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| Preface | |
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| Acknowledgements | |
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| Elementary financial calculus | |
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| Motivating examples | |
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| Cashflows, interest rates, prices and returns | |
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| Bonds and the term structure of interest rates | |
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| Asset returns | |
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| Some basic models for asset prices | |
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| Elementary statistical analysis of returns | |
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| Measuring location | |
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| Measuring dispersion and risk | |
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| Measuring skewness and kurtosis | |
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| Estimation of the distribution | |
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| Testing for normality | |
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| Financial instruments | |
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| Contingent claims | |
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| Spot contracts and forwards | |
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| Futures contracts | |
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| Options | |
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| Barrier options | |
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| Financial engineering | |
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| A primer on option pricing | |
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| The no-arbitrage principle | |
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| Risk-neutral evaluation | |
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| Hedging and replication | |
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| Nonexistence of a risk-neutral measure | |
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| The Black-Scholes pricing formula | |
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| The Greeks | |
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| Calibration, implied volatility and the smile | |
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| Option prices and the risk-neutral density | |
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| Notes and further reading | |
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| References | |
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| Arbitrage theory for the one-period model | |
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| Definitions and preliminaries | |
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| Linear pricing measures | |
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| More on arbitrage | |
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| Separation theorems in R<sup>n</sup> | |
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| No-arbitrage and martingale measures | |
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| Arbitrage-free pricing of contingent claims | |
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| Construction of martingale measures: general case | |
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| Complete financial markets | |
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| Notes and further reading | |
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| References | |
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| Financial models in discrete time | |
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| Adapted stochastic processes in discrete time | |
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| Martingales and martingale differences | |
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| The martingale transformation | |
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| Stopping times, optional sampling and a maximal inequality | |
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| Extensions to R<sup>d</sup> | |
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| Stationarity | |
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| Weak and strict stationarity | |
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| Linear processes and ARMA models | |
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| Linear processes and the lag operator | |
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| Inversion | |
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| ARQ(p) and AR(∞) processes | |
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| ARMA processes | |
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| The frequency domain | |
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| The spectrum | |
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| The periodogram | |
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| Estimation of ARMA processes | |
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| (G)ARCH models | |
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| Long-memory series | |
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| Fractional differences | |
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| Fractionally integrated processes | |
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| Notes and further reading | |
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| References | |
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| Arbitrage theory for the multiperiod model | |
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| Definitions and preliminaries | |
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| Self-financing trading strategies | |
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| No-arbitrage and martingale measures | |
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| European claims on arbitrage-free markets | |
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| The martingale representation theorem in discrete time | |
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| The Cox-Ross-Rubinstein binomial model | |
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| The Black-Scholes formula | |
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| American options and contingent claims | |
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| Arbitrage-free pricing and the optimal exercise strategy | |
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| Pricing american options using binomial trees | |
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| Notes and further reading | |
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| References | |
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| Brownian motion and related processes in continuous time | |
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| Preliminaries | |
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| Brownian motion | |
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| Definition and basic properties | |
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| Brownian motion and the central limit theorem | |
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| Path properties | |
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| Brownian motion in higher dimensions | |
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| Continuity and differentiability | |
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| Self-similarity and fractional Brownian motion | |
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| Counting processes | |
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| The poisson process | |
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| The compound poisson process | |
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| L�vy processes | |
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| Notes and further reading | |
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| References | |
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| It� Calculus | |
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| Total and quadratic variation | |
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| Stochastic Stieltjes integration | |
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| The It� integral | |
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| Quadratic covariation | |
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| It�'s formula | |
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| It� processes | |
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| Diffusion processes and ergodicity | |
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| Numerical approximations and statistical estimation | |
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| Notes and further reading | |
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| References | |
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| The Black-Scholes model | |
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| The model and first properties | |
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| Girsanov's theorem | |
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| Equivalent martingale measure | |
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| Arbitrage-free pricing and hedging claims | |
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| The delta hedge | |
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| Time-dependent volatility | |
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| The generalized Black-Scholes model | |
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| Notes and further reading | |
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| References | |
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| Limit theory for discrete-time processes | |
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| Limit theorems for correlated time series | |
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| A regression model for financial time series | |
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| Least squares estimation | |
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| Limit theorems for martingale difference | |
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| Asymptotics | |
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| Density estimation and nonparametric regression | |
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| Multivariate density estimation | |
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| Nonparametric regression | |
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| The CLT for linear processes | |
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| Mixing processes | |
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| Mixing coefficients | |
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| Inequalities | |
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| Limit theorems for mixing processes | |
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| Notes and further reading | |
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| References | |
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| Special topics | |
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| Copulas - and the 2008 financial crisis | |
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| Copulas | |
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| The financial crisis | |
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| Models for credit defaults and CDOs | |
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| Local Linear nonparametric regression | |
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| Applications in finance: estimation of martingale measures and Ito diffusions | |
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| Method and asymptotics | |
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| Change-point detection and monitoring | |
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| Offline detection | |
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| Online detection | |
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| Unit roots and random walk | |
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| The OLS estimator in the stationary AR(1) model | |
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| Nonparametric definitions for the degree of integration | |
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| The Dickey-Fuller test | |
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| Detecting unit roots and stationarity | |
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| Notes and further reading | |
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| References | |
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| Appendix A | |
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| (Stochastic) Landau symbols | |
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| Bochner's lemma | |
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| Conditional expectation | |
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| Inequalities | |
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| Random series | |
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| Local martingales in discrete time | |
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| Weak convergence and central limit theorems | |
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| Convergence in distribution | |
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| Weak convergence | |
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| Prohorov's theorem | |
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| Sufficient criteria | |
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| More on Skorohod spaces | |
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| Central limit theorems for martingale differences | |
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| Functional central limit theorems | |
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| Strong approximations | |
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| References | |
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| Index | |