Asset pricing, hedging and portfolio optimization

Abstract

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.