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Martingale Methods in Financial Modelling

Authors: Marek Musiela, Marek Rutkowski

Publisher: Springer Berlin Heidelberg

Book Series : Stochastic Modelling and Applied Probability

Part of: Springer Professional "Wirtschaft+Technik" , Springer Professional "Wirtschaft" , Springer Professional "Technik"

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About this book

In the 2nd edition some sections of Part I are omitted for better readability, and a brand new chapter is devoted to volatility risk. As a consequence, hedging of plain-vanilla options and valuation of exotic options are no longer limited to the Black-Scholes framework with constant volatility.

In the 3rd printing of the 2nd edition, the second Chapter on discrete-time markets has been extensively revised. Proofs of several results are simplified and completely new sections on optimal stopping problems and Dynkin games are added. Applications to the valuation and hedging of American-style and game options are presented in some detail.

The theme of stochastic volatility also reappears systematically in the second part of the book, which has been revised fundamentally, presenting much more detailed analyses of the various interest-rate models available: the authors' perspective throughout is that the choice of a model should be based on the reality of how a particular sector of the financial market functions, never neglecting to examine liquid primary and derivative assets and identifying the sources of trading risk associated. This long-awaited new edition of an outstandingly successful, well-established book, concentrating on the most pertinent and widely accepted modelling approaches, provides the reader with a text focused on practical rather than theoretical aspects of financial modelling.

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Table of Contents

Frontmatter

Spot and Futures Markets

Frontmatter

1. An Introduction to Financial Derivatives

We shall first review briefly the most important kinds of financial contracts, traded either on exchanges or over-the-counter (OTC), between financial institutions and their clients. For a detailed account of the fundamental features of spot (i.e., cash) and futures financial markets the reader is referred, for instance, to Duffie (1989), Kolb (1991), Redhead (1996), or Hull (1997).

Marek Musiela, Marek Rutkowski

2. Discrete-time Security Markets

This chapter deals mostly with finite markets – that is, discrete-time models of financial markets in which all relevant quantities take a finite number of values. Essentially, we follow here the approach of Harrison and Pliska (1981); a more exhaustive analysis of finite markets can be found in Taqqu and Willinger (1987). An excellent introduction to discrete-time financial mathematics is given by Pliska (1997) and Shreve (2004). A monograph by Föllmer and Schied (2000) is the most comprehensive source in the area.

Marek Musiela, Marek Rutkowski

3. Benchmark Models in Continuous Time

The option pricing model developed in a groundbreaking paper by Black and Scholes (1973), formalized and extended in the same year by Merton (1973), enjoys great popularity. It is computationally simple and, like all arbitrage-based derivative pricing models, does not require the knowledge of an investor’s risk preferences. Option valuation within the Black-Scholes framework is based on the already familiar concept of perfect replication of contingent claims. More specifically, we will show that an investor can replicate an option’s return stream by continuously rebalancing a self-financing portfolio involving stocks and risk-free bonds. At any date t, the current wealth of a replicating portfolio determines the arbitrage price of an option.

Marek Musiela, Marek Rutkowski

4. Foreign Market Derivatives

In this chapter, an arbitrage-free model of the domestic security market is extended by assuming that trading in foreign assets, such as foreign risk-free bonds and foreign stocks (and their derivatives), is allowed. We will work within the classical Black-Scholes framework. More specifically, both domestic and foreign risk-free interest rates are assumed throughout to be nonnegative constants, and the foreign stock price and the exchange rate are modelled by means of geometric Brownian motions. This implies that the foreign stock price, as well as the price in domestic currency of one unit of foreign currency (i.e., the exchange rate) will have lognormal probability distributions at future times. Notice, however, that in order to avoid perfect correlation between these two processes, the underlying noise process should be modelled by means of a multidimensional, rather than a one-dimensional, Brownian motion. Our main goal is to establish explicit valuation formulas for various kinds of currency and foreign equity options. Also, we will provide some indications concerning the form of the corresponding hedging strategies. It is clear that foreign market contracts of certain kinds should be hedged both against exchange rate movements and against the fluctuations of relevant foreign equities.

Marek Musiela, Marek Rutkowski

5. American Options

In contrast to the holder of a European option, the holder of an American option is allowed to exercise his right to buy (or sell) the underlying asset at any time before or at the expiry date. This special feature of American-style options – and more generally of American claims – makes the arbitrage pricing of American options much more involved than the valuation of standard European claims. We know already that arbitrage valuation of American claims is closely related to specific optimal stopping problems. Intuitively, one might expect that the holder of an American option will choose her exercise policy in such a way that the expected payoff from the option will be maximized. Maximization of the expected discounted payoff under subjective probability would lead, of course, to non-uniqueness of the price. It appears, however, that for the purpose of arbitrage valuation, the maximization of the expected discounted payoff should be done under the martingale measure (that is, under risk-neutral probability). Thus, the uniqueness of the arbitrage price of an American claim holds.

Marek Musiela, Marek Rutkowski

6. Exotic Options

In the preceding chapters, we have focused on the two standard classes of options – that is, call and put options of European and American style. The aim of this chapter is to study examples of more sophisticated option contracts. Although the payoffs of exotic options are given by similar expressions for both spot and futures options, the corresponding valuation formulas would not agree. We restrict here our attention to the case of exotic spot options.

Marek Musiela, Marek Rutkowski

7. Volatility Risk

Loosely speaking, a model of financial security can be often identified with the following three components: an underlying variable, a mechanism used to reflect the uncertainty with regard to the future value of this variable and, last but not least, arbitrage-free considerations.

Marek Musiela, Marek Rutkowski

8. Continuous-time Security Markets

This chapter furnishes a summary of basic results associated with continuous-time financial modelling. The first section deals with a continuous-time model, which is based on the Itô stochastic integral with respect to a semimartingale. Such a model of financial market, in which the arbitrage-free property hinges on the chosen class of admissible trading strategies, is termed the standard market model hereafter. We discuss the relevance of a judicious choice of a numeraire asset. On a more theoretical side, we briefly comment on the class of results – informally referred to as a fundamental theorem of asset pricing – which say, roughly, that the absence of arbitrage opportunities is equivalent to the existence of a martingale measure. The theory developed in this chapter applies both to stock markets and bond markets. It can thus be seen as a theoretical background to the second part of this text.

Marek Musiela, Marek Rutkowski

Fixed-income Markets

Frontmatter

9. Interest Rates and Related Contracts

By a fixed-income market we mean that sector of the global financial market on which various interest rate-sensitive instruments, such as bonds, swaps, swaptions, caps, etc. are traded. In real-world practice, several fixed-income markets operate; as a result, many concepts of interest rates have been developed. There is no doubt that management of interest rate risk, by which we mean the control of changes in value of a stream of future cash flows resulting from changes in interest rates, or more specifically the pricing and hedging of interest rate products, is an important and complex issue. It creates a demand for mathematical models capable of covering all sorts of interest rate risks.

Marek Musiela, Marek Rutkowski

10. Short-Term Rate Models

The aim of this chapter is to survey the most popular models of the short-term interest rate. For convenience, we will work throughout within a continuous-time framework; a detailed presentation of a discrete-time approach to term structure modelling is done in Jarrow (1995). The continuous-time short-term interest rate is usually modelled as a one-dimensional diffusion process. In this text, we provide only a brief survey of the most widely accepted examples of diffusion processes used to model the short-term rate. The short-term rate approach to bond price modelling is not developed in subsequent chapters. This is partially explained by the abundance of literature taking this approach, and partially by the difficulty of fitting the observed term structure of interest rates and volatilities within a simple diffusion model (see Pelsser (2000a) or Brigo and Mercurio (2001a)). Instead, we develop the term structure theory for a much larger class of models that includes diffusion-type models as special cases. Nevertheless, it should be made clear that diffusion-type modelling of the short-term interest rate is still the most popular method for the valuing and hedging of interest rate-sensitive derivatives.

Marek Musiela, Marek Rutkowski

11. Models of Instantaneous Forward Rates

The Heath, Jarrow and Morton approach to term structure modelling is based on an exogenous specification of the dynamics of instantaneous, continuously compounded forward rates f(t,T). For any fixed maturity T≤T^*, the dynamics of the forward rate f(t,T) are (cf. Heath et al. (1990a, 1992b))

$$df(t,T)=\alpha (t,T)\,dt+\sigma (t,T)\cdot dW_{t},$$

(1)

where α and σ are adapted stochastic processes with values in ℝ and ℝ^d respectively, and W is a d-dimensional standard Brownian motion with respect to the underlying probability measure ℙ (to be interpreted as the actual probability).

Marek Musiela, Marek Rutkowski

12. Market LIBOR Models

As was mentioned already, the acronym LIBOR stands for the London Interbank Offered Rate. It is the rate of interest offered by banks on deposits from other banks in eurocurrency markets. Also, it is the floating rate commonly used in interest rate swap agreements in international financial markets (in domestic financial markets as the reference interest rate for a floating rate loans it is customary to take a prime rate or a base rate). LIBOR is determined by trading between banks and changes continuously as economic conditions change. For more information on market conventions related to the LIBOR and Eurodollar futures, we refer to Sect. 9.3.4.

Marek Musiela, Marek Rutkowski

13. Alternative Market Models

The BGM market model, examined in some detail in the previous chapter, is clearly oriented toward a particular interest rate, LIBOR, and thus it serves well for the valuation and hedging of the LIBOR related derivatives, such as plain-vanilla and exotic caps and floors. It thus may be see as a good candidate for the role of the Black-Scholes-like benchmark model for this particular sector of the fixed-income market. Interest rate swaps are another class of interest-rate-sensitive contracts of great practical importance. In a generic fixed-for-floating swap, a fixed rate of interest is exchanged periodically for some preassigned variable (floating) rate. Since swaps and derivative securities on the swap rate, termed swap derivatives, are in many markets more liquidly traded than LIBOR derivatives, there is an obvious demand for specific models capable of efficient handling this class of interest rate products. Our goal in this chapter is to present some recent research focused on market models that are alternatives to the market model for LIBORs.

Marek Musiela, Marek Rutkowski

14. Cross-currency Derivatives

In this chapter, we deal with derivative securities related to at least two economies (a domestic market and a foreign market, say). Any such security will be referred to as a cross-currency derivative. In contrast to the model examined in Chap. 4, all interest rates and exchange rates are assumed to be random. It seems natural to expect that the fluctuations of interest rates and exchange rates will be highly correlated. This feature should be reflected in the valuation and hedging of foreign and cross-currency derivative securities in the domestic market. Feiger and Jacquillat (1979) (see also Grabbe (1983)) were probably the first to study, in a systematic way, the valuation of currency options within the framework of stochastic interest rates (they do not provide a closed-form solution for the price, however). More recently, Amin and Jarrow (1991) extended the HJM approach by incorporating foreign economies. Frachot (1995) examined a special case of the HJM model with stochastic volatilities, in which the bond price and the exchange rate are assumed to be deterministic functions of a single state variable.

Marek Musiela, Marek Rutkowski

Backmatter

Title: Martingale Methods in Financial Modelling
Authors: Marek Musiela
Marek Rutkowski
Copyright Year: 2005
Publisher: Springer Berlin Heidelberg
Electronic ISBN: 978-3-540-26653-2
Print ISBN: 978-3-540-20966-9
DOI: https://doi.org/10.1007/b137866