Mathematical methods for foreign exchange. A financial engineer's approach (Q2763689)

It is said that the money never sleeps. This is true apparently with the inner working of the world of financial markets. The best conformation can be found in the foreign exchange (forex) markets because of their depth, versatility and transparency. Changes in foreign exchange rates (FXRs) are caused by both deep structural shifts in the respective economics and a variety of less fundamental factors which have profound impact on the world economy at large. An adequate formalism for the dynamics of FXRs has been developed in this book. The author presents an exhaustive and systematic study of mathematical instruments with applications to the FXRs. It deals with various problems that financial engineers face in the forex market place and gives a detailed account of mathematical methods necessary for their solution. NEWLINENEWLINENEWLINEThe contribution of the author in this book consists in the following: NEWLINENEWLINENEWLINE(i) The reader is introduced to the key problems of contemporary financial engineering in the forex context with particular emphasis on pricing and risk management of derivative instruments. NEWLINENEWLINENEWLINE(ii) The author describes and develops a whole spectrum of mathematical tools which are needed for solving these problems. NEWLINENEWLINENEWLINE(iii) It is demonstrated how to use these tools in a variety of practical problems of forex markets and FXRs. NEWLINENEWLINENEWLINEThe book is divided into four parts. The first part introduces a brief overview of the historic development of financial engineering. The properties of forex as an asset class, and that of spot FXRS are also discussed. The derivatives in the forex context are discussed in this part. NEWLINENEWLINENEWLINEPart two contains three chapters describing the mathematical tools which are needed in order to understand the theory of discrete -- and continuous -- time processes, and, by implication, the dynamics of FXRs. In the first chapter a beautiful account with illustrative examples of the basic definitions and the concepts of probability theory, the convergence of sequences of random variables and the limit theorems for sums of independent random variables have been given. In the second chapter the author describes the properties of the ``stochastic engines'' needed in order to construct discrete-time models of forex in single- and multi-period markets. These engines, which are called binomial and multinomial, Bernoulli processes and Wiener processes have been tested in a variety of situations. The third chapter is devoted to describe the mathematical tools that are required for the investigation of continuous-time markets with a particular emphasis on the stochastic engines, which are used as sources of randomness in continuous time. The finite-dimensional distributions associated with a random processes and Kolmogoroff's fundamental theorem are discussed. The transition probability functions associated with Markov processes have been introduced and the Chapman-Kolmogoroff equation is formulated. An account of diffusion processes has been given and Kolmogoroff's backward and the Fokker-Planck forward equations are derived which govern the transition probability density function for diffusion processes. NEWLINENEWLINENEWLINEPart III is devoted to discrete-time models. This part consists of two chapters discussing deep problems of financial engineering in the context of single- and multi-period markets considering such markets as collections of different countries each with its own currency. The first chapter deals with the binomial single-period two countries market with non-risk, risk and ideal world money market accounts and solves the corresponding forex problem. The author derives an idealized world-market in its generality and formulates the forex problem. The relevant economic constraints are derived on the relative FXRs with guarantee that the market is logically constraint. It has been shown how these constraints can be formulated in a long usage of linear algebra and FXRS by virtue of the theorem of alternative and separative hyperplane is constructed. Complete and incomplete markets are defined and shown how to price contingent claims terms of underlying assets. Here the elementary portfolio theory with forex treated as an asset class has been also discussed. The optimal investment problem for a representative investor market under an equilibrium approach to find the FXRs has been discussed. NEWLINENEWLINENEWLINEThe next chapter deals with multiperiod market and overcomes the drawbacks of single period market by extending the concept of the space of elementary events with appropriate information structures and objectives and risk neutral probability by measures and predictable self-financing strategies. The stationary binomial multi-period two country markets with investors as time progresses and choice of trading strategies have been discussed. A model for the non-stationary market which are non re-combining and combining have been discussed and a general model for the multi-normal multi-country market is given. NEWLINENEWLINENEWLINEThe finest part of the book is the Part IV, which consists of 9 chapters on continuous models. In the first chapter the author introduces several models describing the continuous time dynamics of the fixed income instrument and FXRs in two and multi country economics, assuming that these follow geometric Brownian motion, which are used in subsequent chapters. NEWLINENEWLINENEWLINEThe second chapter deals with the group-theoretical approach of European options showing how to price and hedge European options on forex, assuming that in both countries bond prices are deterministic while the FXRs follow the standard geometric Brownian motion with constant volatility and derives the homogeneous Black-Scholes equation. NEWLINENEWLINENEWLINEIn the next chapter the author discusses various derivations from the Black-Scholes paradigm and their nullification for pricing and hedging. NEWLINENEWLINENEWLINEThe stochastic volatility plays an outstanding role in the forex context. The author discusses in detail various approaches which can be used in order to price options on FXRs with stochastic volatility including analytical perturbative and numerical ones. NEWLINENEWLINENEWLINEOne chapter is devoted to American options on Forax which can be exercised at any time between their inception and maturity. The pricing of American options is more difficult than their European counterparts since the corresponding pricing problem is nonlinear due to the possibility of an early exercise. NEWLINENEWLINENEWLINEThe author presents several complementary formulations of the pricing problem. NEWLINENEWLINENEWLINE1. Introducing the inhomogeneous Black-Scholes American option which replaces the homogeneous problem for the European option NEWLINENEWLINENEWLINE2. Formulating the valuation problem as a linear complementarity problem. NEWLINENEWLINENEWLINE3. Formulating the valuation problem as linear program. It is shown that switching from the domestic to foreign country transforms the valuation problem for calls into the valuation problem for puts thus confirming formally put-call symmetry relations which are established on financial ground. NEWLINENEWLINENEWLINETwo chapters are devoted to path dependent option: I Barrier option . II Look back, Asian and other option. The Barrier option which have weak path dependent feature and cheaper alternative for ordinary calls and puts have been discussed. It is shown that out-style barrier options are deactivated (while in-style options are activated) if FXRs hits a certain predetermined level at some time between the inception of the option and it maturity. NEWLINENEWLINENEWLINEOne chapter is devoted to discuss effects of imperfect hedging and market frictions, such as transaction cost, liquidity, constraints, default risk etc. as pricing and hedging of derivatives. NEWLINENEWLINENEWLINEIn the last chapter, the author outlines some future research directions which in his opinion are particularly promising and briefly summarise the main contributions which are made in this book. NEWLINENEWLINENEWLINEThe book is a useful textbook for students of financial engineering and a valuable reference book of the research work in financial engineering and so it is expected to be self-contained in all respects. The bibliography is exhaustive and is suitable for those who desire to pursue their further studies in this branch of financial engineering.