Methods And Applications Of Time Deformation In Financial Market

Price and time both are the fundamentals of financial problems. But current researches only concern the first variable-price and neglect the second one-time. Actually time is important. It just like coordinate system. Usually Descartes rectangular coordinate system is used. But sometime other coordinate system is preferred. Coordinate transformations make things more simple. For example, a lot of integrals are very sophisticated in rectangular coordinate system while there are very simple in polar one. In similar reasons that we can use other time (named economic time) instead of the usual time (named calendar time). The transformation from calendar time to economic time is so called time deformation. Just like giving curves equations after choosing a coordinate system in mathematics, we should first choose a proper time to the prices in finances.Actually, methods of time deformations already exist in some economic problems. There are economic times in practice. For example, the monthly or quarterly GDPs. Because every month or quarter have different numbers of days, the monthly or quarterly time series already involve time deformation. Another example, many exchanges close on holidays. So it's natural to delete holidays in the time series of daily prices. So comes the transaction time. It is also an economic time. The last example, in auto insurance, distance is used instead of ages.The other defection in the treatment of time is that history is neglected. Because of the assumption of independent increments, there is little memory or no memory in prices. So history has no or little impact on future. It can be neglected. But the existence of long memories in prices makes history can't be neglected.Because reasons above, we must pay great attention to the factor of time in asset pricing or risk analysis. This can be done in two aspects:(1) Time Scale: It is the problem of choosing proper time. Usually when time is mentioned, what we really mean is calendar time. But in nature time is just an increasing and irreversible process. Any such process can be called time. Time is chosen as a means to describe the speed and direction of single systems. Then how to choose a new time, what time is good and what is the statistical relation between different times?(2) Time Range: It is the persistence of impacts. That is to say, the extent of memories. Intuitively long memory means what happens today always influences what will happen in future. And the impact decays so slowly that history can't be neglected. The existence of long memory let us face the correlation of events. There are two puzzles. First, why prices can't act like the EMH and what causes the long memory property? Second, how to price assets and manage risk under long memory conditions? Is VaR still a proper model? If not, how to revise?It should be noted that there are two aspects in the definition of time deformation. In the narrow sense, it refers to time scale. While in the broad sense, both time scale and time range belong to time deformation. Because each problem is studied under a specific time scale.The whole paper focuses on the two sections above. Chapter3 and Chapter 4 concerns the sale of time and statistical properties under time deformation, while Chapter 5 and Chapter 6 cares the range of time and the problems such as why prices have long memory and how to manage risk then. Time deformation method is the basis of the whole thesis. From the introduction of dependent-subordination, to asset pricing under time deformation and the impact of time deformation on memories, everything supports that time is an important factor in asset pricing and risk analysis. The result is that risk or memory is a relative concept. They rely on the specific time scale. Time deformation can change both risk and memory. In details:Chapter One describes the backgrounds and gives a simple introduction of the contents, structure and innovations of this thesis.Chapter Two summarizes researches on time scale and time range, points out the shortcomings and give the main objects of the thesis.Chapter Three gives the definition of dependent-subordination, statistical properties of dependent-subordination and the impact of time deformation on kurtosis and skewness. Results support the introduction of dependence. Chapter Four focuses on the assets pricing under time deformation. There are 3 parts. Part one gives results of assets pricing in economic time, which assemble CAPM or APT. Empirical tests find when there is no riskless asset in the market, economic asset pricing models have better fitness than calendar ones. Part two and Part three are applications of time deformation in insurance. Part two gives life actuary models under time deformation and an empirical example. Part three gives a new method of designing premiums in automobile insurance with time deformation.Chapter Five focuses on time deformation and memory. It contains 3 parts. Part one gives a simple introduction of WICK-ITO fractional integrals. Part two concerns the portfolios of processes with different memories and finds that diversification isn't a way to distinguish long memories. And also gives the first way of changing short memory process into long memory one. Part three focuses on the impact of time deformation on memories and gives the second reason. Finally, Part four gives empirical tests and finds the introduction of economic time can increase the memories of processes.Chapter Six focuses on the management of risk under long memories and a new VaR model is proposed. It contains 3 parts. Part one gives a simple summarization of VaRs. Part two proposes the new model named HVaR to long memory risk management. The formula under FBMs, the properties and also the estimates of the HVaR are considered. Finally, Part three gives empirical research and finds HVaRs can do a better job than VaRs.At last, Chapter Seven concludes the results and gives further problems.Innovations:The primary innovations of the dissertation are the systematic research of time in asset pricing and risk manage. In details:(1) A concept of'dependent-subordination'is proposed. The current research is based on the definition of subordination and it requires independence. But there are correlations everywhere. For example, prices and volumes. So comes the dependent-subordination. The corresponding statistical properties are considered, especially about diffusions. It's very necessary to introduce correlations in subordination, for example to skewness and kurtosis.(2) Applied time deformation to three kinds of asset pricing. They are economic time pricing, life insurance and automobile insurance problems. The current research never concerns stochastic time deformation in economic time pricing and insurance. So we give these applications. Under the condition of dependent-subordination, we get the results of economic time pricing including multi- factor one. With time deformation, new models of pricing pure premiums and annuities in life insurance and new models of BMS in auto insurance are given.(3) The relations among processes with different memories are studied and method of time scale is connected with that of time range. The current research only concerns on a single process and neglects the relations of different processes, and never uses time deformation in long memory processes. And it usually uses discrete times series to describe long memory processes. So we based on continuous processes and consider the portfolio of different memories and time deformation of long memories.(4) A new model named HVaR is proposed to managing risk. The current VaR (the standard VaR) is based on unconditional distributions and neglects the influence of history. So it isn't applicable to long memory processes. Using history information we give the HVaRs. It is based on conditional distributions. Thus the impact of past is considered. So it is useful for long memory processes.