Contingent Claims in Incomplete Markets

1. Brief review of contingent claims and of the notion of market (in)completeness. The hypotheses underlying to the Black-Merton-Scholes model and the associated pricing kernel. 2. The volatility smile model-free volatility measures (RV and VIX) and option-implied distributions. We will revisit the Black-Scholes implied volatility smile and some of its potential explanations. We will then consider different model-free volatility measures and see how variance swaps and/or option data can be used to extract information about the risk-adjusted distribution of returns. Finally we will introduce the notion of variance risk premia a concept that we will further revisit when we study dynamic volatility models.

Selected papers: Andersen Bollerslev Christoffersen and Diebold (2006); The CBOE Volatility Index - VIX (2009); Demeterfi Derman Kamal and Zou (1999); Martin (2013); Bollerslev and Todorov (2011) . 3.Stochastic volatility models and the pricing kernel. Conditionally normal GARCH models and the pricing kernel in discrete time. Component models.

We will introduce stochastic volatility and GARCH models highlighting the pricing kernel used or implied by the different models. We will review some stylized facts on volatility and see how the different models accommodate these (or not). In particular we will discuss volatility models in which volatility process is separat