INTEREST RATE THEORY THE BGM MODEL

Transcription

1 INTEREST RATE THEORY THE BGM MODEL By Igor Grubišić SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE AT LEIDEN UNIVERSITY LEIDEN,THE NETHERLANDS AUG 2002 c Copyright by Igor Grubišić, 2002

2 LEIDEN UNIVERSITY DEPARTMENT OF MATHEMATICS The undersigned hereby certify that they have read and recommend to the Faculty of Graduate Studies for acceptance a thesis entitled Interest Rate Theory The BGM Model by Igor Grubišić in partial fulfillment of the requirements for the degree of Master of Science. Dated: Aug 2002 Supervisor: S. M. Verduyn Lunel Readers: ii

3 LEIDEN UNIVERSITY Date: Aug 2002 Author: Igor Grubišić Title: Interest Rate Theory The BGM Model Department: Mathematics Degree: M.Sc. Convocation: Aug Year: 2002 Permission is herewith granted to Leiden University to circulate and to have copied for non-commercial purposes, at its discretion, the above title upon the request of individuals or institutions. Signature of Author THE AUTHOR RESERVES OTHER PUBLICATION RIGHTS, AND NEITHER THE THESIS NOR EXTENSIVE EXTRACTS FROM IT MAY BE PRINTED OR OTHERWISE REPRODUCED WITHOUT THE AUTHOR S WRITTEN PERMISSION. THE AUTHOR ATTESTS THAT PERMISSION HAS BEEN OBTAINED FOR THE USE OF ANY COPYRIGHTED MATERIAL APPEARING IN THIS THESIS (OTHER THAN BRIEF EXCERPTS REQUIRING ONLY PROPER ACKNOWLEDGEMENT IN SCHOLARLY WRITING) AND THAT ALL SUCH USE IS CLEARLY ACKNOWLEDGED. iii

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5 Table of Contents Table of Contents List of Tables List of Figures Abstract Acknowledgements v vii viii i ii Introduction 1 1 Definitions and notations Bank Account, Zero-Coupon Bonds and Spot Interest Rates Forward Rates, Interest-Rate Swaps and Interest Rates Options No-Arbitrage Pricing and Numeraire Change Introduction Market, Portfolio and Arbitrage Change-of-Numeraire Technique Change-of-Numeraire Toolkit Brace-G atarek-musiela Model (BGM) Introduction Forward-Rate Dynamics under Different Numeraires Black formula for caplets Instantaneous Volatility and Correlation of Forward rates Introduction Specification of Instantaneous Volatility Forward rate correlations Calibration to forward rate correlation v

6 4.3.2 Iteration of projections PCA Method Application of the BGM model Introduction Monte Carlo Method Pricing of Contingent claim by Monte Carlo Ratchets Model Input Calibration Pricing of Ratchets Concluding Remarks 46 A Summery of Stochastic Calculus 48 A.1 Filtration,Usual Conditions and Martingale A.2 Itô process A.3 Radon-Nikodym and Girsanov s Theorem A.4 Solutions and approximation of SDE A.4.1 Stochastic Euler scheme A.4.2 Convergence of stochastic Euler scheme Bibliography 54 B Discussion of Bibliography and justification of presented results 56 B.1 Chapter 1,2 & B.2 Chapter B.3 Chapter B.4 Appendix A vi

7 List of Tables 4.1 Piecewise-constant inst. volatility Initial forward rates and caplet volatilities Calibration to Σ Specifications of calibration performance Specification of the Monte Carlo pricing of a LIBOR spread option Results of the Monte Carlo pricing of a LIBOR spread option vii

8 List of Figures 1 Example Monte Carlo simulation Overview of the chapters Illustration of a forward rate structure associated with a set of four discount bonds Evolution of the time-homogeneous volatility term structure for a LIBOR market consisting of 10 forward rates, and τ i = 1 2 year Example of a projection algorithm between linear subspace X and closed set Y Calibration to the historical correlation matrix for a LIBOR market consisting of 10 forward rates, and τ i = 1 2 year Plotted surfaces of matrix differences. C P CA and C proj are calculated for d = 2. O is given by (o i,j ) = 1 for i, j = 1,..M viii

9 Abstract In this thesis the theory of the BGM Market Model is presented, as well as practical issues arising from computer implementation. To this end, we consider the system of coupled stochastic differential equations driven by a d-dimensional Wiener process with the parameter functions σ i (t) which, together with the boundary conditions given by the market, constitute the BGM Market Model. The solution of this system represents the forward LI- BOR rates. The aim of the BGM is to price interest rate options. The price of an interest rate option is the expectation of a function of the forward LIBOR rates. We study the numerical procedures of determing σ i (t) in such a way that the BGM Model resembles the market prices as closely as possible. Also, we have derived a new method for reduction of the dimension of the Wiener process used in the BGM Model. This method is incorporated into the model, thereby extending it. As an illustration, this new method, called the Projection Method, is applied to price ratchets. Its performance is compared to the most commonly used method for reduction of the dimension of the Wiener process, namely the PCA. i

10 Acknowledgements This thesis has been submitted to obtain the Master s Degree in Mathematics at the University of Leiden. Research for this Thesis took place during an internship at Bank Nederlandse Gemeente N.V. Portfolio management, from 16 September 2001 till 16 May I would like to thank, Drs. Tim Segboer, for his guidance and the confidence he had in me. Furthermore, Hans, Freek, Tiemo, Roger and all the others at BNG, for pleasant working atmosphere and their practical help. Also, I want to thank my supervisor, Prof. Dr. S. M. Verduyn Lunel, for his guidance and significant contribution in my mathematic forming. Special thanks goes to my parents, who where always there when I needed them. I also want to mention Raoul Pietersz from ABN AMRO Bank. He read various drafts and gave valuable comments and suggestions. I am very grateful to my friend Robert Vermeulen, who agreed to read this thesis from beginnig to end. His comments allowed me to make many improvements in the English. Last, but not least, I would like to thank Leontien for all her loving support and reminding me that there is more to my life than mathematics. Leiden, The Netherlands January 13, 2005 Igor Grubišić ii

11 Introduction The concept of interest rates belongs to our every-day reality. It is something familiar, something we know how to deal with. When depositing a certain amount of money into a bank account, everybody expects that the money amount will grow with time. The fact that lending money must be rewarded somehow, so that receiving a given amount of money tomorrow is not equivalent to receiving exactly the same amount today, is indeed common knowledge and wisdom. However, expressing such concepts in mathematical terms may be less immediate. Many definitions have to be introduced to develop a consistent theoretical apparatus. In this introduction we will motivate some definitions and give a rough sketch of an mathematical Interest Rate Model. 1 euro B(t) euro P(0,T) P(s,T) 1 euro present T present s T a) b) Figure 1: Example We will start our discussion with an example which is illustrated by figure 1. Suppose we deposit 1 euro at the bank. For obvious reasons, this is called a riskless investment. Question: What should be the fair amount of money that we receive at future time T? This amount which we receive at time T is called the bank account and is denoted by B(t) (see figure 1a). It is obvious that B(t) should depend on the interest rate prevailing in the market during 1

12 2 the time period [0, T ]. This leads to the following differential equation: db(t) = r t B(t)dt, B(0) = 1 with the solution B(t) = e t 0 rsds. In the equation above, r t represents the short rate at time t. The short rate is the interest rate for the deposit of 1 euro for an infinitely short time interval dt. In view of uncertainty about the future course of the interest rate, it is natural to model it as a random variable. Furthermore, it should be a positive function of time, for otherwise lending money will cost us money. This means that the riskless investment of 1 euro will yield B(T ) = e T 0 rsds euro at time T, which is a stochastic quantity. We can also reverse the situation, see figure 1b. This can be illustrated by the question What is the present value of 1 euro at future time T? 1 If r t is a deterministic function, the answer will be simply B(T ) euro. However r t is stochastic and instead of B(t) we consider the expectation E(B(T )). This yields that the 1 present value of 1 euro is E(B(T )) euro. The expression 1 E(B(T )) is called the zero-bond and denoted by P (0, T ). Note that P (0, T ) is deterministic quantity at the present time. We can go even further and ask what is the value at time s of 1 euro which is at our disposal at time T (with T t). As indicated in figure 1b, the answer is P (s, T ) := E(e T s rtdt ) euro. Notice that P (s, T ) is now a stochastic quantity. It represents the expectation of the market at time s of the short rate between times s and T. The big difference between r s and P (s, T ) is the following. The short rate is an unobservable in the market, for people do not trade in short rate. On the other hand, the market express its vision of rate evolution through the prices of zero-bonds. For this reason the zero-bond is an observable in the market. Another important feature in the Interest rate theory is the forward rate. It is a constant rate to be paid if you want to borrow money at time s for the future time period between t and T. The forward rate is denoted by F (t, T ). It is a simple expression of zero-bonds and is given by Interest rate models F (s; t, T ) := 1 ( P (s, t) ) (T t) P (s, T ) 1 An interest rate model is designed to value interest rate options. It postulate certain market behavior, thereby creating a hypothetical interest rate market. Throughout this thesis we refer to the real market as the LIBOR market.

13 3 An interest rate option is a function of forward rates. As an example, we consider the most simple option, namely the caplet. A caplet gives the owner the right, but not an obligation to borrow money for the specified time period (in this case [t, T ]) at the pre-negotiated strike rate K. The caplet payoff can be expressed as f(f (t; t, T )) = max{f (t; t, T ) K, 0}. Because F (t; t, T ) is stochastic, the price of the caplet is given by E(max{F (t; t, T ) K, 0}) 1. To be able to calculate this expectation we must know the probability distribution of F at time s where s [t, T ]. The market prices caplets with the Black formula. This is equivalent to the assumption that F s is lognormally distributed with variance σ(s) at time s. Thus, from the caplet prices we can recover variance σ s, which reflects the uncertainty of the market about the forward rate during the period [0, s]. In summery, the information about interest rates given by the market consists basically of the set {F (0, T n, T n+1 )} n and the corresponding set of caplet prices. The central question in interest rate theory is: Given the present market information, what is the price of an arbitrary option? or more formally Given the set of {F (0, T n, T n+1 )} n and the corresponding set of caplet prices, what is the value of E(f(F s ))? where F s = (F (s 1, T 1, T 2 ),..., F (s n, T n, T n+1 )) and f is the payoff of the option. To answer this question we need to know the probability distribution of F (s) at the corresponding times. To this end we must start with a model for evolution in time of interest rates. Thereby we have to ensure that there are no arbitrage opportunities in our model. Roughly speaking, absence of arbitrage is equivalent to the impossibility of investing zero today and receiving tomorrow a nonnegative amount that is positive with positive probability. Because interest rate is an stochastic quantity, its dynamics have to be modelled by means of the stochastic differential equations. Researchers spent much time in developing such models, resulting in many different approaches. We can distinguish two different classes, namely the Short Rate Models, and the Market Models. In the Short Rate Models, one tries to derive the stochastic differential equation for the short rate, i.e. to specify the coefficient s functions F, G in the expression dr t = F (r t, t)dt + G(r t, t)dw. For example, we arrive at the Hull & Withe model if we assume the following functional forms for the coefficient functions; G(r t, t) = σ(t) and F (r t, t) = ϑ(t) a(t)r(t) where ϑ, a, σ 1 Actually, we have to multiply this expression by P (0, T ) and T t in years, because this amount is received at time T. However these are known constants at time zero, and are not important for our discussion at this stage.

14 4 are deterministic functions of time. Furthermore, such a model has to be calibrated to prevailing market conditions, i.e., we have to give algebraic expressions for ϑ, a, σ from the available information. Modelling such dynamics directly is very convenient since all fundamental quantities (rates and zero bonds) are readily defined, by no-arbitrage arguments, as the expectation of a functional of the process r. However short rate models, such as Hull & White model, commonly specify - explicitly or implicitly - the stochastic behavior of unobservable financial quantities, such as the instantaneous short rate or its variance. Therefore, the calibration of a model to a set of market quantities requires a transformation of the dynamics of these unobservable quantities into dynamics of observable quantities. For this, complicated numerical procedures are needed and the results are not always satisfactory. The biggest disadvantage of these models is the inconsistency with the standard market pricing formula, namely the Black formula. The BGM model This shortcoming has inspired the second type of approach, which takes real market interest rates, like forward LIBOR 2 or swap rates, as model building blocks, earning the approach the title of Market Model. The market model arose from the general framework of Heath-Jarrow-Morton [HJM92]. Rather than working on the short rates as in the Hull & Withe model, Brace, G atarek and Musiela ([BGM97]), Jamshidian ([Jam97]), and Miltersen, Sandmann and Sondermann ([MSS97]), took the forward term rates as the model primitives. In their novel approach, they modelled forward rates, in such a way that the Black formula ([Bla76]) is recovered for the price of a European-style option on the market price. Since the new model works on the observable interest rates in the LIBOR market, it is often termed LIBOR Market Model. In this paper it will be called BGM model, referring to the authors of the first paper where the approach was introduced rigorously. For example, when modelling an n-year LIBOR market we have to derive a system of 2n 1 coupled stochastic differential equations ( since there are 2n 1 forward rates in n years) df k (t) = µ k (F, t)dt + σ k (F, t) dw (t), for k = 1,.., 2n 1 where F k (t) = F (t; T k 1, T k ), µ k : [0, T ] Ω R, σ k : [0, T ] Ω R d and dw (t) is 2n 1 dimensional Wiener process. The µ k and σ k must be chosen in agreement with the Black formula and the no-arbitrage conditions. The term dw (t) represents the uncertainties which underlie the LIBOR market. It is, in general, 2n 1 dimensional and the mutual correlation between i and k factor of dw (t) can be determined from historical data. However, in some cases the dimensionality of dw (t) can be reduced. For instance, if all forward rates are perfectly correlated, the 2n 1-dimensional dw (t) can be replicated, in a trivial way, by a 1-dimensional Wiener process. This corresponds with the situation that there is only one cause of uncertainty in the market. 2 The k-forward LIBOR rate, denoted by F k is defined as a forward rate where t = T k 1, T = T k and T n+1 T n = 1/2 year

15 5 Once the BGM Market Model has been calibrated, e.g., once we have determined the coefficient functions, we can move on to the process of pricing. Pricing is usually done by means of the Monte Carlo approximation. The Monte Carlo approximation is given by E[g(F s1,..f sm )] N j=1 g ( (Fs t 1,..., Fs t ) m ), ω j N where N denotes number of simulations, F t denote the Euler discretization of process F and g denotes the payoff of the option. Monte Carlo pricing: An example For example, consider a 5-years BGM model. Suppose we want to price a cap ( a sum of all caplets in the market) with the strike at 6 %. This means we have a system of 9 stochastic differential equations and f(f ) = 9 j=1 max{f t j (T j ) 6}. In figure 2 we present one simulation, e.g., F t (ω 1 ) = (F t 1 (T 1, ω 1 ),..F t 9 (T 9, ω 1 )). The blue flat surface represent the strike level. The stippled curves at times 0, 1,..9 represent the respective forward curves 3 at times 0, , 9 2 in years. The curves at times , 9 2 are the trajectories of the Euler discretization of the F = (F 1,...F 9 ). In this scenario the option payoff is the sum of the positive differences in height between the star-marked point which is first from the right on the forward curves( except the forward curve at time zero) and the blue surface. Next, we repeat the same procedure N 1-times (because we already have one scenario) thereby creating N different scenarios in the same manner and thus producing F t (ω 2 ), F t (ω 3 ),..., F t (ω N ) which corresponds with producing N 1 different pictures. The price of the cap is just the mean of the N payoff s. If we were asked to calculate some other option with payoff g we would replace the blue flat surface by a surface generated by function g. Remark From figure 2 we can note that the 5-year BGM Market is modelled through a multi-dimensional Wiener process, for the forward curve changes its shape through time. If it was modelled through a 1-dimensional Wiener process, all points on a forward curve would be perfectly correlated and the forward curve would just shift up or down as time passes by. However, the price of the k-caplet is independent of the dimensionality of the Wiener process used in the model because we consider only one point on the k + 1- forward curve. 3 The curve through points F 0 (T k ), F 1 (T k ),..., F 9 (T k ) at time T k.

16 6 Figure 2: Monte Carlo simulation The main problem of the thesis As we have just seen, the Monte Carlo algorithm is very simple to implement. But its main weakness is its speed which is directly related to the dimensionality of the Wiener process used in the model. To complete the task of pricing as fast as we can, we can reduce the dimensionality of the Wiener process for the following reasons: As regards the calibration objects (usually the cap prices) the market will quote biding and offering prices. Any price that is between the bid and the offering prices is said to be within the spread. This is the error margin within which our input data is known. The option price has to be calculated to a predetermined level of precision. The minimal dimensionality of a Wiener process is dependent on the kind of option. For example, if we have to calculate caplet prices it is enough to use a 1-dimensional Wiener process. On the other hand, the options which depend on several different forward rates ( for instance f = max{f k (T k ) F k 1 (T k 1 ), 0}) must be priced through a model with a multi-dimensional Wiener Process. However, the procedure of reducing the dimensionality of the Wiener process, except for the trivial reduction to 1-dimension, was not clear until now. Let us at the problem more closely. An n-year BGM market is modelled through a 2n 1 dimensional Wiener process where the mutual correlation between the factors is given by the historical (2n 1) (2n 1)

17 7 correlation matrix C. If we want to implement the BGM model with an r dimensional Wiener process (where 1 r 2n 1) we have to solve the following problem: Find the matrix C model such that C C Model inf C C i C i R r where R r denotes the space of correlation matrices with rank r. Or in other terms: what is the nearest matrix C Model R r to the given historical matrix C. The choice of the norm will reflect what is meant by nearness of the two matrices. Structure of this thesis Herewith we provide a description of the contents of each chapter. Figure 3 gives a schematic overview of the structure of this thesis. It can also be helpful as a reading guide. Chapter I Market definitions Chapter II General Interest Rate Theory Chapter III Fundations of BGM Model Chapter IV Calibration of BGM Model Chapter V Applications of BGM Model Appendix Stochastic calculus & its numerics Figure 3: Overview of the chapters 1. Definitions and Notation. This chapter is devoted to standard definitions and concepts in the world of interest-rate. We introduce the standard language used by the market people. We define the terms: bank-account, Zero-bond, LIBOR rate, day-count etc.. This chapter can be skipped and used as a dictionary when needed. 2. No-Arbitrage Pricing and Numeraire Change. The chapter introduces the theoretical apparatus needed for a general interest rate model. It starts with definitions of economy, portfolio, arbitrage and numeraire. These definitions are fundamental for a financial model. As indicated in figure 3 this section relies heavily on Stochastic Calculus which can be found in Appendix. At the end of this chapter we introduce the change of numeraire technique that will be used in Chapter 4 to derive the dynamics of the BGM model.

18 8 3. The BGM Model. This chapter presents one of the most popular Interest rate models: The BGM model. To start with, we derive the BGM dynamics under forward measure. At the end we give the connection between BGM dynamics and caplet prices. 4. The Calibration of the BGM. In this chapter we discuss the methods of calibration of a M-dimensional BGM model, when there are two sets of calibration objects, namely caplet prices and the historical correlation matrix. The chapter starts with a presentation of calibration to the caplet prices. After this we give a new way of calibration to the historical correlation matrix named Projection method, which is the answer to the central question of this paper. For this reason it forms the heart of the thesis. The derivation of this method is based on some properties of Hilbert space. At the end we review the old PCA method and give the algorithm for the Projection method. 5. Application of the BGM model. In this chapter we apply the theory described in chapters 3 & 4. We start with a brief review of the Monte Carlo method. Finally, from a set of market data, the 5.5 year BGM model is calibrated through the Projection method and the PCA method. These two methods are then tested by valuation of a spread option and the results are compared. 6. Appendix. In this chapter we review some basic results that are mentioned and applied through this thesis. This includes definitions (like filtration, martingale, Itô process ) needed to model an economy and the notion of fairness. We introduce the Itô formula together with some helpful corollaries about Stochastic Calculus. We also give two important theorems: The Radon-Nikodym theorem and the Girsanov theorem. The latter theorem is used in particular to derive the change of numeraire toolkit. The end of this chapter is devoted to the numerical solutions of SDE. The stochastic version of the Euler scheme is introduced. This is essential in Monte Carlo simulation.

19 Chapter 1 Definitions and notations In this chapter we present the main definitions that will be used throughout this thesis. 1.1 Bank Account, Zero-Coupon Bonds and Spot Interest Rates By interest rates we denote rates at which deposits are exchanged between banks, and at which swap transactions between banks occur. The most important interbank rate usually considered as a reference for contracts is the LIBOR rate, fixing daily in London. However, there exist analogues interbank rate fixing in other markets (e.g. the EURIBOR rate, fixing daily in Brussels), and when we refer to LIBOR we actually mean any of these interbank rates. Definition Bank Account (Money-market account). Define B(t) to be the value of a bank account at time t 0. Assume B(0) = 1 and that the bank account evolves according to the following differential equation: db(t) = r t B(t)dt, B(t) = 1, (1.1.1) where r t is a positive function of time. As a consequence: t B(t) = e 0 r sds (1.1.2) The above definition tells us that investing a unit amount at time 0 yields the value (1.1.2)at time t, and r t is the instantaneous rate at which the bank account accrues. This instantaneous rate is usually referred to as instantaneous spot rate, or short rate. Definition Stochastic discount factor D(t, T ) between two time instants t and T is the amount at time t that is equivalent to one unit of currency payable at time T, and is given by D(t, T ) = B(t) B(T ) = T e t r s ds. (1.1.3) 9

20 10 Definition Zero-Coupon bond with maturity T is a contract that guarantees its holder the payment of one unit currency at time T, with no intermediate payments. The contract value at time t < T is denoted by P (t, T ). Clearly, P (T, T ) = 1, for all T. The difference between D(t, T ) and P (t, T ) lies in the fact that D(t, T ) is an equivalent amount of currency which is a random quantity at time t, and P (t, T ) a value of a contract which has to be known at time t. As can be seen 1 later on, P (t, T ) can be viewed as the expectation of D(t, T ) under a particular probability measure. Definition Time to maturity T t is the amount of time (in years) from the present time t to the maturity time T > t. Definition Year fraction denoted by τ(t, T ) is the chosen time measure between t and T. The particular choice that is made to measure the time between two dates is known as the day-count convection. In this paper we use actual/360 so that the corresponding year fraction is D 2 D where D 2 D 1 is the actual number days between the two dates. Definition Simply-compounded spot interest rate prevailing at time t for the maturity T is denoted by L(t, T ) and is constant rate at which an investment has to be made to produce an amount of one unit of currency at maturity, starting from P (t, T ) units of currency at time t, when accruing occurs proportionally to the investment time. In a formula: L(t, T ) := 1 P (t, T ) (1.1.4) τ(t, T )P (t, T ) The market LIBOR rates are simply-compounded rates, which motivates why we denote such rates by L. 1.2 Forward Rates, Interest-Rate Swaps and Interest Rates Options Forward rates are characterized by three time instants, namely time t at which the rate is considered, its expiry T and its maturity S, with t T S. They can be locked today for an investment in a future time period, and are set consistently with current term structure(see below) of discount bonds. 1 A simple application of proposition (2.3.1)

21 11 Definition Term structure model. Any mathematical model that determines, at least theoretically, the stochastic processes P (t, T ), 0 t T, T (0, T ) where T is the maturity of the last bond in consideration, also known as horizon. Definition Simply-compounded forward interest rate prevailing at time t for the expiry T > t and maturity S > T is denoted by F (t; T, S) and is defined by F (t; T, S) := 1 ( P (t, T ) ) τ(t, S) P (t, S) 1 (1.2.1) The forward rate F (t; T, S) may be viewed as an estimate of the future spot rate L(T, S), which is random at time t, based on the market conditions at time t. In particular, we will see later on that F (t; T, S) is the expectation of L(T, S) at time t under a suitable measure (follows from equation (3.1.1)). Definition An interest rate swap is a contract where two parties agree to exchange a set of floating rate payments for a set of fixed interest rate payments. The set of floating rate payments is based on forward LIBOR rates and is called the floating leg. The set of fixed payments is called the fixed leg. In a payer swap you pay the fixed side, in a receiver swap you receive the fixed side. Definition A swaption gives its owner the right, not an obligation, to enter into a swap with fixed rate K at time T n. A receiver swaption gives the right to enter into a receiver swap, a payer swaption gives the right to enter into a payer swap. Definition A caplet is a call option on a forward LIBOR rate. It can be viewed as a swaption with the swap defined on a set consisting of only one interest rate payment.

22 Chapter 2 No-Arbitrage Pricing and Numeraire Change The fundamental economic assumption is the absence of arbitrage opportunities in the financial market considered. Roughly speaking, absence of arbitrage is equivalent to the impossibility to invest zero today and receive a nonnegative amount that is positive with positive probability tomorrow. In other words, two portfolios having the same payoff at a given future date must have the same price today. 2.1 Introduction In this chapter we briefly consider the case of the continuous-time economy. The reader interested in more formal treatment of arbitrage theory in continuous-time is referred, for example, to [MuRu98]. 2.2 Market, Portfolio and Arbitrage We consider an economy E (see below) in which n + 1 non dividend paying assets are continuously traded from time t until time T. Let W be a standard m-dimensional Wiener process on some filtered probability space (Ω, F, F, Q 0 ), satisfying the usual conditions 1 (we need these to ensure that stochastic integrals are well defined). The filtration F = {F t : 0 t T }, called Wiener filtration, is defined to be the Q 0 -augmentation of the F t = σ(w (s) : 0 s t) and all Q 0 -nullsets of F t. The Wiener filtration contains all the information about the Wiener processes which underlie the economy. We will suppress the notational dependance of stochastic variables on ω where no confusion can arise. 1 see appendix, definition (A.1.2) 12

23 13 Definition (i) A economy E is an Ft m -adapted (n+1)-dimensional Itô process S(t) = (S 0 (t), S 1 (t),..., S n (t)) ; 0 t T which we will assume has the form ds 0 (t) = r t S 0 (t)dt; S 0 (0) = 1 (2.2.1) and m ds i (t) = µ i (S i, t)dt + σ i,j (S i, t)dw j (t) = µ i (S i, t)dt + σ i (S i, t)dw (t), j=1 where σ i is a row number of the n n matrix ( σ i,j ) ; 1 i n N. S i (0) = s i (2.2.2) (ii) A portfolio is a (n + 1 dimensional) process φ = {φ t : 0 t T }, whose components φ 0, φ 1,..., φ n are locally bounded and predictable. (iii) The value process associated with a portfolio φ is defined by V t (φ) = φ t S t = n φ k t St k, 0 t T, (2.2.3) k=0 and the gains process associated with a portfolio φ by G t (φ) = t 0 φ u ds u = n k=0 (iv) A portfolio φ is self-financing if V (φ) 0 and t 0 φ k us k u, 0 t T, (2.2.4) V t (φ) = V 0 (φ) + G t (φ), 0 t T, (2.2.5) (v) A portfolio φ which is self-financing is called admissible if the associated value process V t (φ) is almost surely lower bounded, i.e. K R : V t (φ) K, t [0, T ], a.s. (2.2.6) Remark (i) We think of S i (t) as the price of asset number i at time t. The assets number 1,..., n are called risky because of the presence of their diffusion terms. The asset number 0 is called safe due to absence of a diffusion term, and we will insist that S 0 is measurable with respect to F 0, and thus some constant at time zero. This asset represent the bank account.

24 14 (ii) The k-th component φ k t of the strategy φ t at time t, for each t, is interpreted as the number of units of asset k held by an investor at time t. (iii) The V t (φ) and G t (φ) are respectively interpreted as the market value of the portfolio φ t and the cumulative gains realized by the investor until time t by adopting the strategy φ. (iv) This condition is intended to formalize the intuitive idea of a portfolio of which the value changes only due to changes in the asset prices. (v) This condition prohibits portfolios with doubling-up strategies, which contradicts real time situation, see [HuKe00], Example Next we will introduce the connection between the economic concept of absence of arbitrage and the mathematical property of existence of a risk-neutral measure. Definition An admissible portfolio φ t is called an arbitrage (in the economy E) if the associated value process V t (φ) satisfies V 0 (φ) = 0 and V T (φ) 0 a.s. and Q 0 [V T (φ) > 0] > 0. (2.2.7) If there is no such φ then the economy is said to be arbitrage-free. Remark I.e. V t (φ) generates profit without risk. Definition An equivalent martingale measure Q is a probability measure on the space (Ω, F) such that (i) Q 0 and Q are equivalent measures 2, that is Q 0 (A) = 0 Q(A) = 0, A F. (ii) The Radon-Nikodym derivative dq/dq 0 L 2 (Ω, F, Q 0 ). (iii) the discounted asset price process D(0, )S is an (F, Q)-martingale, i.e E[D(0, t)st k F u ] = D(0, u)su, k for all k = 0, 1,.., n, for all 0 u t T. (2.2.8) with E denoting expectation under Q. Theorem Assume that there exists a equivalent martingale measure P. Then the economy E is free of arbitrage. 2 Notation Q 0 Q.

25 15 Proof. Assume that φ is an arbitrage for E. We assumed that there exists a equivalent martingale measure P which implies that V T (φ) is a martingale under P. Thus E P [V T (φ)] = V 0 (φ) = 0, so that E P (V T (φ)) = 0 and V T (φ) = 0 a.s. V T (φ) = 0 a.s. which contradicts the arbitrage condition P[V T (φ) > 0] > 0. Definition A contingent T -claim is any random variable X L 2 (Ω, F T, Q 0 ). 2. It is called attainable if φ : V T (φ) = X a.s. 3. Such a φ is said to generate X, and π t = V t (φ) is the price at time t associated with H. 4. A economy E is called complete if and only if every contingent claim is attainable. Remark The interpretation of this definition is that the contingent claim is a contract which specifies that the stochastic amount, X, of currency is to be paid out to the holder of the contract at time T. Theorem The economy is complete if and only if the martingale measure is unique. Remark The proof can be found in [HaPl81]. Thus, the existence of a unique martingale measure make economy free of arbitrage and also allows the derivation of a unique price associated with any contingent claim. Proposition Assume there exists an equivalent martingale measure Q and let H be an attainable contingent claim. Then, for each time t, 0 t T, there exists a unique price π t associated with H, i.e., π t = E[D(t, T )H F t ] (2.2.9) Proof. From the definition of equivalent martingale measure we obtain D(0, t)π t = E[D(0, T )π T F t ] = E[D(0, T )H F t ]

26 Change-of-Numeraire Technique Formula (2.2.9) gives the unique no-arbitrage price of an attainable contingent claim H in terms of the expectation of the claim payoff under selected martingale measure Q. However, this measure is not necessarily the most natural and convenient measure for pricing the claim H. Definition Any non-dividend paying asset in the model of which the price is always strictly positive a.s. can be taken as numeraire. In general, a numeraire Z is identifiable with a self-financing strategy φ in that Z t = V t (φ) for each t. Intuitively, a numeraire is a reference asset that is chosen so as to normalize all other asset prices with respect to it. Thus, choosing a numeraire Z implies that relative prices S k /Z, k = 0, 1,...n are considered instead of the asset prices themselves, which together with the filtered probability space (Ω, F, F, Q 0 ) defines an economy Ẽ. In the definition of equivalent martingale measure (2.2.3), it has been implicitly assumed that bank account B 0 is numeraire. As pointed previously, this is just one of all possible choices and, as far as the calculation of the claim prices is considered, we should switch to a more convenient numeraire. The following proposition provides a fundamental tool for the pricing of derivatives and is the natural generalization of Proposition (2.2.3) to any numeraire. Proposition Let N be a numeraire, and Q 0 the equivalent martingale measure for the numeraire B. Then Q N defined by Q N B(0, 0) N(T ) (A) := N(0) B(0, T ) dq 0 = 1 N(T )D(0, T )dq 0, A F(T ) (2.3.1) N(0) is an equivalent martingale measure for Q. A Remark Q N and Q 0 are equivalent and Q 0 (A) = N(0) B(0, T ) B(0, 0) N(T ) dqn, A F(T ) An equivalent way of expressing (2.3.1) is to say A A Q N Q 0 with Radon-Nikodym derivative QN B(0, 0)N(t) Q Ft = 0 N(0)B(0, t) (2.3.2) Proof. Because N is the price process for some asset follows from definition (2.2.3) that N/B(0, T ) is martingale under Q 0. Therefore, Q N (Ω) = 1 N(T ) N(0) Ω B(0, T ) dq 0 = 1 [ N(T ) ] N(0) E = 1 N(0) B(0, T ) N(0) B(0, 0) = 1

27 17 And we see that Q N is a probability measure. Let Y be an asset price. Under Q 0, Y/B(0, T ) is a martingale. We must show that under Q N, Y/N is a martingale. For this we will use Lemma (A.1.1) from appendix, with Z(t) = QN Q 0 Ft, which tell as how to combine conditional expectations with change of measure. Then E N[ Y (t) ] N(t) F B(0, s)n(0) [ N(t)B(0, 0) s = N(s)B(0, 0) E B(0, t)n(0) which is the martingale property for Y/N under Q N. Y (t) ] N(t) F s = B(0, s)y (s) N(s)B(0, s) = Y (s) N(s) Change-of-Numeraire Toolkit In this section, we present some useful formulas on the change-of-numeraire technique developed in the previous section, which will be used later on, for the derivation of the asset-price dynamics under different numeraires. The superscripts in the Wiener process will be suppressed when the measure is clear from the context. Proposition Suppose that the two numeraires S and U evolve under Q U according to ds t = (...)dt + σt S CdWt U, Q U du t = (...)dt + σ U t CdW U t, Q U where σ S t and σ U t are an 1 n vectors. Further, consider an n-dimensional Itô process X whose dynamics under Q S and Q U are, respectively, given by dx t = µ S t (X t )dt + σ t (X t )CdW S t, Q S (2.3.3) dx t = µ U t (X t )dt + σ t (X t )CdW U t, Q U (2.3.4) where µ S t and µ U t are an n 1 vector and σ t is an n n diagonal matrix. The n n invertible matrix C is the model correlation, i.e., CdW is equivalent to an n-dimensional Wiener process with instantaneous correlation matrix ρ = CC. Then, the drift of the process X under numeraire U is µ U t (X t ) = µ S t (X t ) σ t (X t ) ρ ( σs t S t σu t U t ) (2.3.5)

28 18 Proof. We can apply Grisanov s theorem, see theorem (A.3.2) from appendix, to deduce the Radon-Nikodym derivative between Q U and Q S from the dynamics of X under the two different measures: ξ t := dqs dq U F t = e 1 t 2 0 (σ s(x s )C) 1 [µ S s (X s ) µ U s (X s )] 2 ds+ t 0 {(σ s(x s )C) 1 [µ S s (X s ) µ U s (X s )]} dws U. The process ξ defines a measure Q S under which X has the desired dynamics, given its dynamics under Q U. Set α t := [µ S t (X t ) µ U t (X t )] (σ t (X t )C) 1 ) Note that ξ is a martingale under Q U, thus its dynamics is driftless, namely: By (2.3.2) and since ξ is a Q U martingale By differentiation, where, S/U is a martingale, where σ S/U t ξ t = E Q U t dξ t = α t ξ t dw U t. (2.3.6) ξ T = QS Q U FT = U 0S T U T S 0 (2.3.7) [ [ξ T ] = E Q U U0 S ] T t = U 0S t (2.3.8) U T S 0 U t S 0 dξ t = U 0 S 0 d( S t U t ) = U 0 S 0 σ S/U t CdW U t, (2.3.9) d( S t ) = σ S/U t CdWt U U t is an 1 n vector. Comparing (2.3.6) and (2.3.9) we deduce that α t ξ t = U 0 S 0 σ S/U t C, and taking into account (2.3.9) and definition of α, we obtain the fundamental result µ U t (X t ) = µ S t (X t ) σ t (X t )ρ U t S t (σ S/U t ) (2.3.10) The only thing remaining to be done is computing σ S/U t. This can be done by means of corollary (A.2.5) from appendix, after which we arrive at σ S/U t which combined with (2.3.10) gives (2.3.5). = σs t U t σu t U t S t U t

29 19 For a lognormal case the previous proposition can be stated as follows Corollary Assume a lognormal functional form for volatilities,i.e. σ S t = v S t S t, σ U t = v U t U t, σ t (X t ) = diag(x t )diag(v X t ), µ S t (X t ) = diag(x t )m S t, where the v s are deterministic 1 n-vector functions of time, m S is a n 1 vector and diag(x) denotes a diagonal matrix of which the diagonal elements are the entries of vector X. We then obtain m U t = m S t d log X, log(s/u) t. dt

30 Chapter 3 Brace-G atarek-musiela Model (BGM) BGM is one of the most popular interest rate models and belongs to the class of models known as market models. These models postulate a geometric Wiener process for the market rates under consideration, such that the Black formula is recovered for the price of a European-style option on the market rate. The Black formula is the standard for calculating prices of European-style interest rate options. The main idea behind this is that caps are very liquid assets and therefore their market price can be taken as correct. From this starting point we aim to price other interest rate options (European style) using numerical procedures (mainly Monte Carlo). 3.1 Introduction The arbitrage-free economy E, see (2.2.1), consisting of M + 1 bonds, driven by a d- dimensional Wiener process constitutes the d-dimensional BGM market. Let t = 0 be the current time. Consider a set Π = {T 0, T 1,..., T M } from which expirymaturity dates (T i 1, T i ) are taken for a family of spanning forward rates and set T 1 := 0. Denote by {τ 0,..., τ M } the corresponding year fractions, meaning that τ i is the year fraction associated with the expiry-maturity pair (T i 1, T i ) for i > 0, and τ 0 is the year fraction from settlement to T 0. Consider the generic forward rate F k (t) = F (t; T k 1, T k ), k = 1,.., M, which is alive up to time T k 1, where it coincides with the simply-compounded spot rate F k (T k 1 ) = L(T k 1, T k ). Consider now the probability measure Q k, often called forward measure with maturity T k, associated with the numeraire P (, T k ). P (, T k ) is the price of the bond whose maturity coincides with maturity of the forward rate. 20

31 21 F 1 F 2 F 3 P(t,T 0 ) T 0 T 1 T 2 T 3 P(t,T 1 ) P(t,T 2 ) P(t,T 3 ) Figure 3.1: Illustration of a forward rate structure associated with a set of four discount bonds. By definition F k (t)p (t, T k ) = P (t, T k 1) P (t, T k ) τ k. Therefore, F k (t)p (t, T k ) is the price of a tradable asset and, by definition of measure associated with the numeraire, it has to be a martingale under measure Q k. But the price F k (t)p (t, T k ) divided by numeraire P (, T k ) is simply F k (t) itself. Therefore, F k is a martingale under Q k and it follows that its dynamics need to be driftless under Q k. Thus its dynamics are given by df k (t) = F k (t) σ k (t), dw k t 2 (3.1.1) where σ k : [0, T k 1 ] R d are deterministic, bounded and piecewise continuous functions for k = 1,.., M and, 2 stands for the standard inner product on R d. 3.2 Forward-Rate Dynamics under Different Numeraires If we want to price a general contingent claim, we need to model the behavior of all forward rates simultaneously under one single measure. This can easily be done by means of proposition (2.3.2). Remark For a definition of lognormal assumption see Corollary (2.3.3). Proposition Under the lognormal assumption, we obtain that the dynamics of F k under the forward measure Q i, in the three cases i < k, i = k and i > k are, respectively, F k (t) k τ j F j (t) σ j (t),σ k (t) 2 j=i+1 1+τ j F j (t) dt + F k (t) σ k (t), dwt i 2, i < k, t T i ; df k (t) = F k (t) σ k (t), dwt i 2, i = k, t T k 1 ; F k (t) i τ j F j (t) σ j (t),σ k (t) 2 j=k+1 1+τ j F j (t) dt + F k (t) σ k (t), dwt i 2, i > k, t T k 1. were W i is the standard d-dimensional Wiener process under Q i.

32 22 Proof. Case i = k has already been proven in the previous section. We will prove i < k and then case i > k can be shown analogously. (ii) Apply (2.3.2) to X = F k ( = F (, Tk 1, T k ) ), where S = P (, T k ), U = P (, T i ) and dynamics of the X under Q k are given by (3.1.1). Then, we obtain the following drift term m i t = d log F k, log ( P (, T k )/P (, T i ) ) t. dt Note that the definition of F in terms of P s implies from which m i t = ( P (t, Tk )/P (t, T i ) ) = k j=i+1 k j=i+1 d log F k, log ( 1 + τ j F j ) t dt P (t, T j ) P (t, T j 1 ) = = k j=i+1 k j=i τ j F j (t) τ j F j (t) σ j (t), σ k (t) τ j F j (t) The following result is taken from [HuKe00], theorem (18.1). Proposition All the equations above admit a unique strong solution if all coefficients σ( ) are bounded. Remark The main idea of the proof of this proposition is to consider modified BGM dynamics which are obtained by means of the strong Markov property for Itô diffusion. Then theorem (A.4.1) can be applied which yields the existence and the uniqueness of the solution. Because of its technicality we have omitted the proof. 3.3 Black formula for caplets The discounted payoff for a caplet with maturity date T i, notional amount A and strike K is given by AD(0, T i )τ i (F i (T i 1 ) K) + Here the function ( ) + : R R is defined by (x) + = max(x, 0) for x R. If the notional amount A is taken to be 10,000 euro, then the payoff is said to be in basispoints (bps). The Black formula for T i -caplet reads C Black i (υ i ) = Aτ i P (0, T i ) ( F i (0)N(d 1 ) KN(d 2 ) ) (3.3.1) d 1 = log ( F i (0)) K υ2 i T i 1 (3.3.2) υ i Ti 1 d 2 = d 1 υ i Ti 1 (3.3.3) where N stands for the standard normal distribution function.

33 23 Remark There is a 1 1 correspondence between Ci Black and υ i. For this reason, if financial derivatives are priced by the Black formula, quoting their price is equivalent to quoting corresponding volatilities. These volatilities are referred to as implied Black volatilities and denoted by υi Black. Proposition The price of the T i -caplet impled by the BGM model coincides with the corresponding Black formula, i.e., C BGM i (F i, K, σ i ) = Ci Black (F i, K, υi Black ) (υi Black ) 2 1 Ti 1 = σ i (s) 2 2 ds T i 1 0 Proof. Without loss of generality, set A = 1 and τ i = 1. Then by proposition (2.2.3) the price of a T i -caplet is given by C BGM i (F i, K, σ i ) = E[D(0, T i ) ( F i (T i 1 ) K ) + ] = P (0, T i)e i [(F i (T i 1 ) K) + ] (3.3.4) where the last equality is achieved by applying proposition (2.3.1). Under numeraire Q i, F i becomes a martingale, see proposition (3.2.1). For this SDE we can explicitly write the solution t F i (t) = F i (0)e 0 σn(s),dw i (s) 2 1 t 2 0 σn(s) 2ds, 0 t T i 1. Set κ 2 = T i 1 0 σ i (s) 2 2 ds. Then, F i(t) = F i (0)e Z where Z is an F(T i 1 )-measurable random variable which is normally distributed under Q i, Z ( 1 2 κ2, κ 2 ). Now E i [(F i (T i 1 ) K) + ] = F i (0)E i [e Z I {Fi (0)e Z K}] KE i [I {Fi (0)e Z K}] = F i (0)E i [I {e κ 2 F i (0)e Z K} ] KEi [I {Fi (0)e Z K}] where in the last step we used the identity E i [e Z f(z)] = E i [f(z + κ 2 )] with f(x) = I {Fi (0)e x K} which is easily verified through substitution y = x κ 2. Note that and {e κ2 F i (0)e Z K} = { Z 1 2 κ2 κ Z 1 2 κ2 κ N (0, 1). log ( F i (0)) K + 1 κ We can conclude that E i [I {e κ 2 F i (0)e Z K} ] = N(d 1), and analogously that E i [I {Fi (0)e Z K}] = N(d 2 ). Substituting this in equation (3.3.4) concludes the proof. 2 κ2 }

34 Chapter 4 Instantaneous Volatility and Correlation of Forward rates In this chapter we discuss methods of calibration, when there are two sets of calibration objects, namely caplet prices and observed historical correlation matrix. To complete calibration we must make some assumptions about volatility function. These assumptions have to be economically plausible. Once the necessary assumptions has been made, we will proceed towards the general calibration. 4.1 Introduction The diffusion term σ k (t) : [0, T k 1 ] R d in dynamics equations of forward rate F k (t) ( see ) is usually interpreted as instantaneous volatility at time t for the forward LIBOR rate F k. Because span{e 1,.., e M } = R d, we can without loss of generality write any volatility structure, as follows: σ k (t) = γ k (t)e k (t), e k R d, k = 1,..M (4.1.1) where e k are unit vectors in R d and γ k : [0, T k 1 ] R +. We can conclude then that separate calibration is always possible, for choice of γ k will only influence caplet prices (see (3.3.1)) and the choice of e k will specify the correlation structure. 4.2 Specification of Instantaneous Volatility In this section we will focus on a specification of γ k. As mentioned previously, only relevant set of calibration objects is the set of implied Black caplet volatilities. Let Σ 0 := {υi Black i = 1,...M} 24

35 Σ 0 Σ 2 Σ 4 Σ Figure 4.1: Evolution of the time-homogeneous volatility term structure for a LIBOR market consisting of 10 forward rates, and τ i = 1 2 year. denote implied Black caplets volatilities at time 0 given by the market. proposition (3.3.1) we have the recursive relation : Further, from (υi Black ) 2 = 1 Ti 1 σ i (t) 2 dt = 1 Ti 1 γi 2 (t)dt (4.2.1) T i 1 0 T i 1 0 which connects γ k with υk Black at time 0. However, to be able to price a contingent claim, we also must determine γ k on every interval (T i, T i+1 ] induced by partition Π. To analyze this problem, we consider the time evolution of the volatility term structure. T Σ 0 T 0 1 T M 2 Σ1... Σ M 1 where Σ k = {υ Black k i i = 1,...M k}, for at time T k there are M (k + 1) forwards alive. In other words, to be able to proceed with calibration, we need knowledge of all Σ k at time 0. One way to resolve this is to assume that υ Black k i = υ Black 0 i for k = 1,..M 1, i = 1,..M k. This implies a time-homogenous volatility structure, which is economically plausible and justified by observation of the volatility term structure which does not change shape much over time (see figure (4.1)). Remark Here we assumed that τ i = τ j, e.g., all intervals have same length. This is not as big a restriction as it appears to be at first glance, for it can be resolved by invoking some interpolation techniques (see for example [BrMe01], section 6.17). This choice and relation (4.2.1) imply γ k (t) = g(t k 1 t), for some g. BGM dynamics require g to be bounded and deterministic 1. Then g L p (0, T M 1 ] for all p 1. However, 1 BGM dynamics can be even set up in more general way, for example with stochastic volatility.