Liquidity Risk, Volatility And Financial Bubbles
The goal of this work is to study and characterize the hedging and pricing of contingent claims and develop a theory of financial bubble origination and termination from a liquidity risk and trading impacts perspective. Our approach is to combine both notions of liquidity risk by hypothesizing the existence of a linear supply curve that evolves randomly in time and by studying the impact of trades on prices. This leads to a simple characterization of self-financing trading strategies in which the profit is directly affected by the level of liquidity. The main goal of Chapter 3 is to study the effect of liquidity risk on the replicating costs of contingent claims. The use of variance swaps will prove to be helpful and mathematically tractable in this context. In Chapter 4, we show that the replicating cost obtained can be represented as the viscosity solution of an associated quasilinear partial differential equation. In Chapter 5, we build on the work of Chapter 3 and study the case of American options. We obtain a general result concerning reflected forward-backward stochastic differential equations, and apply these results to the problem of hedging American options. In Chapter 6, we study the relation between bubbles and liquidity risk. In particular, we use the model presented in Chapter 3 to analyze the formation and the bursting of financial bubbles from a price impact and liquidity risk perspective. The approach differs from the existing theory of bubble birth as it consists in fixing a fundamental process, thereby fixing the equivalent martingale measure used for valuation, and considering conditions under which the mar- ket price gives rise to a bubble. We show how the life of a bubble is determined by quantities such as the trading volume, the resiliency of the order book, the level of liquidity and the speed of price impact decay. We give sufficient conditions for the no arbitrage condition to hold at the time of bubble creation. In the last part of the chapter, we study the implication of positive probability of future bubbles on option prices and show that information about the likelihood of this future event is contained in option prices before the event happens. In Chapter 7, we use the notion of viscosity solutions of integral-partial differential equations (IPDE) studied in previous chapters for the pricing of American options in the stochastic volatility model of Barndorff-Nielsen and Shephard (2001).
dissertation or thesis