The Brody-Hughston-Macrina approach to information-based asset pricing introduces a new way of looking at the mechanisms determining price movements in financial markets. The resulting theory of financial informatics is applicable across a wide range of asset classes and is distinguished by its emphasis on the explicit modelling of market information flows. In the BHM theory, each asset is defined by a collection of cash flows and each such cash flow is associated with a family of one or more so-called information processes that provide partial information about the cash flow. The theory is highly appealing on an intuitive basis: it is directly applicable to trading, investment and risk management - and yet at the same time leads to interesting mathematics. The present volume brings together a collection of 18 foundational papers of the subject by Brody, Hughston, and Macrina, many written in collaboration with various co-authors. There is a preface summarizing the current status of the theory, together with a brief history and bibliography of the subject. This book will be of great interest both to newcomers to financial mathematics as well as to established researchers in the subject.
Author(s): Dorje Brody, Lane Hughston, Andrea Macrina
Publisher: World Scientific Publishing
Year: 2022
Language: English
Pages: 443
City: Singapore
Contents
Preface
About the Editors
Chapter 1 Beyond Hazard Rates: A New Framework for Credit-Risk Modelling
1 Introduction and Summary
2 The Information-Based Approach
3 Defaultable Discount Bond Price Processes
4 Defaultable Discount Bond Dynamics
5 Simulation of Bond Price Processes
6 Digital Bonds and Binary Bonds with Partial Recovery
7 Dynamic Consistency and Model Calibration
8 Options on Credit-Risky Bonds
9 Bond Option Price Processes
10 Coupon Bonds: The X-Factor Approach
11 Credit Default Swaps
12 Baskets of Credit-Risky Bonds
13 Homogeneous Baskets
Acknowledgments
References
Chapter 2 Information-Based Asset Pricing
1. Introduction
2. The Modelling Framework
3. Modelling the Cash Flows
4. Modelling the Information Flow
5. Asset Price Dynamics in the Case of a Single Cash Flow
6. European-Style Call Options
7. Examples of Specific Dividend Structures
8. Market Factors and Multiple Cash Flows
9. Geometric Brownian Motion Model
10. Dividend Growth
11. Assets with Common Factors
12. Origin of Unhedgeable Stochastic Volatility
13. Time-Dependent Information Flow
14. Changes of Measure for Brownian Bridges
15. Derivation of the Conditional Density
16. Consistency Relations
17. Expected Dividend
18. Asset Prices and Derivative Prices
19. Existence of the Information Process
20. Multi-Factor Models with a Time-Dependent Information Flow Rate
Acknowledgments
References
Chapter 3 Dam Rain and Cumulative Gain
1. Introduction
2. Gamma processes and associated martingales
3. Properties of gamma bridge processes
4. Valuation of aggregate claims
5. Valuation of general reinsurance contracts
6. Discretely distributed cash flows
7. Price processes for options on reserves
8. Gamma-distributed terminal gains
References
Chapter 4 Informed Traders
1. Introduction
2. Information and asset pricing
3. Amount of information about the future cash flow contained in the price process
4. Analysis of information measures
5. A model for an informed trader
6. Innovations and the dynamics of informed valuations
7. Additional information held by the informed trader and statistical arbitrage strategies exploiting this
References
Chapter 5 Information of Interest
Market information about future liquidity
An elementary model for bond pricing
Dynamics of the bond price
Calibration procedure
Interpretation of the auxiliary measure
Interpretation of the function ϕ(x)
Conclusion
Chapter 6 Credit Risk, Market Sentiment and Randomly-Timed Default
1 Credit-Risk Modeling
2 Modeling the Market Filtration
3 Credit-Risky Discount Bond
4 Discount Bond Dynamics
5 Hazard Rates and Forward Hazard Rates
6 Options on Defaultable Bonds
Acknowledgements
References
Chapter 7 Lévy Random Bridges and the Modelling of Financial Information
1. Introduction
2. Preliminaries
2.1. Lévy processes
2.2. Lévy bridges
3. Lévy random bridges
3.1. Defining LRBs
3.2. Finite-dimensional distributions
3.3. LRBs as conditioned Lévy processes
3.4. The Markov property
3.4.1. Continuous state-space
3.4.2. Discrete state-space
3.5. Conditional terminal distributions
3.6. Measure changes
3.7. Dynamic consistency
3.8. Increments of LRBs
3.8.1. Increment distributions
3.8.2. The reordering of increments
4. Information-based asset pricing
4.1. BHM framework
4.1.1. Single X-factor market
4.1.2. Multiple X-factor market
4.2. Lévy bridge information
4.3. European option pricing
4.4. Binary bond
Acknowledgements
References
Chapter 8 Modelling Information Flows in Financial Markets
5.1 Cash Flow Structures and Market Factors
5.2 X-factor Analysis
5.3 Information Processes
5.4 Brownian-Bridge Information
5.5 Assets Paying a Single Dividend
5.6 Geometric Brownian Motion Model
5.7 Pricing Contingent Claims
5.8 Volatility and Correlation
5.9 Amount of Information about the Future Cash Flow Contained in the Price Process
5.10 Information Disparity and Statistical Arbitrage
5.11 Price Formation in Inhomogeneous Markets
Acknowledgements
References
Chapter 9 Heat Kernel Interest Rate Models with Time-Inhomogeneous Markov Processes
1. Introduction
2. Heat Kernels for Supermartingales
3. Pricing with Time-Inhomogeneous Markov Information
4. Explicit Pricing Kernel Models
5. Fixed-Income Derivatives with Brownian Bridge Information
Acknowledgments
References
Chapter 10 Lévy Information and the Aggregation of Risk Aversion
1. Introduction
2. Random risk aversion and market heterogeneity
3. Modelling the pricing kernel
4. Information-based estimation of market risk aversion
5. Geometric Lévy models
6. On the aggregation of jump-risk aversion
Appendix A. Conditional distribution of the risk aversion factor
Appendix B. Emergence of the Brownian driver
Appendix C. Existence and construction of the hidden variables X and Bt
References
Chapter 11 Signal Processing with Lévy Information
1. Introduction
2. Lévy information
3. Properties of Lévy information
4. Examples of Lévy information processes
References
Chapter 12 Heat Kernel Models for Asset Pricing
1. Introduction
2. Pricing Kernel Models and the Pricing of Bonds, Caplets and Swaptions
3. Closed-Form and Explicit Price Models
3.1. Quadratic and exponential quadratic models
3.2. Caplets and swaptions
3.3. Boundedness of prices
4. Dynamical Equations
4.1. Model risk
5. Incomplete Market Models Driven by LRBs
5.1. Exponential linear two-factor model with jumps
5.2. Useful formula
6. Pricing of a Generic Asset
6.1. Asset price diffusion with stochastic discounting
6.2. Dynamical equations
6.3. Asset price dynamics with heavy tails and stochastic discounting
7. Spiraling Debt and Its Impact on International Bond Markets
7.1. Dependence in international markets
7.2. Simulation of contagion
8. Conclusions
Acknowledgments
References
Chapter 13 Randomized Mixture Models for Pricing Kernels
1 Introduction
2 Randomised Esscher Martingales
3 Filtered Esscher Martingales
3.1 Filtered Brownian Martingales
3.2 Filtered Gamma Martingales
3.3 Filtered Compound Poisson and Gamma Martingales
4 Filtered Esscher Martingales with Lévy Information
5 Pricing Kernel Models
6 Pricing Kernel Models Driven by Filtered Brownian Martingales
7 Bond Prices Driven by Filtered Gamma Martingales
8 Bond Prices Driven by Filtered Variance-Gamma Martingales
9 Chameleon Random Mixers
10 Model-Generated Yield Curves
11 Pricing of European-Style Bond Options
12 Randomised Heat Kernel Interest Rate Models
12.1 Weighted Heat Kernel Approach
12.2 Quadratic Model Driven by the Ornstein–Uhlenbeck Process
Acknowledgments
Open Access
References
Chapter 14 Stochastic Modelling with Randomized Markov Bridges
1. Introduction
2. Setup and results
2.1. Filtering results
3. Skew-normal randomized diffusion bridge
4. Applications to commodity pricing and securitization of greenhouse gas risk
4.1. Commodity pricing
4.2. GHG risk securitization
Notes
Acknowledgments
Disclosure statement
Funding
References
Chapter 15 Modulated Information Flows in Financial Markets
1. Introduction
2. Modulated Information Processes
2.1. Endogenous jump-diffusion
2.2. Multiple point fields
2.3. Modulation as projection
3. Applications
3.1. Merton-type jump-diffusion models for vanilla options
3.2. Information asymmetry and market competition
3.3. Conclusions
Acknowledgments
References
Chapter 16 Pricing with Variance Gamma Information
1. Introduction
2. Gamma Subordinators
3. Normalized Variance-Gamma Bridge
4. Variance Gamma Information
5. Information Based Pricing
6. Examples
7. Conclusions
Author Contributions
Acknowledgments
Conflicts of Interest
References
Chapter 17 On the Pricing of Storable Commodities
1. Introduction
2. Information-based commodity pricing
3. Properties of the Ornstein–Uhlenbeck process
4. Markov property of market information
5. Commodity pricing formula
6. Pricing commodity derivatives
Acknowledgments
References
Chapter 18 Mathematical Models for Fake News
1. Introduction
2. Fake news and communication theory
3. From communication theory to phenomenology
4. Modelling fake news
5. Estimating the arrival times of fake news
6. Representative voter framework
7. Application to opinion-poll statistics in an election
8. Election-microstructure models
9. Opinion polls in the microstructure model
10. Discussion and outlook
Acknowledgments
References
