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Markov Representation of the Heath-Jarrow-Morton Model

Markov Representation of the Heath-Jarrow-Morton Model
Author: Oren Cheyette
Publisher:
Total Pages: 13
Release: 1999
Genre:
ISBN:
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This paper provides a derivation of an arbitrage free approximation to any HJM model as a continuous time Markov model with a finite number of state variables. Arbitrage freedom is maintained exactly at the cost of approximating any particular term structure of volatility. Using a large enough set of state variables, any ``reasonable'' term structure of volatility can be approximated to any desired level of accuracy in a systematic fashion. The state variables fall into two sets. One set specifies the values of a finite number of forward rates of different maturities. The other set determines how to ``interpolate'' to other maturities; their values depend on the integrated covariances of the variables in the first set. A restriction of the model to volatility term structures dependent only on time and relative maturity is derived: a general volatility structure with this property is approximated as a sum of exponentials in the relative maturity, weighted by arbitrary functions of the time and the current term structure. A three factor, 19 state version of this model is fitted to 12 years of weekly US Treasury yield curve data. The results indicate that it is possible to achieve an accurate representation of the empirical term structure factors with only two or three exponential terms per factor. NOTE: This paper is available in TeX (with macros) or postscript form.

Single Factor Heath-Jarrow-Morton Term Structure Models Based on Markov Spot Interest Rate Dynamics

Single Factor Heath-Jarrow-Morton Term Structure Models Based on Markov Spot Interest Rate Dynamics
Author: Andrew Jeffrey
Publisher:
Total Pages:
Release: 2000
Genre:
ISBN:
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This paper considers the class of Heath-Jarrow-Morton term structure models where the spot interest rate is Markov and the term structure at time t is a function of time, maturity and the spot interest rate at time t. A representation for this class of models is derived and I show that the functional forms of the forward rate volatility structure and the initial forward rate curve cannot be arbitrarily chosen. I provide necessary and sufficient conditions indicating which combinations of these functional forms are allowable. I also derive a partial differential equation representation of the term structure dynamics which does not require explicit modeling of both the market price of risk and the drift term for the spot interest rate process. Using the analysis presented in this paper a class of intertemporal term structure models is derived.

Stochastic Calculus for Finance II

Stochastic Calculus for Finance II
Author: Steven E. Shreve
Publisher: Springer Science & Business Media
Total Pages: 586
Release: 2004-06-03
Genre: Business & Economics
ISBN: 9780387401010
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"A wonderful display of the use of mathematical probability to derive a large set of results from a small set of assumptions. In summary, this is a well-written text that treats the key classical models of finance through an applied probability approach....It should serve as an excellent introduction for anyone studying the mathematics of the classical theory of finance." --SIAM

Understanding and Managing Model Risk

Understanding and Managing Model Risk
Author: Massimo Morini
Publisher: John Wiley & Sons
Total Pages: 452
Release: 2011-10-20
Genre: Business & Economics
ISBN: 0470977744
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A guide to the validation and risk management of quantitative models used for pricing and hedging Whereas the majority of quantitative finance books focus on mathematics and risk management books focus on regulatory aspects, this book addresses the elements missed by this literature--the risks of the models themselves. This book starts from regulatory issues, but translates them into practical suggestions to reduce the likelihood of model losses, basing model risk and validation on market experience and on a wide range of real-world examples, with a high level of detail and precise operative indications.

Dynamic Asset Pricing Theory

Dynamic Asset Pricing Theory
Author: Darrell Duffie
Publisher: Princeton University Press
Total Pages: 488
Release: 2010-01-27
Genre: Business & Economics
ISBN: 1400829208
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This is a thoroughly updated edition of Dynamic Asset Pricing Theory, the standard text for doctoral students and researchers on the theory of asset pricing and portfolio selection in multiperiod settings under uncertainty. The asset pricing results are based on the three increasingly restrictive assumptions: absence of arbitrage, single-agent optimality, and equilibrium. These results are unified with two key concepts, state prices and martingales. Technicalities are given relatively little emphasis, so as to draw connections between these concepts and to make plain the similarities between discrete and continuous-time models. Readers will be particularly intrigued by this latest edition's most significant new feature: a chapter on corporate securities that offers alternative approaches to the valuation of corporate debt. Also, while much of the continuous-time portion of the theory is based on Brownian motion, this third edition introduces jumps--for example, those associated with Poisson arrivals--in order to accommodate surprise events such as bond defaults. Applications include term-structure models, derivative valuation, and hedging methods. Numerical methods covered include Monte Carlo simulation and finite-difference solutions for partial differential equations. Each chapter provides extensive problem exercises and notes to the literature. A system of appendixes reviews the necessary mathematical concepts. And references have been updated throughout. With this new edition, Dynamic Asset Pricing Theory remains at the head of the field.

Arbitrage Theory in Continuous Time

Arbitrage Theory in Continuous Time
Author: Tomas Björk
Publisher: Oxford University Press
Total Pages: 486
Release: 2004-03
Genre: Arbitrage
ISBN: 019153384X
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The second edition of this popular introduction to the classical underpinnings of the mathematics behind finance continues to combine sounds mathematical principles with economic applications. Concentrating on the probabilistics theory of continuous arbitrage pricing of financial derivatives, including stochastic optimal control theory and Merton's fund separation theory, the book is designed for graduate students and combines necessary mathematical background with a solid economic focus. It includes a solved example for every new technique presented, contains numerous exercises and suggests further reading in each chapter. In this substantially extended new edition, Bjork has added separate and complete chapters on measure theory, probability theory, Girsanov transformations, LIBOR and swap market models, and martingale representations, providing two full treatments of arbitrage pricing: the classical delta-hedging and the modern martingales. More advanced areas of study are clearly marked to help students and teachers use the book as it suits their needs.

Arbitrage Theory in Continuous Time

Arbitrage Theory in Continuous Time
Author: Tomas Bjork
Publisher: Oxford University Press, USA
Total Pages: 584
Release: 2020-01-16
Genre: Arbitrage
ISBN: 0198851618
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The fourth edition of this widely used textbook on pricing and hedging of financial derivatives now also includes dynamic equilibrium theory and continues to combine sound mathematical principles with economic applications. Concentrating on the probabilistic theory of continuous time arbitrage pricing of financial derivatives, including stochastic optimal control theory and optimal stopping theory, Arbitrage Theory in Continuous Time is designed for graduate students in economics and mathematics, and combines the necessary mathematical background with a solid economic focus. It includes a solved example for every new technique presented, contains numerous exercises, and suggests further reading in each chapter. All concepts and ideas are discussed, not only from a mathematics point of view, but with lots of intuitive economic arguments. In the substantially extended fourth edition Tomas Bjork has added completely new chapters on incomplete markets, treating such topics as the Esscher transform, the minimal martingale measure, f-divergences, optimal investment theory for incomplete markets, and good deal bounds. This edition includes an entirely new section presenting dynamic equilibrium theory, covering unit net supply endowments models and the Cox-Ingersoll-Ross equilibrium factor model. Providing two full treatments of arbitrage theory-the classical delta hedging approach and the modern martingale approach-this book is written so that these approaches can be studied independently of each other, thus providing the less mathematically-oriented reader with a self-contained introduction to arbitrage theory and equilibrium theory, while at the same time allowing the more advanced student to see the full theory in action. This textbook is a natural choice for graduate students and advanced undergraduates studying finance and an invaluable introduction to mathematical finance for mathematicians and professionals in the market.

XVA

XVA
Author: Andrew Green
Publisher: John Wiley & Sons
Total Pages: 536
Release: 2015-10-23
Genre: Business & Economics
ISBN: 1118556755
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Thorough, accessible coverage of the key issues inXVA XVA – Credit, Funding and Capital ValuationAdjustments provides specialists and non-specialists alikewith an up-to-date and comprehensive treatment of Credit, Debit,Funding, Capital and Margin Valuation Adjustment (CVA, DVA, FVA,KVA and MVA), including modelling frameworks as well as broader ITengineering challenges. Written by an industry expert, this booknavigates you through the complexities of XVA, discussing in detailthe very latest developments in valuation adjustments including theimpact of regulatory capital and margin requirements arising fromCCPs and bilateral initial margin. The book presents a unified approach to modelling valuationadjustments including credit risk, funding and regulatory effects.The practical implementation of XVA models using Monte Carlotechniques is also central to the book. You'll also find thoroughcoverage of how XVA sensitivities can be accurately measured, thetechnological challenges presented by XVA, the use of gridcomputing on CPU and GPU platforms, the management of data, and howthe regulatory framework introduced under Basel III presentsmassive implications for the finance industry. Explores how XVA models have developed in the aftermath of thecredit crisis The only text to focus on the XVA adjustments rather than thebroader topic of counterparty risk. Covers regulatory change since the credit crisis includingBasel III and the impact regulation has had on the pricing ofderivatives. Covers the very latest valuation adjustments, KVA and MVA. The author is a regular speaker and trainer at industry events,including WBS training, Marcus Evans, ICBI, Infoline and RISK If you're a quantitative analyst, trader, banking manager, riskmanager, finance and audit professional, academic or studentlooking to expand your knowledge of XVA, this book has youcovered.