Stochastic Volatility in Financial Markets

1 - Introduction.- 1.1 Background and aims of the monograph.- 1.2 Empirical models in discrete time.- 1.3 Theoretical issues.- 1.4 Statistical inference.- 1.5 Plan.- 2 - Continuous time behavior of non linear ARCH models.- 2.1 Introduction.- 2.2 Approximation results for a general class of non linear ARCH models.- 2.3 Interpretation of the moment conditions.- 2.4 Effectiveness of ARCH as diffusion approximations of theoretical models.- 2.5 Limiting behavior of the error process.- 2.6 Continuous time behavior of the volatility switching models.- Appendix A: proofs on convergence issues.- Appendix B: proofs on distributional issues.- Appendix C.- 3 - Continuous time stochastic volatility option pricing: foundational issues.- 3.1 Introduction.- 3.2 The reference model.- 3.3 Applications to stochastic volatility.- 3.4 On mean self-financing strategies and the minimal martingale measure.- Appendix: relative entropy and the minimal martingale measure.- 4 - Models of the term structure with stochastic volatility.- 4.1 Introduction.- 4.2 From the one factor model to the modelling of conditional heteroskedasticity.- 4.3 Searching for affinity.- 4.4 Early equilibrium-based models.- 4.5 A class of equilibrium models of the term structure with stochastic volatility.- 4.6 Concluding remarks: taking account of nonlinearities.- 5 - Formulating, solving and estimating models of the term structure using ARCH models as diffusion approximations.- 5.1 Introduction.- 5.2 Specification of the theoretical models.- 5.3 Econometric strategy.- 5.4 The pure numerical solution of the theoretical models.- 5.5 An illustrative example.- Appendix: a solution method based on the approach of iterated approximations.- References.