Starting from the most famous Black-Scholes model for the underlying asset
price, there has been a large variety of extensions made in recent decades.
One main strand is about the models which allow a jump component in the
asset price. The first topic of this thesis is about the study of jump risk
premium by an equilibrium approach. Different from others, this work provides
a more general result by modeling the underlying asset price as the ordinary
exponential of a L?vy process. For any given asset price process, the equity
premium, pricing kernel and an equilibrium option pricing formula can be
derived. Moreover, some empirical evidence such as the negative variance risk
premium, implied volatility smirk, and negative skewness risk premium can
be well explained by using the relation between the physical and risk-neutral
distributions for the jump component.
Another strand of the extensions of the Black-Scholes model is about the
models which can incorporate stochastic volatility in the asset price. The second
topic of this thesis is about the replication of exponential variance, where
the key risks are the ones induced by the stochastic volatility and moreover it
can be correlated with the returns of the asset, referred to as leverage effect.
A time-changed L?vy process is used to incorporate jumps, stochastic volatility
and leverage effect all together. The exponential variance can be robustly
replicated by European portfolios, without any specification of a model for the
stochastic volatility.
Beyond the above asset pricing and hedging, portfolio optimization is also
discussed. Based on the Merton (1969, 1971)'s reduced portfolio optimization
and the delta hedging problem, a portfolio of an option, the underlying stock
and a risk-free bond can be optimized in discrete time and its optimal solution
can be shown to be a mixture of the Merton's result and the delta hedging
strategy. The main approach is the elasticity approach, which has initially
been proposed in continuous time.
In addition to the above optimization problem in discrete time, the same
topic but in a continuous-time regime-switching market is also presented. The
use of regime-switching makes our market incomplete, and makes it difficult to
use some approaches which are applicable in complete market. To overcome
this challenge, two methods are provided. The first method is that we simply
do not price the regime-switching risk when obtaining the risk-neutral probability.
Then by the idea of elasticity, the utility maximization problem can be
formulated as a stochastic control problem with only a single control variable,
and explicit solutions can be obtained. The second method is to introduce
a functional operator to general value functions of stochastic control problem
in such a way that the optimal value function in our setting can be given by
the limit of a sequence of value functions defined by iterating the operator.
Hence the original problem can be deduced to an auxiliary optimization problem,
which can be solved as if we were in a single-regime market, which is
complete. / published_or_final_version / Statistics and Actuarial Science / Doctoral / Doctor of Philosophy
| Identifer | oai:union.ndltd.org:HKU/oai:hub.hku.hk:10722/167210 |
| Date | January 2012 |
| Creators | Fu, Jun, 付君 |
| Contributors | Yang, H |
| Publisher | The University of Hong Kong (Pokfulam, Hong Kong) |
| Source Sets | Hong Kong University Theses |
| Language | English |
| Detected Language | English |
| Type | PG_Thesis |
| Source | http://hub.hku.hk/bib/B48199345 |
| Rights | The author retains all proprietary rights, (such as patent rights) and the right to use in future works., Creative Commons: Attribution 3.0 Hong Kong License |
| Relation | HKU Theses Online (HKUTO) |
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