following arbitrage argument: One monetary unit is to be invested today (time t) until a specified maturity date T ′. The investor can consider the following two investment strategies: 1. Invest the monetary unit without adjusting the position until the maturity date T ′, or FUNDAMENTAL RISK FACTORS OF FINANCIAL MARKETS 21 Matu rity Spot rate (%) Forward rate (%) for 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 2.20 in Years Years Years Years Years Years Years Years Years Years Years Years Years Years Years 2 2.50 1 2.80 3.25 3.67 4.08 4.36 4.65 4.83 5.02 5.20 5.28 5.37 5.45 5.49 5.52 5.54 3 2.90 2 3.70 4.11 4.51 4.76 5.03 5.18 5.34 5.51 5.56 5.63 5.70 5.72 5.74 5.74 4 3.30 3 4.51 4.91 5.11 5.36 5.47 5.61 5.77 5.80 5.84 5.90 5.90 5.91 5.90 5 3.70 4 5.32 5.41 5.65 5.71 5.83 5.98 5.98 6.01 6.06 6.04 6.04 6.02 6 4.00 5 5.51 5.82 5.85 5.96 6.11 6.10 6.11 6.15 6.12 6.11 6.08 7 4.30 6 6.12 6.01 6.11 6.26 6.21 6.21 6.24 6.20 6.18 6.14 8 4.50 7 5.91 6.11 6.31 6.24 6.23 6.26 6.21 6.18 6.14 9 4.70 8 6.31 6.52 6.35 6.31 6.33 6.26 6.22 6.17 10 4.90 9 6.72 6.36 6.31 6.33 6.25 6.21 6.15 11 5.00 10 6.01 6.11 6.21 6.13 6.10 6.05 12 5.10 11 6.21 6.31 6.17 6.13 6.06 13 5.20 12 6.41 6.15 6.10 6.02 14 5.25 13 5.90 5.95 5.90 15 5.30 14 6.00 5.89 16 5.33 15 5.78 0.00% 1 3 5 7 9 11 2.00% 4.00% 6.00% 8.00% Figure 2.6 Forward rates for periods starting in T = 1, 2, . . . , 15 years for terms T ′ −T = 1, 2, . . . , 15 years. From line to line the start points T of the forward periods change by one year. From column to column the lengths T ′ − T of the forward periods change by one year. The inset graphic shows, from top to bottom, the current term structure along with the forward term structures in 1, 5, and 10 years the formulas): R(T , T ′|t) = ⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ R(t, T ′)(T ′ − t)− R(t, T)(T − t) T ′ − T Continuous compounding[ 1+ R(t, T ′)] T ′−tT−t [1+ R(t, T)] T−tT ′−t − 1 Discrete compounding [ 1+ R(t, T ′)(T ′ − t) 1+ R(t, T)(T − t) − 1 ] /(T ′ − T) Simple compounding[ 1− 1− R(t, T ′)(T ′ − t) 1− R(t, T)(T − t) ] /(T ′ − T) Linear compounding This clearly demonstrates the advantage of using the general notation for discount factors Equation 2.4 introduced above. In Figure 2.6, the forward rates are calculated from the spot rates taken from the Excel sheet Plain- Vanilla.xls (see accompanying CD-ROM) using the above formulas for annual, discrete compounding over a period of 15 years. 2 .2 MA RKE T PR IC E S Let S(t) be the spot price at time t of a stock, a commodity, or a currency. The dividend-adjusted spot price S̃(t, T) at time t is the price net of the value 22 DERIVATIVES AND INTERNAL MODELS of all dividends paid between the times t and T . It is given by S̃(t, T) = ⎧⎨⎩ S(t) no dividendS(t)− D(t′)BR(t, t′) dividend payment D due at time t′S(t)Bq(t, T) dividend yield q (2.5) The dividend adjustment is thus accomplished by subtracting the value of dividends, discounted back to time t at the spot rate R, from the spot price, or – in the case of a dividend yield q – discounting the spot price from T back to t using the dividend yield q. For currencies, q represents the risk-free interest rate in the foreign cur- rency, for commodities, it is the difference between the convenience yield and the cost of carry (expressed as a yield, see later).When considering stock indices, the dividend payments from the assets in the index are commonly averaged out to result in a dividend yield q of the index rather than con- sidering each dividend of each stock as an individual payment. However, performance indices, such as the GermanDAX, which require that dividends be reinvested in the index should not be adjusted for dividend payments.2 2 .3 A N I N T U I T I V E M O D E L F O R F I N A N CI A L RI SK F A C TO R S 2.3.1 Random walks as the basis for pricing and risk models The random walk is a mathematical model which is frequently used to char- acterize the random nature of real processes. The concept of the random walk has attained enormous importance in the modern financial world. Most option pricing models (such as the Black-Scholes model) and several meth- ods used in modern risk management (for example, the variance-covariance method) and, of course, Monte Carlo simulations are based on the assump- tion that market prices are in part driven by a random element which can be represented by a randomwalk. It is thereforeworthwhile to acquire an under- standing of random walks if only to develop an intuitive comprehension of stochastic processes which are at the heart of this book. A random walk can be described as follows: starting from some point in space, we travel a random distance in a randomly selected direction. Having arrived at a new point, another such step of random length and direction is taken. Each individual step in the procedure has a length and direction and 2 Except for effects caused by taxation (corporate tax). FUNDAMENTAL RISK FACTORS OF FINANCIAL MARKETS 23 r1 r2 r3 r4 r5 r6 r7 r8R Figure 2.7 A random walk in two dimensions thus can be represented as a vector as shown in Figure 2.7. The completed random walk is a series of such vectors. Each base point of a vector is the end point of its predecessor. At this point, we ask the following important question: what is the distance from the starting point after having completed a random walk consisting of n steps?3 In other words: how large is the “end-to-end distance” repre- sented by the length of the vector R in Figure 2.7? The length and direction of the vector R are random since R is the sum of random steps. As a result, only statistical statements are available to describe the properties of this vector. For example, the length of this vector cannot be determined with cer- tainty but we could calculate its mean length. This requires a large number of random walks with the same number of steps. For each of these random walks, the square of the Euclidean norm of the end-to-end vector R is deter- mined and used to calculate the mean 〈 R2 〉 . The mean end-to-end distance is then defined as the square root of this value. A Monte Carlo simulation (see Chapter 11) could be carried out to generate the random walks and obtain an estimate for the mean 〈 R2 〉 by measuringR2, the square of the end-to-end vector, of each simulated random walk and then take the average of these. In doing so, it can be observed that the square of the end-to-end vector is, on average, proportional to the number of steps taken in the random walk.4 E [ R2 ] ≈ 〈R2〉 ∝ n. Here E[x] denotes the expectation of a random variable x and 〈R2〉 denotes the mean of the squared Euclidean norm of R. This holds irrespective of the dimension of the space in which the random walk occurs. The same result holds for a random walk on a line, in a plane as in Figure 2.7, or in a 15-dimensional Euclidean space [44]. The expectation of the end-to-end vector itself is equal to zero, i.e., E[R] = 0. This is immediate since the end-to-end vector R points in any direction with equal likelihood and therefore all these vectors cancel each 3 Figure 2.7 for instance shows a random walk with n = 8. 4 The symbol “∝” means “is proportional to”. 26 DERIVATIVES AND INTERNAL MODELS 2.3.2 Risk factors as random walks A random variable z(t), whose values change randomly with time t is called a stochastic process. The process is called Markov if its future behavior is influenced solely by its value at the present time t. This means intuitively that the future behavior is independent of the path taken to reach the present value.Assuming that the current value of a risk factor, such as a stock price or an interest rate, contains all the information about its historical development (this is called weak market efficiency), it follows that the subsequent values taken on by such a risk factor depend only on the current price and other external effects, such as politics, but not on the past prices or rates. Market prices can be then assumed to be Markov processes. In order to derive a model for the Markov process S(t) representing the time-evolution of a risk factor, we assume that the process can be split into a random and a deterministic component. The deterministic component is called the drift. We will begin our discussion with an analysis of the random component. The derivation of the model describing the random component given here is fundamentally different from that which is commonly found in the related literature and is based on the general properties of the random walk, Equations 2.6 and 2.7. The literature usually begins with the introduction of a model and proceeds with a presentation of calculations and results fol- lowing from the model assumption. It is more intuitive, however, to begin with a derivation of the model itself, i.e., to illustrate the steps and describe the motivation leading to its construction. Thus, this section serves the dual purpose of introducing a model for future analysis and an example of the modeling process. What follows is a detailed discussion of how we can con- struct a model describing a real phenomenon by means of abstraction (and intuition!) which is simple enough to allow a mathematical and/or compu- tational analysis but still complex enough to capture the essential features of the real phenomenon. A real process, for example the closing prices of a stock on 500 days, might look like the points shown in Figure 2.8. We propose to model this process. To do so, several questions must first be answered: What is the fundamental idea behind the model? As mentioned above, the random properties of many processes can be traced back to the general concept of the random walk.We therefore use the random walk as the basis for our model. A random walk in which dimension? A market price can rise or fall, i.e., can change in only two directions. A randomwalk in d-dimensional space allows for an upward or downward change in d linearly independent directions, i.e., there are 2d possible FUNDAMENTAL RISK FACTORS OF FINANCIAL MARKETS 27 65 67 69 71 73 75 77 79 81 0 50 100 150 200 250 300 350 400 450 500 S1 S2 S3 S4 S5S0 dt € Days Figure 2.8 End-of-day values of a stock price over a period of 500 trading days. The values after 0, 100, 200, . . . , 500 days are denoted by S0, S1, S2, . . . , S5 changes in direction. Thus, the dimension required for the description of just upward or downward changes is d = 1. Which real parameter is described by the number of steps n in the random walk? In order to observe a change in price (in other words for a step in a random walk to be taken), one thing must occur: time must pass. If the price is observed in regular, fixed time intervals dt (for example, every 100 days as in Figure 2.8, or daily or hourly, etc.), then the amount of time passing between steps is dt. If the entire random walk occurs between t (= today) and a future date T then T − t = ndt (2.8) Since dt is a constant, the number of steps n is proportional to the time in which the random walk occurs, i.e., proportional to T − t. Which real parameter should be modeled by a random walk? At first glance, we might take the market price of a risk factor. The market price evolution S5 − S0 over the entire period in Figure 2.8 can be decomposed into individual steps as follows : S5 − S0 = (S0 − S1)− (S2 − S1)+ (S3 − S2) + (S4 − S3)+ (S5 − S4), 28 DERIVATIVES AND INTERNAL MODELS or more generally, Sn − S0 = n∑ i=1 dSi with dSi = Si − Si−1. If the market price itself were a random walk, then as a result of the self similarity property, the individual steps dSi would also be random walks. The price differences however are real cash amounts given in euros, for example. This would mean that a security costing 1000 euros would experience the same fluctuations (in euros) as one costing only 10 euros. This is surely not the case. It would make much more sense to consider relative fluctuations. Our next candidate for a step in our random walk could therefore be the ratio Si/Si−1. The ratio of the last price to the first is given by S5 S0 = S1 S0 S2 S1 S3 S2 S4 S3 S5 S4 , or more generally, Sn S0 = n∏ i=1 Si Si−1 . This is the product of the individual steps and not their sum. A random walk however, is a vector and as such must always be the sum of its component steps. In light of this fact, the ratios Si/Si−1 are completely unsuitable for the steps of a random walk as they are not even vectors! However, the ratios Si/Si−1, whichmake economic sense, can be utilized by converting the products into sums by taking the logarithm of both sides of the above equation. The functional align for the logarithm is given by ln(a× b) = ln(a)+ ln(b), ln(a/b) = ln(a)− ln(b). Thus taking the logarithm of both sides of the above product yields ln ( S5 S0 ) = ln ( S1 S0 ) + ln ( S2 S1 ) + ln ( S3 S2 ) + ln ( S4 S3 ) + ln ( S5 S4 ) , or more generally, ln ( Sn S0 ) = n∑ i=1 ln ( Si S0i − 1 ) FUNDAMENTAL RISK FACTORS OF FINANCIAL MARKETS 31 Table 2.5 Statistical properties of the logarithm of a risk factor and of the risk factor itself RandomVariable x = ln ( S(t + dt) S(t) ) x = ( S(t + dt) S(t) ) Distribution Normal Distribution Lognormal Distribution Density 1√ 2πσ 2dt e − (x−μdt)2 2σ2dt 1 x √ 2πσ 2dt e − (ln(x)−μdt)2 2σ2dt P(x ≤ a) 1√ 2πσ 2dt a∫ −∞ e − (x−μdt)2 2σ2dt dx 1√ 2πσ 2dt ln(a)∫ −∞ e − (x−μdt)2 2σ2dt dx Expectation μdt e(μ+ σ2 )dt Variance σ 2dt e2μdt(e2σ 2dt − eσ 2dt) of a risk factor can be represented as the appropriately discounted expected future value. In order to ensure that this property is satisfied when model- ing market movements with a random walk, two parameters are required. To provide a motivation8 for the second parameter in our random walk, we consider the following: the end-to-end vector in our 1-dimensional random walk is given by R = ln [S(T)] − ln [S(t)]. If our random walk is to serve as a model for a stock price which, in the long term, should show a positive mean return, then the expectation of the end-to-end vector should not be equal to zero but should grow with time. This is incorporated into the model by introducing a deterministic term, called the drift: d ln [S(t)] = μdt + σdW (2.13) Since no additional random component has been introduced into the model, ln(S(t)) retains the normal distribution with variance σ 2(T − t) after the passage of time from t to T . The expectation, however, is now given by μ(T− t). The addition of the drift into our randomwalk has the consequence that the mean of the random walk is no longer 0 but rather μ(T − t). A random variable whose logarithm has a normal distribution has a log- normal distribution. The price S(t) is thus lognormally distributed. Themost important properties of ln(S) and S are summarized in Table 2.5; see also Section A.4. In Figure 2.9, the density functions and cumulative probabilities of the normal and lognormal distribution are shown. Here, the parameters μdt and σ √ dt have been set equal to 0 and 1, respectively. The random walk model just derived can, of course, be generalized by allowing for nonconstant drifts and volatilities. If σ is expressed as in 8 This is only amotivation based on the example and not a general explanation for a drift. For example, when considering an interest rate as a risk factor, there is no reason to believe that the change in the interest rate should show a “mean return.” The more general and profounder explanation for the drift is that the 32 DERIVATIVES AND INTERNAL MODELS 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1 −3 −2 ,8 −2 ,6 −2 ,4 −2 ,2 −2 −1 ,8 −1 ,6 −1 ,4 −1 ,2 −1 −0 ,8 −0 ,6 −0 ,4 −0 ,2 0 0, 2 0, 4 0, 6 0, 8 1 1, 2 1, 4 1, 6 1, 8 2 2, 2 2, 4 2, 6 2, 8 3 Random number P ro ba bi lit ie s cumulative normal normal density cumulative lognormal lognormal density Figure 2.9 The distribution from Table 2.5 with μdt = 0 and σ√dt = 1. With these parameters the normal distribution has mean 0 and variance 1 while the lognormal distribution has mean √ e ≈ 1, 65 and variance e2 − e ≈ 4, 67. Equation 2.10 in terms of the variance and if the yield is left in its most general form as a function of time t and the market price S(t) at time t, Equation 2.13 can be expressed in the following generalized form: d ln (S(t)) = μ(S(t), t)dt + X√var[d ln (S(t))] with X ∼ N(0, 1) (2.14) This equation is now in the general form of a stochastic diffusion process as given by Equation 2.15 and is the starting point for more general stochastic models for the market parameters. These will receive a detailed treatment in the following chapter. Concrete examples of such generalized stochastic processes can be found in Section 31.1. 2 .4 I T O PRO CE S S E S A N D S TOC HA S T IC A N A L YSI S In the previous section, risk factors in financial markets were introduced and a motivation and derivation of an intuitive model (the random walk) was risk factors must be modeled in such a way that there exists a probability measure in which all (tradeable and properly normalized) financial instruments are martingales. FUNDAMENTAL RISK FACTORS OF FINANCIAL MARKETS 33 provided to describe them. Now, this section will be devoted to the more theoretical fundamentals underlying these concepts, namely to stochastic analysis. Stochastic analysis is the branch of mathematics dealing with the investigation of stochastic processes. Particularly close attentionwill be paid to results which find application in finance. This subject is naturally quite theoretical, but should give the interested reader a deeper understanding of relationships between different fundamental concepts in financialmathemat- ics. Nevertheless, the readerwho is less interested in themathematical details may skip over the rest of this chapter and continue directly to Chapter 3 and continue directly on to Chapter 4. 2.4.1 General diffusion processes All stochastic processes in this book which will be used to model risk factors satisfy – as long as there is only one single random variable involved – an equation of the following form: dS(t) = a (S, t) dt + b(S, t) dW with dW ∼ X√dt, X ∼ N(0, 1) (2.15) Here, W denotes – as always – a Brownian motion and X a standard nor- mally distributed random variable. Processes satisfying an equation of this type are called diffusion processes or Ito processes. These quite general stochastic processes have long since been the subject of research in the field of stochastic analysis. The parameters a(S, t) and b(S, t) are called the drift rate and the volatility of the Ito process. They may depend on the time t, on the stochastic process S or on both. The interpretation of the variable S depends on the particular application under consideration. In the simplemodels derived in Section 2.3, for example in Equation 2.13, the logarithm of the risk factor wasmodeled as the stochastic variable. These models are of the form 2.15 with the stochastic variable being given by ln(S), where b(S, t) = σ and a (S, t) = μ. The firstmoments of the conditional probability distribution of the general Ito process are E [dS(t)] = a(S, t) dt var [dS(t)] = E [(dS − E [dS])2] = E [(b(S, t) dW)2] = b(S, t)2dt E [ dS(t)2 ] = E [(a(S, t)dt + b(S, t)dW)2] ≈ b(t, S)2dt + higher order terms in dt 36 DERIVATIVES AND INTERNAL MODELS In the literature, an equivalent approach is sometimes taken to arrive at this equation. A random walk model for the risk factor S(t) is introduced in the first place (and not for its logarithm as done here) with a subsequent application of Ito’s lemma to derive the process for ln(S(t)). Explicitly, one begins with the process dS(t) = S(t)μ̃ dt + S(t)σdW (2.20) obtaining from Ito’s lemma10 d ln(S(t)) = ( μ̃− σ 2 2 ) dt + S(t)σdW (2.21) This result corresponds to Equations 2.19 and 2.13 in our development with a somewhat modified drift. μ̃ = μ+ σ 2 2 (2.22) The model is thus exactly the same, only the drift must be reinterpreted; see Equations 2.26 and 2.28 below. The formulation of 2.19 and 2.20, in which the risk factor (and not its logarithm) is directly modeled as a stochastic process, of course also has the form of a diffusion process as in Equation 2.15, with the choice of a and b given by b(S, t)= S(t)σ and a(S, t)= S(t)μ̃ or equivalently, a(S, t) = S(t)(μ+ σ 2/2). The Process for the risk factor over a finite time interval Equation 2.15 describes the infinitesimal change in S and thus determines the differential of S. We are therefore dealing with a (partial) differential equation. Because it contains the stochastic component dW , it is referred to as a stochastic partial differential equation, often abbreviated by SPDE. Special cases such as Equation 2.19, for example, are SPDEs as well. With the aid of Ito’s lemma and the general diffusion process, Equation 2.15, an equation for finite changes in S (over a finite, positive time span δt) can be derived by solving the SPDE 2.19 (which holds for infinitesimal changes dS). For this purpose, we use the process 2.15 with a(y, t) = 0 and b(y, t) = 1, i.e., simply11 dy(t) = dW(t). Now we construct 10 Equation 2.17 with f (S, t) = ln(S(t)). 11 In order to avoid confusion in the notation, we denote the stochastic variable by y. FUNDAMENTAL RISK FACTORS OF FINANCIAL MARKETS 37 a function S of the stochastic variable y by S(y, t) := S0 exp (μt + σy) where y(t) = W(t) is the value of the Wiener process at time t, and S0 is an arbitrary factor. Ito’s lemma gives us the process for S induced by the process dy: dS = ⎡⎢⎢⎢⎣ ∂S∂y︸︷︷︸ σS a(y, t)︸ ︷︷ ︸ 0 + ∂S ∂t︸︷︷︸ μS + 1 2 b(y, t)2︸ ︷︷ ︸ 1 ∂2S ∂y2︸︷︷︸ σ 2S ⎤⎥⎥⎥⎦ dt + ∂S∂y︸︷︷︸ σS b(y, t)︸ ︷︷ ︸ 1 dW = (μ+ σ 2 2 )Sdt + σSdW . This corresponds exactly to the process in Equation 2.19. Thismeans that the process S thus constructed satisfies the stochastic differential equation 2.19, i.e., is a solution of this SPDE. Simply making the substitution t → t + δt we obtain S(t + δt) = S0 exp (μt + μδt + σy(t + δt)) = S0 exp (μδt + σW(t + δt)) = S0 exp (σW(t)+ μδt + σδW) where in the second step we absorbed exp (μt) in the (still arbitrary) S0 and in the third step we adopted the notation δW for a change in a Brownian motion after the passing of a finite time interval δt: δW := W(t + δt)−W(t) =⇒ δW ∼ N(0, δt) (2.23) The first term in the exponent refers to (already known) values at time t. It can also be absorbed into the (still arbitrary) factor S0, i.e., S(t + δt) = S0 exp (μδt + σδW) . Finally, the (still arbitrary) S0 is chosen so that S(t+δt) δt→0= S(t) holds. This corresponds to the initial condition for the solution of the SPDE. Thus, we obtain the change in S corresponding to Equation 2.19 for a finite, positive time span δt: S(t + δt) = S(t) exp (μδt + σδW) where δW ∼ N(0, δt) (2.24) 38 DERIVATIVES AND INTERNAL MODELS Analogously, Equation 2.20 gives the corresponding change in S over a finite, positive time span δt as: S(t + δt) = S(t) exp (( μ̃− σ 2 2 ) δt + σδW ) where δW ∼ N(0, δt) (2.25) The drift and the expected return With the risk factor evolution over finite time spans at our disposal, we are now able to answer a question which often confuses market participants: which is the mean (or expected) return of the risk factor over a finite time span? Is it the drift μ of ln S as in Equation 2.13 or the drift μ̃ of S itself as in Equation 2.20? The answer to this question does in fact depend on the compounding methods used. In continuous compounding, if a security is worth S(t) at time t and worth S(t + δt) at time t + δt then the return R of this security over this time span is defined by the equation S(t + δt) = S(t)eRδt , i.e., R ≡ 1 δt ln ( S(t + δt) S(t) ) = 1 δt ln ( S(t) exp (μδt + σδW) S(t) ) = μ+ σ δt δW where we have used Equation 2.24 for S(t + δt). The mean (or expected) return is of course simply the expectation of this return. Since δW∼N(0, δt), i.e., E[δW ] = 0, this expectation is E[R] = μ = μ̃− σ 2 2 (2.26) Thus, in continuous compounding the drift parameter μ is themean return.12 In other compounding methods, things are different. We look at linear compounding as an example, since this method is especially important for the following reason: Very often, the return R is defined as the relative price change over a time period. This very natural definition is in fact equivalent 12 Note that the expectation of the logarithm of the market price was required for the determination of the mean return not the expectation of the market price itself. FUNDAMENTAL RISK FACTORS OF FINANCIAL MARKETS 41 from the “sharp” delta function. The forward equation is useful in situations where S has an established value at a fixed time and we are interested in the probability distribution of S at a later time, in other words, when information available now is to be used to calculate forward in time. The backward equation In mathematical finance, we are more often interested in the opposite sit- uation: information at a future point in time is known (for example, at the maturity of an option) and we wish to calculate values backward to an ear- lier point in time (today, say). To do so, we require a differential equation involving derivatives with respect to the earlier time variables S and t which answers the question: given that the stochastic variable is equal to S′ at time t′, how is the stochastic process distributed at an earlier point in time t < t′? This question can be answered using the backward orKolmogorov equation. This is given explicitly by ∂p ∂t + 1 2 b(S, t)2 ∂2p ∂S2 + a(S, t) ∂p ∂S = 0 (2.31) The associated initial condition is given by p(S′, t′ ∣∣S, t = t′ ) = δ(S − S′). Note that for all processes whose parameters a and b are not dependent on S, the only difference between the forward and backward equation is the sign of the term involving the second derivative. A derivation of the forward and backward equations Both the forward equation 2.30 and the backward equation 2.31 are of such a fundamental nature that it is possible to complete their derivation on the basis of first principles. This derivation involves quite extensive calculations (essentially, taking products of Taylor series) which the reader need not necessarily work through (the reader who does not wish to take the time is recommended to continue on to Section 2.4.4). However, experience shows that one tends to feel somewhat ill at ease when such fundamental equations simply appear out of thin air and are accepted without a sound explanation. Therefore, the derivation of the forward and backward equations is presented explicitly here in greater detail than is commonly found in the literature on mathematical finance. This gives the reader the opportunity to obtain a real understanding of these two fundamental equations. We begin by deriving the forward equation. We split the probability that the stochastic process will travel from (S, t) to (S′, t′) into the probability 42 DERIVATIVES AND INTERNAL MODELS that by time t′ − δt it will have arrived at S′ − δS, S′ + δS or S′ and then will proceed to S′ in the remaining time δt with the probabilities δp+, δp−, and δp0, respectively, thus p(S′, t′ |S, t ) = p(S′ − δS, t′ − δt |S, t )δp+(S′ − δS, t′ − δt) + p(S′, t′ − δt |S, t )δp0(S′, t′ − δt) + p(S′ + δS, t′ − δt |S, t )δp−(S′ + δS, t′ − δt) (2.32) where δp±(S, t) := p(S ± δS, t + δt |S, t ) δp0(S, t) := p(S, t + δt |S, t ) = 1− δp+(S, t)− δp−(S, t). Intuitively δp+ is the probability that the process will increase by δS in the short time span δt, δp− the probability that it will decrease by δS in the time span δt, and δp0 is the probability that it will remain constant in the time span δt. Other possibilities for this stochastic process within this time span are not admitted. This split of the probabilities is graphically represented by the trinomial tree in Figure 2.10. The relationship between these δp and the stochastic process 2.15 results from the requirement that the first two moments, i.e., the expectation and the variance of the discrete process defined for the trinomial tree, agree with the first two moments of the continuous Ito process in the limit as δt → dt → 0. The expectation of δS is E [δS] = δp+δS + δp00− δp−δS = (δp+ − δp−)δS. By definition, its variance is given by var [δS] = E [(δS − E [δS])2] = δp+(δS − E [δS])2 + δp−(−δS − E [δS])2 + δp0(0− E [δS])2. t ′ t ′− δt S ′ − δS S ′ + δS S ′ Figure 2.10 The trinomial tree used to derive the forward equation FUNDAMENTAL RISK FACTORS OF FINANCIAL MARKETS 43 Substituting E[δS] = (δp+− δp−)δS and δp0 = 1− δp+− δp− yields, after several simple algebraic manipulations var [δS] = [δp+(1− δp+ + δp−)2 + δp−(1+ δp+ − δp−)2 + (1− δp+ − δp−)(δp+ − δp−)2]δS2 = (δp+ + δp−)(1− (δp+ + δp−))δS2 ≈ (δp+ + δp−)δS2 where only terms up to order δpδS2 were considered in the last step. The requirement that the expectation and the variance be equal to those of the Ito process 2.15 as δt → dt → 0, i.e., up to linear order in δt means that (δp+ − δp−)δS = E [δS] δS→dS−→ E [dS] = a(S, t)dt (δp+ + δp−)δS2 ≈ var [δS] δS→dS−→ var [dS] = b(S, t)2dt. This is obviously achieved by choosing δp±(S, t) = 1 2 δt δS2 [±a(S, t)δS + b(S, t)2] (2.33) And, in order to retain the equality δp0 = 1− δp+ − δp− we have δp0(S, t) = 1− δt δS2 b(S, t)2. Substituting these values in our original equation 2.32 and expanding all terms appearing in this expression in a Taylor series about the point (S′, t′) up to order δt finally yields the forward equation. As a result of Ito’s lemma we have δS2 ∼ δt. This means that all terms of order 1, δS, δS2, and δt are retained and all higher order terms can be neglected. Note that δt/δS2 in Equation 2.33 is of order 1. Explicitly, the Taylor series for p is given by: p(S′ ± δS, t′ − δt |S, t ) = p(S′, t′ |S, t )± ∂p ∂S′ δS + 1 2 ∂2p ∂S′2 δS2 − ∂p ∂t′ δt + · · · p(S′, t′ − δt |S, t ) = p(S′, t′ |S, t )− ∂p ∂t′ δt + · · · . 46 DERIVATIVES AND INTERNAL MODELS plus the probability of being S′ − δS at time t′ + δt multiplied by the probability of having made a “down move” in the time interval from t′ to t′ + δt, plus the probability of beingS′ at time t′+δt multiplied by the probability of having made a “null move” in the time interval from t′ to t′ + δt. We can now see immediately that the backward equation is more eas- ily derived than its counterpart since the probabilities δp are needed at the point (S′, t′), and consequently, a Taylor series expansion about this point is unnecessary. We only need the Taylor series expansion of p given by p = [ p+ ∂p ∂S′ δS + 1 2 ∂2p ∂S′2 δS2 + ∂p ∂t′ δt ] 1 2 δt δS2 [ b2 + aδS] + [ p+ ∂p ∂t′ δt ] [ 1− δt δS2 b2 ] + [ p− ∂p ∂S′ δS + 1 2 ∂2p ∂S′2 δS2 + ∂p ∂t′ δt ] 1 2 δt δS2 [ b2 − aδS] . All terms appearing in the above expression are evaluated at the point (S′, t′). Taking the products and neglecting all terms of order∼ δSδt or smaller gives, in linear order of δt, p = p+ ∂p ∂S′ δS δt δS2 aδS + 1 2 ∂2p ∂S′2 δS2 δt δS2 b2 + ∂p ∂t′ δt which, after rearranging the terms, immediately yields the backward equation 2.31. 2.4.4 Forward and backward equations in the Black-Scholes world We consider the simple example of the process given by Equation 2.20, namely dS(t) = S(t)μ̃ dt + S(t)σdW , i.e., we simply have a(S′, t′) = S′μ̃ b(S′, t′) = S′σ . This process, or its equivalent 2.13, is, as has been shown above, the simplest process of relevance in mathematical finance. Reasonable and often even analytical expressions for the prices of financial instruments can be obtained by solving these equations. For example, we will later show that such pro- cesses form the basis for the famous Black-Scholes option pricing formula. We thus speak of the Black-Scholes world when referring to the description FUNDAMENTAL RISK FACTORS OF FINANCIAL MARKETS 47 of market parameters using such processes. With these assumptions, the backward equation becomes ∂p ∂t + 1 2 σ 2S2 ∂2p ∂S2 + μ̃S ∂p ∂S = 0, and likewise the forward equation is simply15 ∂p ∂t′ − σ 2 2 S′2 ∂ 2p ∂S′2 + [μ̃− σ 2] S′ ∂p ∂S′ + [μ̃− σ 2] p = 0. The solution to this differential equation with the above initial condi- tion p(S′, t | S, t )= δ(S′ − S) provides an explicit example for a transition probability: p(S′, t′ | S, t ) = 1 S′ √ 2πσ 2(t′ − t) × exp { − [ln(S′/S)− (μ̃− σ 2/2)(t′ − t)]2 2σ 2(t′ − t) } (2.34) The corresponding equation and its solution for the equivalent process 2.19 can of course be found by simply making the substitution μ̃ = μ+ σ 2/2 in accordance with 2.22. 15 The derivation of this equation is a simple application of the product rule: ∂p ∂t′ − 1 2 ∂2 ∂S′2 [ S′2σ 2p ] + ∂ ∂S′ [ S′μ̃ p ] = 0 ∂p ∂t′ − σ 2 2 ∂ ∂S′ [ 2S′p+ S′2 ∂p ∂S′ ] + μ̃ [ p+ S′ ∂p ∂S′ ] = 0 ∂p ∂t′ − σ 2 2 [ 2p+ 2S′ ∂p ∂S′ + S ′2 ∂2p ∂S′2 ] + μ̃ [ p+ S′ ∂p ∂S′ ] = 0. CHAPTER 3 Financial Instruments: A System of Derivatives and Underlyings As mentioned in the introduction, trading can be defined as an agreement between two parties in which one of the two consciously accepts a financial risk in return for the receipt of a specified payment or at least the expecta- tion of such a payment at same future time from the counterparty. Financial instruments, also called financial products, are instrumentswhichmake such a risk mitigation or risk transfer possible. The purpose of this section is to present a classification of such instruments in a system of underlyings and derivatives, specifically for interest rate instruments. Interest rate risk is by far the most complex market risk. Correspondingly, interest rate instru- ments, i.e., instruments having interest rates as their underlying risk factors, are among the most complex financial instruments traded on the market. Instruments on other risk factors such as stocks or foreign exchange rates can be classified analogously and will be discussed in detail in later sections of the book. The main characteristics of interest rate instruments is that they can be represented as a right to one or several payments (cash flows) which may be either fixed or variable and will occur at some future dates. The amount of these cash flows is determined from the length of the respec- tive interest periods and the associated interest rates quoted according to the convention particular to the market under consideration. The following parameters may be used to define the cash flow structure of an instru- ment, in particular, to establish the times on which payments are to be made: valuation date, rollover date, maturity date, payment date, reset or fixing date, frequency of the coupon payments per year (e.g., 1, 3, 6, and 12 months), long or over-long first period, partial final period, 48 FINANCIAL INSTRUMENTS: DERIVATIVES AND UNDERLYINGS 51 Interest bearing securities Papers having one ormore regular interest payments are referred to as interest bearing securities or coupon bearing securities. They are issued at face value. Certificates of deposit, floating rate notes, reverse floaters, and short- term capital market papers are included in this category. Certificates of deposit (CD), are tradable money market papers issued by banks having a term between 30 days and 4 years. Essentially, CDs are securatized time deposits in banks. Floating rate notes (also called floaters orFRNs) are bondswhose interest payments vary depending on a current market rate. The interest rates are regularly adjusted to a reference rate (for example, theLIBOR orEURIBOR), in general, to a money market rate. Hence, floaters, despite often having longer terms are also assigned to the category of money market papers. As opposed to other floaters (reverse floaters, for example), these normal floaters are referred to as plain vanilla. Plain vanilla is anAmerican term for simple papers without any particular attributes such as options or convertibility rights. These simple floaters are to be considered as floating rate notes in their most basic form. The most important difference between a floater and a fixed rate instru- ment is the nominal interest rate. While for a fixed rate paper, the nominal interest rate is fixed over the entire lifetime of the paper, the nominal rate of the floater fluctuates. The nominal interest rate changes according to the current level of the reference market rate; a decrease in short-term rates results in a corresponding decline in the interest yield of a floater. Investors expecting a decline in money market rates will therefore usually close out their positions in plain vanilla floaters. Amore complex variant of plain vanilla floaters are reverse floater, some- times referred to as bull floating rate notes. Like plain vanilla floaters, reverse floaters are papers paying interest at a variable rate, depending on the level of the current money market rate. Also like normal floaters, the interest is paid in regular intervals, e.g., every six months. However, while the interest yields of a plain vanilla floater increases with rising money mar- ket rates, the opposite is true of the reverse floater (hence the name). For a reverse floater, the current money market rate is subtracted from a fixed base rate. Long-term papers on the capital market such as federal bonds and obli- gations, mortgage bonds, jumbo bonds, and municipal bonds are indeed issued with terms than money market papers. However, the residual time to maturity naturally decreases constantly with the passing of time. Therefore, these papers, initially designated as capital market papers, are comparable with money market papers in the last year of their lifetime. 52 DERIVATIVES AND INTERNAL MODELS Time and notice deposits The name time deposit signifies a deposit at a bank which is invested for a certain time ( fixed deposits) or with an agreed upon term of notice (deposits at notice). Fixed deposits are deposits at a bank or savings institution with individual, fixed and unalterable maturities, amounts, and interest rates, all agreed upon in advance. The conditions depend on current interest rates as well as the term of the deposit. The amount deposited also plays a deciding role; usually, the more invested, the higher the interest paid. Investors gain somewhat more flexibility in comparison to fixed deposits by placing their savings in deposits at notice. For these deposits, a period is agreed upon which the investor must give notice in advance if he or she wants to withdraw the invested funds. Withdrawal notice can be one day, 48 hours, seven days or even three months. In comparison to fixed deposits, deposits at notice play a subordinate role. Trading conventions for money market instruments Similar to papers sold on the capital markets, the yields of money mar- ket papers are quoted publicly. Yields are always per year, in other words, they are annualized. However, different conventions for calculating yields are observed on the different international markets, as was illustrated in Section 2.1. The annualized yields are influenced by the day count con- vention and compounding method used. The day count conventions vary in the calculation of the number of days in the year as well as the days of the month. The usual day count convention in the money markets is the Act/360 convention. Examples of instruments whose interest is calculated in accordance with this day count convention are commercial papers, trea- sury bills, and floating rate notes in the United States. In Germany, quotes have been published in accordance to this convention since July 1, 1990 for inter-bank trading. Act/360 is employed elsewhere in Europe as well. Floaters are also calculated employing this method. The method Act/360 is also referred to as the international or French method. It is frequently called the euro interest convention as well. Alternatively, Act/365 is used in several markets. Sterling commercial papers, for instance, are quoted in accordance with this method; it is sometimes referred to as the English method. In addition, the 30/360 convention was employed in particular in trading bonds and obligations; with the introduction of the euro, it was phased out and replaced with the Act/Act convention. The 30/360 convention assumes that each year has 360 days, each month 30 days. The 30/360 convention is also referred to as the German method. The Act/Act method calculates with the actual number of days of the month and year. In Germany, the 30/360 FINANCIAL INSTRUMENTS: DERIVATIVES AND UNDERLYINGS 53 convention is still applied, in particular for mortgage bonds issued before 1999. Some existing issues have already changed over to theAct/Act conven- tion (for example, all government bonds and all jumbo bonds). New issues in the euro zone are computed in accordance with the Act/Act convention. If yields are determined for money market instruments which pay interest several times within a single year (or in its lifetime if this happens to be less than one year in duration), a conversion into an annual yield (annual- ization) can be accomplished with or without compounding interest effects. Yields which do not take compounded effects into consideration (linear and simple compounding) are referred to as nominal yields or nominal rates. These methods of computing yields predominate in most money markets. We already provided a detailed description of the different compounding conventions in Section 2.1. On the international money markets, money market papers are traded in accordance with linear (discount rate) or simple compounding (money market yields). Discount papers are generally traded on the basis of the discount rate (DR). The discount rate is not a yield since, in contrast to a yield, it is not calculated by reference to invested capital, but on the nominal principal of the instrument. Examples of discount papers traded in foreign markets are treasury bills in the United States. The money market yield (MMY ) or CD-equivalent yield is computed for money market instruments having interest payments. An example is the Certificate of Deposit (CD). Table 3.1 provides an overview of the most important details of several money market instruments on the international money markets. 3.1.2 Capital market securities Capital market papers are long-term interest rate instruments with a term (time to maturity) longer than one year. Papers with terms longer than Table 3.1 Some examples of international money market instruments Instrument Yield calculation Day count Typical term on issue Treasury Bill Discount rate Act/360 13, 26, 52 weeks EURIBOR MMY Act/360 1, 2, 3, . . . , 12 months Commercial Paper USD (CP) Discount rate Act/360 7 days to 2 years Commercial Paper GBP (CP) Discount rate Act/365 7 days to 2 years Certificate of Deposit USD (CD) MMY Act/360 7 days to 1 year Federal Financing Treasuries Discount rate Act/Act 1 or 2 years 56 DERIVATIVES AND INTERNAL MODELS Table 3.2 Interest payment modes in international bond markets National bond market Coupon frequency Yield computation Australia Semi-annual Semi-annual yield Belgium Annual Annual yield Federal republic of Germany Annual Annual yield France Annual Annual yield United Kingdom Semi-annual Semi-annual yield Japan Semi-annual Simple yield-to-maturity Italy Semi-annual Semi-annual yield Canada Semi-annual Semi-annual yield Netherlands Annual Annual yield Austria Annual Annual yield Sweden Annual Annual yield Switzerland Annual Annual yield Spain Annual Annual yield United States Semi-annual Semi-annual yield While the fixed leg of the swap has the character of a fixed rate bond, the float- ing leg refers to a variable, short-term interest rate (the 6-month EURIBOR for instance) which is “fixed” at some agreed upon future dates. Thus, swaps combine two portfolios (the legs) and the interest is usually earned on the same nominal principal for both legs. However, different compounding and payment conventions might be observed. A standard or plain vanilla swap consists of series of coupon payments, called the coupon strip in exchange for a series of EURIBOR payments, the EURIBOR strip. For a plain vanilla swap in euros, both strips have the same maturity but may roll over at different dates. The fixed leg pays interest annually in accordance with the 30/360 compounding convention. The floating leg, in contrast, pays semi-annually and the Act/360 compounding convention is observed. TheTARGETholiday calendar is used to determine bank holidays. 3 .2 FO RW A RD TR A N S A C TION S Derivative securities, or short derivatives, can be divided into conditional and unconditional forward transactions. A conditional forward transaction grants one party of the contract certain rightswhereas the other party assumes certain obligations. In contrast, an unconditional forward transaction is an agreement that is binding on both parties. Warrants, options, and instru- ments similar to them, e.g., caps and floors, can be classified as conditional FINANCIAL INSTRUMENTS: DERIVATIVES AND UNDERLYINGS 57 derivatives. Futures and forwards, on the other hand, are assigned to the unconditional derivatives. 3.2.1 Forward rate agreements and forwards Forward rate agreements (abbreviated FRAs) are the oldest and thus prob- ably the most simple interest rate instruments. A forward rate agreement is an agreement between parties to lend or borrow short-term money at a fixed rate of interest at some time (usually a few months) in the future. The agreed upon fixed rate is referred to as the FRA rate or forward rate. If the reference interest rate, e.g., EURIBOR, exceeds the agreed FRA rate at maturity, the buyer realizes a profit and receives a cash settlement payment from the seller. However, if the reference rate at maturity is lower, the buyer suffers a loss and compensates the seller (in form of a cash settlement). It follows that the buyer pays the FRA rate and receives in return the current interest rate from the seller. A FRA is usually purchased in combination with a loan. The buyer of a FRA protects him or herself against rising interest rates. If the reference interest rate (for example, the EURIBOR) at maturity lies above the FRA rate, the buyermakes a profit (which can be used to compensate for the higher interest rate on the loan). If the reference rate lies under the FRA rate, the buyer takes a loss (but on the other hand has to pay lower interest on the loan). Conversely, the seller of a FRA protects him or herself against falling interest rates. He or she could for instance sell a FRA to lock in (at least for the FRA period) a certain interest yield on a floating rate investment. If the reference rate (e.g., EURIBOR) at the FRA’s maturity is lower than the FRA rate, the seller receives a cash settlement which can be used to increase the (then low) yield from the investment. If the EURIBOR at FRA maturity is higher than the FRA rate, the seller has to pay a cash settlement (but then will have a high yield from his investment). If the reference rate at the fixing time is equal to the FRA rate, neither a profit nor a loss is realized. Two additional factors should be taken into account with respect to FRAs. Both parties of the contract face a counterparty credit risk. This is because depending on the interest rate level at maturity of the FRA, either party receives or is obliged to make a cash settlement payment. The second aspect is that no liquid funds are exchanged at the transaction or during the life of the contract. Only at maturity of the FRA will the contract be settled if the current interest rate does not equal the FRA rate. If the underlying in the contract is not an interest rate (e.g., LIBOR, EURIBOR) but a fixed rate security (for example, a government obligation, government bond, mortgage bond, promissory note, etc.) then such a forward transaction is simply referred to as a forward. 58 DERIVATIVES AND INTERNAL MODELS 3.2.2 Financial futures Although commodity futures have been traded on organized exchanges since 1860, financial futures are a rather recent addition to the market. In the United States, active trading in commodity futures began at the beginning of the nineteenth century. It was no coincidence that the Chicago Board of Trade (CBOT ) was founded as early as 1848; this waswherewheat contracts were first traded. In the course of the following years, additional commodity exchanges were founded, for example in NewYork and London. Trade in financial forward transactions is relatively recent; trading in these instruments commenced in Chicago in 1972 on the International Monetary Market (IMM). The first transactions were concluded for foreign currencies. InGermany, futures have been traded since the establishment of theDeutsche Terminbörse (DTB, now called the EUREX) in 1990. The basic idea behind futures is identical to that of forward rate agreements and forwards. Futures are a further development in forward transactions, which go back as far as the seventeenth century. Hence, futures have the same payoff profile as forwards. The essential difference between forwards and futures is that contract elements are not individually negotiated.A future is a standardized forward transaction. The underlying security, the volume, the time of settlement, and other payment and delivery conditions are stan- dardized and are set by the exchange (for example, EUREX, LIFFE). Futures are usually not exercised but closed out prior to maturity by entering into a trade opposite to the original one. Similar to forward transactions, both parties to a futures contract take on a counterparty risk. The counterparty risk in the case of futures, how- ever, is lower by orders of magnitude; it is in fact negligible. Futures markets have developed two mechanisms for the elimination of counter- party risk. The first involves the settlement of a potential payment not at maturity but daily. If, for example, the forward price of the futures underly- ing increases, the holder of a futures contract makes a profit. For a forward transaction, the profit in the form of a settlement payment cannot be real- ized until the maturity of the forward transaction, i.e., at the end of the forward’s lifetime. For futures, in contrast, the profit is credited directly to a so-called margin account on a daily basis. Similarly, in the case of a loss, the corresponding amount is debited from the account. In the language of the futures market, these daily profit and loss payments are referred to as daily settlement. The daily sum credited or debited to the margin account is called the variation margin. Since the time to repayment is reduced to one day by the daily settlement, the counterparty risk is accordingly smaller. In addition to the variation margin, a second mechanism reducing the counterparty risk is the risk-based margin which every market participant has to set aside. These are securities which must be set aside by the investors FINANCIAL INSTRUMENTS: DERIVATIVES AND UNDERLYINGS 61 The desired result is attained by both strategies: the bank accomplishes the hedge against rising short-term rates since it can compensate for the higher interest rates it has to pay for the floater through the profits resulting from the long call on the 6-month EURIBOR or the long put on the 3-month Euro future. There has been a marked increase in demand for caps and floors in recent years since an increasing number of investors wish to protect themselves against fluctuations in the money market rates or the term structure curve, or to profit from these fluctuations. While an investor would wish to invest in caps if he or she expected increasing money market rates, an increasingly steep interest rate term structure or an upward parallel shift of the term structure, floors are interesting for the investor wishing to profit from falling money market rates, a flatter term structure, or a downward parallel shift of the term structure. A cap is an agreement between the seller of the cap and the buyer according to which the seller pays the buyer the difference in the interest earned on a nominal with respect to a reference rate (e.g., a floating market rate such as EURIBOR) and that earned on an agreed upon fixed strike rate, should this fixed rate be exceeded by the reference rate. Different modes of payment are used in practice: in some cases, the discounted payoff is made at thematurity of the option (“early”), otherwise, payment is made upon maturity of the respective EURIBOR period (“standard”). Viewed analytically, a cap can be interpreted as a series (portfolio) of European call options on a reference rate with various maturities. The buyer of a cap (long cap) profits from an increase in interest rates. Caps are employed primarily to hedge against interest rate risk arising in consequence of variable financing (floaters, for instance). They can also be used when speculating on rising interest rates. Moreover, variations in the term structure effect a corresponding change in the value of the option. Floors are the counterpart of a cap. While caps limit a floating rate from above, the floor limits them from below (lower interest rate limit). A floor is an agreement between the seller of the floor (short floor) and the buyer (long floor) inwhich the seller pays the buyer the difference in interest on a nominal calculated with respect to a floating market rate (for example, the 6-month EURIBOR) and a fixed strike rate established in the floor, should the market rate fall below this fixed rate. Floors are employed in hedging against risks involved in floating rate investments (such as floating rate notes). From the analytical point of view, a floor can be interpreted as a series (a portfolio) of European put options on the reference market rate with various maturities. Collars are a combination of caps and floors. The purchase of a collar (long collar) corresponds to simultaneously buying a cap (long cap) and selling a floor (short floor) where the strike of the floor is lower than that of the cap.With the purchase of a collar, the buyer obtains the right to receive a 62 DERIVATIVES AND INTERNAL MODELS settlement payment from the seller should the reference rate exceed the cap rate; he must however make payment to the seller should the reference rate sink below the floor rate. In this way, the buyer of a collar ensures that for him the interest rate effectively stays within a specified range whose upper and lower limits are established by the cap and floor rates, respectively. The purpose of a collar is to reduce the costs of the cap involved. The long position in the collar receives a premium through the short floor position which lowers the price of the long cap position. However, an investor can then only profit from falling interest rates up to the lower interest rate limit of the collar. Swaptions are interest rate instruments having a swap as an underlying. They thus give the holder the right to enter into the underlying swap. The strike of the swap is quoted in the form of the specified swap rate. Both “cash” and “swap” are typical as modes of settlement; upon exercise of the swap, either a cash payment is made in settlement or the underlying swap is actually entered into. In Part III, the structure and function of these and other financial instru- ments will be discussed in far greater detail. Perhaps the most important point in understanding financial instruments is their valuation and in this connection, the concept of hedging. Hence, the most important methods of pricing and hedging will be introduced in the following Part II before we subsequently apply them to today’s most common financial products in Part III. GH Par" DJ) Methods 66 DERIVATIVES AND INTERNAL MODELS dependent on the instrument’s seller. This is a good approximation for products traded on an exchange which also acts as a clearing house. This, of course, is not the case when dealing with OTC transactions. The quan- titative treatment of credit and counterparty risks is presently a rapidly growing and increasingly important branch of risk management. 5. There is no “friction” in the market. Market friction is a general term referring to all costs involved in trad- ing. These include transaction costs, bid/ask spreads, (opportunity) costs of margins, taxes, etc. All such costs not belonging to the actual invest- ment are neglected. This is by no means a bad approximation for large institutions. 6. Continuous trading is possible. This means that the time between two trades can be arbitrarily small as can the differences in price and the number of instruments traded. The assumption that trading can take place in arbitrarily short time intervals is an approximation for the simple reason that no exchange is always open for trade (weekends, bank holidays). Furthermore, infinitely many adjustments in a continuous hedging strategy would cause infinitely high transaction costs if Assumption 5 did not hold. This means that Assumption 5 must hold if Assumption 6 is to make sense. The assump- tion that arbitrarily small price differences are possible can be problematic for options which are well out of the money. For example, the smallest possible price change of bond futures (“tick size”) usually is 0.01 which is small in comparison to the futures quote, say 99.8. Therefore, the assumption of continuously changing prices is good for futures. How- ever, for an option on the future which is far out of the money, a change in the futures price by one tick can result in a change in the price of the option of 50% of its value. Thus, such options do not have continuous price changes. 7. The logarithm of the relative price change of a risk factor is a random walk. In light of Section 2.3, this implies that the price of a risk factor is lognormally distributed. 8. Interest rates are not stochastic. Intuitively, this means that the evolution of interest rates over time does not involve a random component but is completely deterministic. In such a world, future interest rates are known today. 9. Interest rates are constant. This means that future interest rates are not only known today, they have today’s value as well. Interest rates do not change in the course of time. OVERVIEW OF THE ASSUMPTIONS 67 10. The volatilities of risk factors are not stochastic. Intuitively, this means that the volatilities as a function of time do not involve a random component but are completely deterministic. Future volatilities are known today. 11. The volatilities of risk factors are constant. This means that future volatilities are not only known today, they have the same value as today as well. Volatilities do not change in the course of time. 12. Dividend yields of risk factors are constant. In many pricingmodels for derivatives, the dividend (yield) of an under- lying is an input parameter which is assumed to be constant throughout the term of the derivative contract. 13. Correlations between the risk factors are not stochastic. As with volatilities, the evolution of the correlations over time has no random component. Future correlations are known today. As in the case of volatilities, this is not often the case in reality. CHAPTER 5 Present Value Methods, Yields, and Traditional Risk Measures Present value methods determine the value of a financial instrument by dis- counting all future cash flows resulting from the instrument. Applying this method requires few assumptions. Only Assumptions 1, 2, 3, 4, and 5 from Chapter 4 are necessary. 5 .1 PRE SEN T VA L U E A N D YI E L D T O MA T U R ITY Suppose that between the time t and maturity T of a security, there are n dates ti on which cash flows C(ti), i = 1, . . . , n are due. The present value VR(t, T) of a security1 is the sum of all future cash flows due between t and T , where each cash flow C(ti) is discounted at the spot rate belonging to the maturity ti of the corresponding cash flow: VR(t, T) = n∑ i=m+1 BR(t, ti)C(ti) (5.1) where TE = t0 < t1 < · · · < tm−1 < tm︸ ︷︷ ︸ Past ≤ t < tm+1 < tm+2 < · · · < tn = T︸ ︷︷ ︸ Future . 1 In this book, the value of a financial instruments will usually be denoted by V . In general, V will often have an index (R, for example) to signify its dependence on a specific parameter. 68 PRESENT VALUE METHODS, YIELDS, AND RISK MEASURES 71 the instrument depends on the market situation and appears in the above equation (with a negative sign since it is a payment). Consider for example a simple bond with a term of 6.5 years paying an annual rate of 6% on its nominal. Let’s assume that the bond paid its last coupon 6 months previous to the present date and was purchased 2 months ago for 102.5% (of its nominal value) plus the accrued interest for the four months (in this example, 2%).All future cash flows (coupons and redemption upon maturity) are incoming and thus positive. In this example, the internal rate of return must be chosen so that the discounted future cash flows are equal to the purchase payment compounded at the same rate. The internal rate of return can be interpreted as the constant interest rate at which the portfolio yield compounds the investments in the portfolio for the life of the portfolio (i.e., from purchase to sale or to the due date of the last cash flow).2 Starting from the above equation defining the internal rate of return, we can define a further quantity, known as the net present value or NPV. The NPV of an instrument is its current present value plus all the past cash flows compounded at the internal rate of return: NPV(t, T) = m∑ j=1 BRE (tj, t) −1C(tj)︸ ︷︷ ︸ Past cash flows + n∑ j=m+1 BR(ti, t)C(ti)︸ ︷︷ ︸ Present value where ti < · · · < tm ≤ t ≤ tm+1 < · · · < tn = T and 1 ≤ m,m + 1 ≤ n (5.4) The difference between this and Equation 5.3 is that future cash flows are not discounted at the IRR but at the spot rates and thus represent the present value.According toEquation 5.3, the sumof thepast cashflows compounded at the IRR is exactly the negative value of the sum of the future cash flows discounted at the IRR. Thus, the NPV can be interpreted as the difference between the sum of the future cash flows discounted at the IRR on the one hand and at the spot rates on the other: NPV(t, T) = n∑ j=m+1 [ BR(ti, t)− BRE (tj, t) ] C(ti). 2 Often in definitions of the internal rate of return (and of theNPV) the future cash flows are discounted, while the past cash flows are not compounded. This inconsistency makes a reasonable interpretation of the NPV and IRR practically impossible. The definitions of NPV and IRR introduced here are more general and accurate. 72 DERIVATIVES AND INTERNAL MODELS According to the definition of the YTM given in Equation 5.2, the sum of future cash flows discounted at theYTM instead of the spot rates also yields the present value. Thus, if theYTM equals the IRR, the NPV will be zero: The instrument’s internal rate of return equals its yield to maturity if and only if the net present value is zero. If the IRR is higher (lower) than the YTM, the instrument will have a positive (negative) NPV. In Figure 5.1 (from the sheet IRR in the Excel workbook Plain- Vanilla.xls on the accompanying CD-ROM), the YTM, IRR, and NPV have been calculated for a portfolio. Only the cash flows for the portfolio are required. The actual instruments generating these cash flows are irrelevant. In the column “DiscountedUsing Spot Rate,” the future cash flows have been discounted at the current spot rates. The sumof these values yields the present Yield curve Portfolio cash flows Payment Date Time 30/E360 Spot rate (%) Cash flow Discounted using spot rates Discounted using YTM Discounted/ Compounded using IRR Dec. 31, 99 −1,142 Jun. 30, 00 −0,642 Dec. 31, 00 −0,142 Feb. 21, 01 Jun. 30, 01 0,358 Dec. 31, 01 0,861 Jun. 30, 02 1,358 Dec. 31, 02 1,861 Jun. 30, 03 2,358 Dec. 31, 03 2,861 Jun. 30, 04 3,358 Dec. 31, 04 3,861 Jun. 30, 05 4,358 Dec. 31, 05 4,861 Jun. 30, 06 5,358 Dec. 31, 06 5,861 Jun. 30, 07 6,358 Dec. 31, 07 6,861 Jun. 30, 08 7,358 Values as at YTM and IRR 7.20 4.69% Feb. 21, 01 Past cash flows −472,887 Future cash flows 446,487 446,487 472,887 0 −323,784 −333,439 50,000 50,325 −189,774 −189,774 50,000 49,612 48,769 49,186 50,000 48,662 47,093 48,067 60,000 56,518 54,590 56,382 60,000 54,310 52,714 55,099 60,000 51,949 50,922 53,858 70,000 57,526 57,367 61,404 70,000 54,225 55,417 60,022 70,000 51,086 53,513 58,656 −15,000 −10,303 −11,077 −12,286 0 0 0 40,000 24,596 27,554 31,297 0 0 0 0 0 0 30,000 15,796 18,614 21,912 −15,000 −7,489 −8,991 −10,709 2.20 3.20 4.50 5.50 6.30 7.10 7.90 8.50 9.00 9.30 9.50 9.60 9.70 9.80 9.90 Figure 5.1 Cash flow table of a portfolio PRESENT VALUE METHODS, YIELDS, AND RISK MEASURES 73 value of the portfolio. In the column “Discounted Using YTM,” the future cash flows are discounted at theYTM. In accordance with Equation 5.2, the YTM was adjusted until the sum of the discounted cash flows was equal to the present value calculated using the current spot rates (we have utilized the Excel function “Goal Seek” to make the field “PV Difference” equal zero). In the column “Discounted/Compounded Using IRR” all cash flows were discounted or compounded, respectively, at the IRR, adjusting the IRR until the sum of all cash flows in this column equaled zero (see Equation 5.3). Finally, the NPV was determined according to Equation 5.4 as the sum of the present value and the past cash flows compounded at the IRR. This is the difference between the future cash flows discounted at the IRR and YTM, respectively. The NPV is negative and the IRR of the portfolio lies significantly under the currentYTM. The IRR and NPV are suitable for measuring the performance of a port- folio (or a trader), but are by no means parameters relevant to valuation or risk management. This is due to the fact that the past cash flows influence these parameters. But the value of an instrument (and thus its risk resulting from a potential change in its value) is independent of its history. The market establishes a single price for an instrument independent of the cash flows that may have been generated by the instrument previously. Since the risk of an instrument is by definition a reflection of the potential future change in the its price, the same argument holds for risk management. For example, a single instrument could have different NPVs and IRRs, if a trader bought the instrument in the past at a cheaper price than another trader at the same time, or if two traders bought the same instrument at the same price but on different dates. It is in fact quite unlikely that the same instrument in different portfolios will have exactly the same NPV and IRR since for this to happen, the instruments must be purchased at exactly the same time and price (or the price difference must correspond exactly to compounding at the IRR for the period between the differing purchase dates). 5 .3 A CCRU ED I NTE R E S T , R ES ID UA L D EBT , A N D PA R RA T ES The residual debt of an instrument is the amount of outstandingdebt onwhich interestmust be paid, i.e., the open claims against the issuer of the instrument. The accrued interest is the outstanding interest having accumulated at a contractually agreedfixed interest rateK in the timebetween the last payment date tm and t (= today). Accrued interest only makes sense when dealing with fixed rate instruments or for those instruments whose interest rates are established at the beginning of each interest period. In our general notation Accrued interest = N(tm) [ BK(tm, t) −1 − 1] where tm ≤ t < tm+1 76 DERIVATIVES AND INTERNAL MODELS To conclude this section, wewant to “prove” this statement: the par rate of a fixed rate instrument is defined as the coupon rate K which the instrument should have for the present value to be equal to the residual debt. On the other hand, the present value can be expressed in terms of theYTM through Equation 5.2. For an instrument whose coupon rate is equal to the par rate we have on a payment date, i.e., for t = tm: N(tm) = n∑ i=m+1 BR(tm, ti)C(ti). Multiplying both sides of the equation by the coupon rate compounding factor BK(tm, t)−1, we find that the left-hand side corresponds to the residual debt at time t (see Equation 5.5). Only on the condition that K is equal to the YTM can we make use of Equation 2.4 to transform the residual debt into the present value at time t (in accordance with Equation 5.2): BK(tm,t) −1N(tm)︸ ︷︷ ︸ Present value for R = Par rate = BK(tm,ti)−1 n∑ i=m+1 BR(tm,ti)C(ti) = n∑ i=m+1 BR(t,ti)C(ti)︸ ︷︷ ︸ Present value for R = YTM (5.7) Thus, it follows that if the coupon rate of an instrument is equal to itsYTM, the present value is equal to the residual debt, i.e., the at-par condition given by Equation 5.6 is satisfied and therefore, the coupon rate must be equal to the par rate. Conversely, the at-par condition can only be satisfied for all times t if the par rate is equal to theYTM. 5 .4 T RA DI T I O N A L S E N S IT IV IT IES O F I N T E RE ST RA T E IN S TR U M E NTS 5.4.1 Average lifetime and Macaulay duration The simplest feature characterizing a security whose value derives from future cash flows is its mean (residual) lifetime. The mean is the weighted average taken over the time periods until each future coupon payment. The weight accorded to a particular period is exactly equal to the contribution of the corresponding cash flow to the present value of the security. This mean PRESENT VALUE METHODS, YIELDS, AND RISK MEASURES 77 lifetime is called the Macaulay duration. DMacaulay(t) = n∑ i=m+1 (ti − t)BR(t, ti)C(ti) VR(t, T) = n∑ i=m+1 (ti − t) ⎡⎢⎢⎢⎣ BR(t, ti)C(ti)n∑ k=m+1 BR(t, tk)C(tk) ⎤⎥⎥⎥⎦ (5.8) The Macaulay duration has an interesting property: If a time interval of length dt (during which no cash flow payment was due) has passed since the last calculation of the Macaulay duration, the new Macaulay duration is simply given by the difference of the old value less the elapsed time dt. DMacaulay(t + dt) = DMacaulay(t)− dt if t + dt < tm+1. The derivation of this result is trivial. We only need to make use of the fact that the sum over all weights equals one: DMacaulay(t + dt) = n∑ i=m+1 (ti − t − dt) [ BR(t, ti)C(ti) VR(t, T) ] = n∑ i=m+1 (ti − t) [ BR(t, ti)C(ti) VR(t, T) ] ︸ ︷︷ ︸ DMacaulay(t) − dt [ ∑n i=m+1 BR(t, ti)C(ti)∑n k=m+1 BR(t, tk)C(tk) ] ︸ ︷︷ ︸ 1 . 5.4.2 Modified duration and convexity The methods for pricing financial instruments introduced in this book are applied in risk management when determining the value of a portfolio in any given scenario. This can be extremely time consuming. In order to obtain a quick risk assessment without repeating time-consuming pricing calculations for each different scenario, approximation techniques have been developed which are all based on the same procedure. Let V be the price of a financial instrument and R the price of the under- lying risk factor (for instance an interest rate), also called the underlying. One obvious measure for the risk of this instrument is the sensitivity of 78 DERIVATIVES AND INTERNAL MODELS its price with respect to changes in the underlying. The price is gener- ally a complicated function of R which, however, admits a Taylor series representation. VR+dR(t, T) = VR(t, T)+ ∞∑ k=1 1 k! ∂kVR(t, T) ∂Rk (dR)k . In such a Taylor series, the price of the instrument whose underlying has changed by a factor of dR is calculated by adding the infinite series above to the price corresponding to the original value of the underlying R. The terms of the sum involve all powers (dR)k of the change in the underlying dR. If this change is small (for example, 1 basis point = 0.01% = 0.0001), its powers are particularly small for large k (for example, (0.0001)2 = 0.00000001, (0.0001)3 = 0.000000000001, etc.). Thus, for small changes in R, we can obtain a good approximation of the price by truncating the series after the first few terms.3 In this way the infinite sum is approximated by dVR(t, T) = ∞∑ k=1 1 k! ∂kVR(t, T) ∂Rk (dR)k ≈ ∂VR(t, T) ∂R dR+ 1 2 ∂2VR(t, T) ∂R2 (dR)2 (5.9) where dVR = VR+dR − VR denotes the difference between the old and new price of the instrument. In the traditional terminology of interest rate man- agement and if R represents the yield to maturity, the coefficients of the first two powers of dR (divided by the original price V ) are referred to as the modified duration and the convexity. In modern risk management, these coefficients are called delta and gamma, whereR is any arbitrary underlying. More precisely, the modified duration is defined as the (negative) deriva- tive of the price with respect to R divided by the price, i.e., the (negative) relative sensitivity of the security to linear changes in the yield to maturity, see Equation 5.10. The minus sign reflects the effect that bond prices decline as interest rates rise. Assuming that the cash flows themselves are indepen- dent of the interest rate, Equation 5.2 gives the following expression for the 3 Themethod of expanding a function in a Taylor series for small changes in its parameters, neglecting all but the parameter termsup to a certain power (here, the secondpower) iswell established and frequently used in practice with verifiable correctness (assuming the variables under consideration are continuous). This is not simply an academic “mathematical trick” but a method which finds constant application in the financial world. Almost all hedging strategies as well as the variance-covariance methods used in calculating the value at risk, for example, find their basis in this procedure. The same method has already been applied when we presented Ito’s lemma in Section 2.4.2.