1.Some background on the stochastic analysis
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
1.Some background on the diffusion market models
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
1.Some special market models
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
1.Pathwise inference for parameters of market models
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 models 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
1.Some background on bonds pricing
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
1.Implied volatility and other implied market parameters
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
1.Inference of implied parameters from option prices
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
1.Forecast of short rate based on the CIR model
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
1.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
