Valuation and Hedging of Contracts with Funding Costs and Collateralization

The research presented in this work is motivated by recent papers by Brigo et al. (2011), Burgard and Kjaer (2009), Cr\'epey (2012), Fujii and Takahashi (2010), Piterbarg (2010) and Pallavicini et al. (2012). Our goal is to provide a sound theoretical underpinning for some results presented in these papers by developing a unified framework for the non-linear approach to hedging and pricing of OTC financial contracts. We introduce a systematic approach to valuation and hedging in nonlinear markets, that is, in markets where cash flows of the financial contracts may depend on the hedging strategies. Our systematic approach allows to identify primary sources of and quantify various adjustment to valuation and hedging, primarily the funding and liquidity adjustment and credit risk adjustment. We propose a way to define no-arbitrage in such nonlinear markets, and we provide conditions that imply absence of arbitrage in some specific market trading models. Accordingly, we formulate a concept of no-arbitrage price, and we provide relevant (non-linear) BSDE that produces the no-arbitrage price in case when the contract's cash flows can be replicated

Dynamic Limit Growth Indices in Discrete Time

We propose a new class of mappings, called Dynamic Limit Growth Indices, that are designed to measure the long-run performance of a financial portfolio in discrete time setup. We study various important properties for this new class of measures, and in particular, we provide necessary and sufficient condition for a Dynamic Limit Growth Index to be a dynamic assessment index. We also establish their connection with classical dynamic acceptability indices, and we show how to construct examples of Dynamic Limit Growth Indices using dynamic risk measures and dynamic certainty equivalents. Finally, we propose a new definition of time consistency, suitable for these indices, and we study time consistency for the most notable representative of this class -- the dynamic analog of risk sensitive criterion