Mathematical Methods for Foreign Exchange : A Financial Engineer's Approach

by Lipton, Alexander

• ISBN13: 9789810246150
• ISBN10: 9810246153
• Format: Hardcover
• Copyright: 2002-01-01
• Publisher: WORLD SCIENTIFIC PUB CO INC

Summary

This comprehensive book presents a systematic and practically oriented approach to mathematical modeling in finance, particularly in the foreign exchange context. It describes all the relevant aspects of financial engineering, including derivative pricing, in detail. The book is self-contained, with the necessary mathematical, economic, and trading background carefully explained. In addition to the lucid treatment of the standard material, it describes many original results.

The book can be used both as a text for students of financial engineering, and as a basic reference for risk managers, traders, and academics.

Table of Contents

Preface	ix
I Introduction	1	(18)
Foreign exchange markets	3	(16)
Introduction	3	(1)
Historical background	4	(3)
Forex as an asset class	7	(1)
Spot forex	8	(1)
Derivatives: forwards, futures, calls, puts, and all that	9	(8)
References and further reading	17	(2)
II Mathematical preliminaries	19	(100)
Elements of probability theory	21	(22)
Introduction	21	(1)
Probability spacs	22	(4)
Random variables	26	(12)
Convergence of random variables and limit theorems	38	(4)
References and further reading	42	(1)
Discrete-time stochastic engines	43	(16)
Introduction	43	(1)
Time series	44	(2)
Binomial stochastic engines for single- and multi-period markets	46	(9)
Multinomial stochastic engines	55	(2)
References and further reading	57	(2)
Continuous-time stochastic engines	59	(60)
Introduction	59	(2)
Stochastic processes	61	(2)
Markov processes	63	(2)
Diffusions	65	(11)
Wiener processes	76	(5)
Poisson processes	81	(3)
SDE and Mappings	84	(7)
Linear SDEs	91	(5)
SDEs for jump-diffusions	96	(2)
Analytical solution of PDEs	98	(8)
Introduction	98	(1)
The reduction method	98	(5)
The Laplace transform method	103	(1)
The eigenfunction expansion method	104	(2)
Numerical solution of PDEs	106	(6)
Introduction	106	(1)
Explicit, implicit, and Crank-Nicolson schemes for solving one-dimensional problems	107	(2)
ADI scheme for solving two-dimensional problems	109	(3)
Numerical solution of SDEs	112	(4)
Introduction	112	(1)
Formulation of the problem	113	(1)
The Euler-Maruyama scheme	113	(1)
The Milstein scheme	114	(2)
References and further reading	116	(3)
III Discrete-time models	119	(94)
Single-period markets	121	(50)
Introduction	121	(2)
Binomial markets with nonrisky investments	123	(17)
Binomial markets without nonrisky investments	140	(5)
General single-period markets	145	(2)
Economic constraints	147	(7)
Pricing of contingent claims	154	(8)
Elementary portfolio theory	162	(4)
The optimal investment problem	166	(2)
Elements of equilibrium theory	168	(1)
References and further reading	169	(2)
Multi-period markets	171	(42)
Introduction	171	(1)
Stationary binomial markets	172	(22)
Non-stationary binomial markets	194	(8)
Introduction	194	(1)
The nonrecombining case	195	(2)
The recombining case	197	(5)
General multi-period markets	202	(5)
Contingent claims and their valuation and hedging	207	(1)
Portfolio theory	208	(2)
The optimal investment problem	210	(2)
References and further reading	212	(1)
IV Continuous-time models	213	(434)
Stochastic dynamics of forex	215	(24)
Introduction	215	(1)
Two-country markets with deterministic investments	216	(7)
Two-country markets without deterministic investments	223	(4)
Multi-country markets	227	(3)
The nonlinear diffusion model	230	(2)
The jump diffusion model	232	(1)
The stochastic volatility model	233	(3)
The general forex evolution model	236	(1)
References and further reading	237	(2)
European options: the group-theoretical approach	239	(54)
Introduction	239	(1)
The two-country homogeneous problem, I	240	(10)
Formulation of the problem	240	(5)
Reductions of the pricing problem	245	(3)
Continuous hedging and the Greeks	248	(2)
Forwards, calls and puts	250	(13)
Definitions	250	(1)
Pricing via the Feynman-Kac formula	250	(6)
A naive pricing attempt	256	(1)
Pricing via the Fourier transform method	257	(2)
Pricing via the Laplace transform method	259	(2)
The limiting behavior of calls and puts	261	(2)
Contingent claims with arbitrary payoffs	263	(3)
Introduction	263	(1)
The decomposition formula	263	(1)
Call and put bets	264	(1)
Log contracts and modified log contracts	265	(1)
Dynamic asset allocation	266	(9)
The two-country homogeneous problem, II	275	(2)
The multi-country homogeneous problem	277	(4)
Introduction	277	(1)
The homogeneous pricing problem	278	(1)
Reductions	279	(1)
Probabilistic pricing and hedging	280	(1)
Some representative multi-factor options	281	(11)
Introduction	281	(1)
Outperformance options	282	(2)
Options on the maximum or minimum of several FXRs	284	(2)
Basket options	286	(4)
Index options	290	(1)
The multi-factor decomposition formula	291	(1)
References and further reading	292	(1)
European options, the classical approach	293	(26)
Introduction	293	(1)
The classical two-country pricing problem, I	294	(4)
The projection method	294	(2)
The classical method	296	(1)
The impact of the actual drift	297	(1)
Solution of the classical pricing problem	298	(12)
Nondimensionalization	298	(1)
Reductions	298	(1)
The pricing and hedging formulas for forwards, calls and puts	299	(7)
European options with exotic payoffs	306	(4)
The classical two-country pricing problem, II	310	(5)
The multi-country classical pricing problem	315	(2)
Introduction	315	(1)
Derivation	315	(1)
Reductions	315	(2)
Pricing and hedging of multi-factor options	317	(1)
References and further reading	317	(2)
Deviations from the Black-Scholes paradigm I: nonconstant volatility	319	(86)
Introduction	319	(2)
Volatility term structures and smiles	321	(8)
Introduction	321	(1)
The implied volatility	321	(2)
The local volatility	323	(2)
The inverse problem	325	(4)
How to deal with the smile	329	(1)
Pricing via implied t.p.d.f.'s	329	(12)
Implied t.p.d.f.'s and entropy maximization	329	(3)
Possible functional forms of t.p.d.f.'s	332	(3)
The chi-square pricing formula, I	335	(3)
The Edgeworth-type pricing formulas	338	(3)
The sticky-strike and the sticky-delta models	341	(3)
The general local volatility model	344	(9)
Introduction	344	(1)
Possible functional forms of local volatility	345	(3)
The hyperbolic volatility model	348	(2)
The displaced diffusion model	350	(3)
Asymptotic treatment of the local volatility model	353	(6)
The CEV model	359	(12)
Introduction	359	(1)
Reductions of the pricing problem	360	(2)
Evaluation of the t.p.d.f.	362	(2)
Derivative pricing	364	(4)
ATMF approximation	368	(3)
The jump diffusion model	371	(4)
Introduction	371	(1)
The pricing problem	371	(1)
Evaluation of the t.p.d.f.	372	(1)
Risk-neutral pricing	373	(2)
The stochastic volatility model	375	(12)
Introduction	375	(1)
Basic equations	376	(3)
Evaluation of the t.p.d.f.	379	(5)
The pricing formula	384	(2)
The case of zero correlation	386	(1)
Small volatility of volatility	387	(11)
Introduction	387	(1)
Basic equations	388	(1)
The martingale formulation	388	(1)
Perturbative expansion	389	(4)
Summary of ODEs	393	(1)
Solution of the leading order pricing problem	394	(1)
The square-root model	394	(3)
Computation of the implied volatility	397	(1)
Multi-factor problems	398	(6)
Introduction	398	(1)
The chi-square pricing formula, II	399	(5)
References and further reading	404	(1)
American Options	405	(46)
Introduction	405	(2)
General considerations	407	(8)
The early exercise constraint	407	(1)
The early exercise premium	408	(2)
Some representative examples	410	(1)
Rational bounds	411	(3)
Parity and symmetry	414	(1)
The risk-neutral valuation	415	(1)
Alternative formulations fo the valuation problem	416	(5)
Introduction	416	(1)
The inhomogeneous Black-Scholes problem formulation	416	(2)
The linear complementarity formulation	418	(2)
The linear program formulation	420	(1)
Duality between puts and calls	421	(1)
Application of Duhamel's principle	422	(3)
The value of the early exercise premium	422	(1)
The location of the early exercise boundary	423	(2)
Asymptotic analysis of the pricing problem	425	(9)
Short-dated options	425	(5)
Long-dated and perpetual options	430	(4)
Approximate solution of the valuation problem	434	(8)
Introduction	434	(1)
Bermudan approximation and extrapolation to the limit	434	(6)
Quadratic approximation	440	(2)
Numerical solution of the pricing problem	442	(3)
Bermudan approximation	442	(1)
Linear complementarity	443	(1)
Integral equation	444	(1)
Monte-Carlo valuation	444	(1)
American options in a non-Black-Scholes framework	445	(1)
Multi-factor American options	445	(4)
Formulation	445	(1)
Two representative examples	446	(3)
References and further reading	449	(2)
Path-dependent options I: barrier options	451	(76)
Introduction	451	(1)
Single-factor, single-barrier options	452	(17)
Introduction	452	(1)
Pricing of single-barrier options via the method of images	453	(9)
Pricing of single-barrier options via the method of heat potentials	462	(7)
Static hedging	469	(3)
Single-factor, double-barrier options	472	(12)
Introduction	472	(1)
Formulation	473	(1)
The pricing problem without rebates	474	(3)
Pricing of no-rebate calls and puts and double-no-touch options	477	(5)
Pricing of calls and puts with rebate	482	(2)
Deviations from the Black-Scholes paradigm	484	(14)
Introduction	484	(1)
Barrier options in the presence of the term structure of volatility	484	(2)
Barrier options in the presence of constant elasticity of variance	486	(6)
Barrier options in the presence of stochastic volatility	492	(6)
Multi-factor barrier options	498	(1)
Options on one currency with barriers on the other currency	499	(15)
Introduction	499	(1)
Formulation	499	(2)
Solution via the Fourier method	501	(8)
Solution via the method of images	509	(4)
An alternative approach	513	(1)
Options with one barrier for each currency	514	(6)
General considerations	514	(2)
The Green's function	516	(4)
Two-factor, double-no-touch option	520	(1)
Four-barrier options	520	(6)
References and further reading	526	(1)
Path-dependent options II: lookback, Asian and other options	527	(90)
Introduction	527	(1)
Path-dependent options and augmented SDEs	528	(10)
Description of path dependent options	528	(6)
The augmentation procedure	534	(3)
The pricing problem for augmented SDEs	537	(1)
Risk-neutral valuation of path-dependent options	538	(1)
Probabilistic pricing	539	(3)
Lookback calls and puts	542	(11)
Description	542	(1)
Pricing via the method of images	543	(4)
Similarity reductions	547	(1)
Pricing via the Laplace transform	548	(2)
Probabilistic pricing	550	(2)
Barriers	552	(1)
Asian options	553	(13)
Description	553	(1)
Geometric averaging	553	(3)
Arithmetic averaging	556	(2)
Exact solution via similarity reductions	558	(3)
Pricing via the Laplace transform	561	(1)
Approximate pricing of Asian calls revisited	562	(3)
Discretely sampled Asian options	565	(1)
Timer, fader and Parisian options	566	(12)
Introduction	566	(1)
Timer options	566	(7)
Fader options	573	(1)
Parisian options	573	(5)
Standard passport options	578	(8)
Description	578	(1)
Similarity reductions and splitting	579	(2)
Pricing via the Laplace transform	581	(1)
Explicit solution for zero foreign interest rate	582	(4)
More genral passport options	586	(5)
General considerations	586	(1)
Explicit solution for zero foreign interest rate	587	(4)
Variance and volatility swaps	591	(11)
Introduction	591	(1)
Description of swaps	591	(2)
Pricing and hedging of swaps via convexity adjustments	593	(6)
Log contracts and robust pricing and hedging of variance swaps	599	(3)
The impact of stochastic volatility of path-dependent options	602	(6)
The general valuation formula	602	(1)
Evaluation of the t.p.d.f.	603	(5)
A transformed valuation formula	608	(1)
Forward starting options (cliquets)	608	(3)
Options on volatility	611	(5)
The pricing problem	611	(2)
Pricing of variance swaps	613	(1)
Pricing of general swaps and swaptions	614	(2)
References and further reading	616	(1)
Deviations from the Black-Scholes paradigm II: market frictions	617	(28)
Introduction	617	(1)
Imperfect hedging	618	(9)
P&L distributions	618	(3)
Stop-loss start-gain hedging and local times	621	(1)
Parameter misspecification	622	(5)
The uncertain volatility model	627	(3)
Transaction costs	630	(3)
Liquidity risk	633	(2)
Default risk	635	(8)
Introduction	635	(1)
The pricing model	635	(2)
Pricing of defaultable European calls	637	(3)
Pricing of defaultable forward contracts	640	(3)
References and further reading	643	(2)
Future directions of research and conclusions	645	(2)
Introduction	645	(1)
Future directions	645	(1)
Conclusions	646	(1)
References and further reading	646	(1)
Bibliography	647	(22)
Index	669