Pathwise Estimation and Inference for Diffusion Market Models: 1st Edition (Hardback) book cover

Pathwise Estimation and Inference for Diffusion Market Models

1st Edition

By Nikolai Dokuchaev, Lin Yee Hin

Chapman and Hall/CRC

224 pages | 20 B/W Illus.

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pub: 2019-03-14
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Description

Pathwise estimation and inference for diffusion market models discusses contemporary techniques for inferring, from options and bond prices, the market participants' aggregate view on important financial parameters such as implied volatility, discount rate, future interest rate, and their uncertainty thereof. The focus is on the pathwise inference methods that are applicable to a sole path of the observed prices and do not require the observation of an ensemble of such paths.

This book is pitched at the level of senior undergraduate students undertaking research at honors year, and postgraduate candidates undertaking Master’s or PhD degree by research. From a research perspective, this book reaches out to academic researchers from backgrounds as diverse as mathematics and probability, econometrics and statistics, and computational mathematics and optimization whose interest lie in analysis and modelling of financial market data from a multi-disciplinary approach. Additionally, this book is also aimed at financial market practitioners participating in capital market facing businesses who seek to keep abreast with and draw inspiration from novel approaches in market data analysis.

The first two chapters of the book contains introductory material on stochastic analysis and the classical diffusion stock market models. The remaining chapters discuss more special stock and bond market models and special methods of pathwise inference for market parameter for different models. The final chapter describes applications of numerical methods of inference of bond market parameters to forecasting of short rate.

Nikolai Dokuchaev is an associate professor in Mathematics and Statistics at Curtin University. His research interests include mathematical and statistical finance, stochastic analysis, PDEs, control, and signal processing.

Lin Yee Hin is a practitioner in the capital market facing industry. His research interests include econometrics, non-parametric regression, and scientific computing.

Table of Contents

  1. Some background on the stochastic analysis
  2. Basics of probability theory

    Probability space

    Random variables

    Expectations

    Conditional probability and expectation

    The _-algebra generated by a random vector

    Basics of stochastic processes

    Special classes of processes

    Wiener process (Brownian motion)

    Basics of the stochastic calculus (Ito calculus)

    Ito formula

    Stochastic differential equations (Ito equations)

    Some explicit solutions for Ito equations

    Diffusion Markov processes and related parabolic equations

    Martingale Representation Theorem

    Change of measure and Girsanov Theorem

  3. Some background on the diffusion market models
  4. Continuous time model for stock price

    Continuous time bond-stock market model

    The discounted wealth and stock prices

    Risk-neutral measure

    Replicating strategies

    Arbitrage possibilities and arbitrage-free market

    A case of complete market

    Completeness of the Black-Scholes model

    Option pricing

    Options and their prices

    Option pricing for complete market

    Black-Scholes formula

    Pricing for an incomplete market

    A multi-stock market model

  5. Some special market models
  6. Mean-reverting market model

    Basic properties of mean-reverting model

    Absence of arbitrage and Novikov condition

    Proofs

    A market model with delay in coefficients

    Existence, regularity, and non-arbitrage properties

    Time discretisation and restrictions on the growth

    A market model with stochastic numéraire

    Model setting

    Replication of claims: strategies and hedging errors

    On selection of _ and the equivalent martingale measure

    Markov case

    Proofs

    Bibliographic notes and literature review

  7. Pathwise inference for parameters of market models
  8. Estimation of volatility

    Representation theorems for the volatility

    Estimation of discrete time samples

    Reducing the impact of the appreciation rate

    The algorithm

    Some experiments

    Modelling the impact of the sampling frequency

    Analysis of the model’s parameters

    Monte-Carlo simulation of the process with delay

    Examples for dependence of volatility on sampling frequency for historical data

    Matching delay parameters for historical data

    Inference for diffusion parameters for CIR type models

    The underlying continuous time model

    A representation theorem for the diffusion coefficient

    Estimation based on the representation theorem

    Numerical experiments

    On the consistency of the method

    Some properties of the estimates

    Estimation of the appreciation rates

    Bibliographic notes and literature review

  9. Some background on bonds pricing
  10. Zero-coupon bonds

    One-factor model

    Dynamics of discounted bond prices

    Dynamics of the bond prices under the original measure

    An example: the Cox-Ross-Ingresoll model

    Vasicek Model

    An example of a multi-bond market model

  11. Implied volatility and other implied market parameters
  12. Risk neutral pricing in Black-Scoles setting

    Implied volatility: the case of constant r

    Correction of the volatility smile for constant r

    Imperfection of the volatility smile for constant r

    A pricing rule correcting the volatility smile

    A class of volatilities in Markovian setting

    Unconditionally implied volatility and risk free rate

    Two calls with different strike prices

    Bond price inferred from option prices

    Definitions

    Inferred _ from put and call prices

    Application to a special model

    A dynamically purified option price process

    The implied market price of risk with random numéraire

    The risk-free bonds for the market with random numéraire

    The case of complete market

    The case of incomplete market

    Bibliographic notes

  13. Inference of implied parameters from option prices
  14. Sensitivity analysis of implied volatility estimation with respect to discount rate uncertainty

    An under-defined system of nonlinear equations

    Numerical analysis using cross-sectional S&P call options data

    Numerical analysis using longitudinal S&P call options data

    A brief review of evolutionary optimization

    The original differential evolution algorithm

    The Zhang-Sanderson adaptive differential evolution algorithms

    Inference of implied parameters from overdefined systems

    An over-defined system of nonlinear equations

    Computational implementation

    Construction of the estimation uncertainty bounds for the estimated implied discount rates and implied volatilities

    Numerical experiment with synthetic test data

    Numerical analysis using historical S&P call options data

    Bibliographic notes and literature review

  15. Forecast of short rate based on the CIR model
  16. The model framework

    General setting

    The CIR model

    Inference of the implied CIR model parameters based on cross sectional zero coupon bond prices

    Numerical framework for the inference

    Computational implementation

  17. Forecast of short rate using the implied CIR model parameters

Forecast within the multi-curve framework

Forecast within the single-curve framework

Numerical analysis using the historical US STRIPS data and the effective Federal Funds rate

Short rate prediction in the multi-curve framework

Short rate prediction in the single-curve framework

Bibliographic notes and literature review

Legend of Notations and Abbreviations

About the Authors

Nikolai Dokuchaev is an associate professor in Mathematics and Statistics at Curtin University. His research interests include mathematical and statistical finance, stochastic analysis, PDEs, control, and signal processing.

Lin Yee Hin is a practitioner in the capital market facing industry. His research interests include econometrics, non-parametric regression, and scientific computing.

Subject Categories

BISAC Subject Codes/Headings:
MAT000000
MATHEMATICS / General
MAT029000
MATHEMATICS / Probability & Statistics / General