Pathwise Estimation and Inference for Diffusion Market Models

Cover
Half Title
Title Page
Copyright Page
Table of Contents
Legend of Notations and Abbreviations
Preface
1: Some background on stochastic analysis
	1.1 Basics of probability theory
		1.1.1 Probability space
		1.1.2 Random variables
		1.1.3 Expectations
		1.1.4 Conditional probability and expectation
		1.1.5 The σ-algebra generated by a random vector
	1.2 Basics of stochastic processes
		1.2.1 Special classes of processes
		1.2.2 Wiener process (Brownian motion)
	1.3 Basics of the stochastic calculus (Ito calculus)
		1.3.1 Ito formula
		1.3.2 Stochastic differential equations (Ito equations)
		1.3.3 Some explicit solutions for Ito equations
		1.3.4 Diffusion Markov processes and related parabolic equations
		1.3.5 Martingale representation theorem
		1.3.6 Change of measure and Girsanov theorem
2: Some background on diffusion market models
	2.1 Continuous time model for stock price
	2.2 Continuous time bond-stock market model
	2.3 Discounted wealth and stock prices
	2.4 Risk-neutral measure
	2.5 Replicating strategies
	2.6 Arbitrage possibilities and the arbitrage-free market
	2.7 The case of a complete market
	2.8 Completeness of the Black–Scholes model
	2.9 Option pricing
		2.9.1 Options and their prices
		2.9.2 Option pricing for a complete market
		2.9.3 Black–Scholes formula
	2.10 Pricing for an incomplete market
	2.11 A multi-stock market model
3: Some special market models
	3.1 Mean-reverting market model
		3.1.1 Basic properties of a mean-reverting model
		3.1.2 Absence of arbitrage and the Novikov condition
		3.1.3 Proofs
	3.2 A market model with delay in coefficients
		3.2.1 Existence, regularity, and non-arbitrage properties
		3.2.2 Time discretization and restrictions on growth
	3.3 A market model with stochastic numéraire
		3.3.1 Model setting
		3.3.2 Replication of claims: Strategies and hedging errors
		3.3.3 On selection of θ and the equivalent martingale measure
		3.3.4 Markov case
		3.3.5 Proofs
	3.4 Bibliographic notes and literature review
4: Pathwise inference for the parameters of market models
	4.1 Estimation of volatility
		4.1.1 Representation theorems for the volatility
		4.1.2 Estimation of discrete time samples
		4.1.3 Reducing the impact of the appreciation rate
		4.1.4 The algorithm
		4.1.5 Some experiments
	4.2 Modeling the impact of the sampling frequency
		4.2.1 Analysis of the model’s parameters
		4.2.2 Monte Carlo simulation of the process with delay
		4.2.3 Examples for dependence of volatility on sampling frequency for historical data
		4.2.4 Matching delay parameters for historical data
	4.3 Inference for diffusion parameters for CIR-type models
		4.3.1 The underlying continuous time model
		4.3.2 A representation theorem for the diffusion coefficient
		4.3.3 Estimation based on the representation theorem
		4.3.4 Numerical experiments
		4.3.5 On the consistency of the method
		4.3.6 Some properties of the estimates
	4.4 Estimation of the appreciation rates
	4.5 Bibliographic notes and literature review
5: Some background on bond pricing
	5.1 Zero-coupon bonds
	5.2 One-factor model
		5.2.1 Dynamics of discounted bond prices
		5.2.2 Dynamics of the bond prices under the original measure
		5.2.3 An example: The Cox–Ross–Ingresoll model
	5.3 Vasicek model
	5.4 An example of a multi-bond market model
6: Implied volatility and other implied market parameters
	6.1 Risk-neutral pricing in a Black–Scoles setting
	6.2 Implied volatility: The case of constant r
	6.3 Correction of the volatility smile for constant r
		6.3.1 Imperfection of the volatility smile for constant r
		6.3.2 A pricing rule correcting the volatility smile
		6.3.3 A class of volatilities in a Markovian setting
	6.4 Unconditionally implied volatility and risk-free rate
		6.4.1 Two calls with different strike prices
	6.5 Bond price inferred from option prices
		6.5.1 Definitions
		6.5.2 Inferred ρ from put and call prices
		6.5.3 Application to a special model
	6.6 A dynamically purified option price process
	6.7 The implied market price of risk with random numéraire
		6.7.1 The risk-free bonds for the market with random numéraire
		6.7.2 The case of a complete market
		6.7.3 The case of an incomplete market
	6.8 Bibliographic notes
7: Inference of implied parameters from option prices
	7.1 Sensitivity analysis of implied volatility estimation
		7.1.1 An under-defined system of nonlinear equations
		7.1.2 Numerical analysis using cross-sectional S&P 500 call options data
		7.1.3 Numerical analysis using longitudinal S&P500 call options data
	7.2 A brief review of evolutionary optimization
		7.2.1 The original differential evolution algorithm
		7.2.2 The Zhang–Sanderson adaptive differential evolution algorithms
	7.3 Inference of implied parameters from over-defined systems
		7.3.1 An over-defined system of nonlinear equations
		7.3.2 Computational implementation
		7.3.3 Construction of the estimation uncertainty bounds for the estimated implied discount rates and implied volatilities
		7.3.4 Numerical experiment with synthetic test data
		7.3.5 Numerical analysis using historical S&P500 call options data
	7.4 Bibliographic notes and literature review
8: Forecast of short rate based on the CIR model
	8.1 The model framework
		8.1.1 General setting
		8.1.2 The CIR model
	8.2 Inference of the implied CIR model parameters based on cross-sectional zero coupon bond prices
	8.3 Numerical framework for the inference
	8.4 Computational implementation
	8.5 Forecast of short rate using the implied CIR model parameters
		8.5.1 Forecast within the multi-curve framework
		8.5.2 Forecast within the single-curve framework
	8.6 Numerical analysis using historical data
		8.6.1 Short rate prediction in the multi-curve framework
		8.6.2 Short rate prediction in the single-curve framework
	8.7 Bibliographic notes and literature review
Bibliography
Index