Arbitrage-free valuation in nonlinear financial models

Abstract

Abstract In the decade or so following the sub-prime crisis, there has been a significant shift in the modus operandi of derivatives markets. Nowadays, the traditional assumption that firms can borrow and lend at a unique risk-free rate can no longer be taken for granted, collateralisation has become widespread practice and exogenous risks have become increasingly important to price into contracts.

In this new context, it has become apparent that the classical arbitrage-free valuation paradigm is no longer appropriate because it relies fundamentally on linear structures of an underlying market model. However, in a post-crisis world, derivatives desks must simultaneously manage a multitude of interdependent factors that can feedback on themselves. Therefore, the practicalities of risk management are nonlinear. Although various ad hoc approaches have been developed by some practitioners, only a few attempts to develop a consistent nonlinear arbitrage-free valuation theory have been undertaken (e.g. Bielecki et al. (2018) and Bielecki and Rutkowski (2015)). Moreover, to date, there has been no general attempt to consistently deal with post-crisis realities for contracts with stopping features (i.e. American- and game-style contracts).

In this thesis, we build on the nonlinear arbitrage-free valuation theory developed by Bielecki et al. (2018} and Bielecki and Rutkowski (2015) and extend it to include contracts with stopping features. In the development of this theory, (doubly) reflected backward stochastic differential equations, nonlinear optimal stopping problems and nonlinear Dynkin games play a fundamental role.