Roch, Alexandre. 2009. Liquidity Risk, Volatility and Financial Bubbles. Doctoral
Dissertation, Cornell University.
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 quasi-linear 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).
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