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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.

- Some background on the stochastic analysis
- Some background on the diffusion market models
- Some special market models
- Pathwise inference for parameters of market models
- Some background on bonds pricing
- Implied volatility and other implied market parameters
- Inference of implied parameters from option prices
- Forecast of short rate based on the CIR model
- Forecast of short rate using the implied CIR model parameters

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

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

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

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

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

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

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

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

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