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Postgraduate Thesis: Asset pricing, hedging and portfolio optimization
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TitleAsset pricing, hedging and portfolio optimization
 
AuthorsFu, Jun
付君
 
Issue Date2012
 
PublisherThe University of Hong Kong (Pokfulam, Hong Kong)
 
AbstractStarting 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.
 
AdvisorsYang, H
 
DegreeDoctor of Philosophy
 
SubjectCapital assets pricing model.
Hedging (Finance) - Mathematical models.
Portfolio management - Mathematical models.
 
Dept/ProgramStatistics and Actuarial Science
 
DC FieldValue
dc.contributor.advisorYang, H
 
dc.contributor.authorFu, Jun
 
dc.contributor.author付君
 
dc.date.hkucongregation2012
 
dc.date.issued2012
 
dc.description.abstractStarting 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.
 
dc.description.naturepublished_or_final_version
 
dc.description.thesisdisciplineStatistics and Actuarial Science
 
dc.description.thesisleveldoctoral
 
dc.description.thesisnameDoctor of Philosophy
 
dc.identifier.hkulb4819934
 
dc.languageeng
 
dc.publisherThe University of Hong Kong (Pokfulam, Hong Kong)
 
dc.relation.ispartofHKU Theses Online (HKUTO)
 
dc.rightsThe author retains all proprietary rights, (such as patent rights) and the right to use in future works.
 
dc.rightsCreative Commons: Attribution 3.0 Hong Kong License
 
dc.source.urihttp://hub.hku.hk/bib/B48199345
 
dc.subject.lcshCapital assets pricing model.
 
dc.subject.lcshHedging (Finance) - Mathematical models.
 
dc.subject.lcshPortfolio management - Mathematical models.
 
dc.titleAsset pricing, hedging and portfolio optimization
 
dc.typePG_Thesis
 
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<item><contributor.advisor>Yang, H</contributor.advisor>
<contributor.author>Fu, Jun</contributor.author>
<contributor.author>&#20184;&#21531;</contributor.author>
<date.issued>2012</date.issued>
<description.abstract>&#65279;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)&apos;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&apos;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.</description.abstract>
<language>eng</language>
<publisher>The University of Hong Kong (Pokfulam, Hong Kong)</publisher>
<relation.ispartof>HKU Theses Online (HKUTO)</relation.ispartof>
<rights>The author retains all proprietary rights, (such as patent rights) and the right to use in future works.</rights>
<rights>Creative Commons: Attribution 3.0 Hong Kong License</rights>
<source.uri>http://hub.hku.hk/bib/B48199345</source.uri>
<subject.lcsh>Capital assets pricing model.</subject.lcsh>
<subject.lcsh>Hedging (Finance) - Mathematical models.</subject.lcsh>
<subject.lcsh>Portfolio management - Mathematical models.</subject.lcsh>
<title>Asset pricing, hedging and portfolio optimization</title>
<type>PG_Thesis</type>
<identifier.hkul>b4819934</identifier.hkul>
<description.thesisname>Doctor of Philosophy</description.thesisname>
<description.thesislevel>doctoral</description.thesislevel>
<description.thesisdiscipline>Statistics and Actuarial Science</description.thesisdiscipline>
<description.nature>published_or_final_version</description.nature>
<date.hkucongregation>2012</date.hkucongregation>
<bitstream.url>http://hub.hku.hk/bitstream/10722/167210/1/FullText.pdf</bitstream.url>
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