http://swrc.ontoware.org/ontology#Thesis
Statistical Models in Finance: Applications to Price Change and Credit Risk
en
高橋 久尚
タカハシ ヒサナオ
TAKAHASHI Hisanao
総研大甲第647号
The aim of this study is to make simple statistical models to analyze the risks in finance. Comparing with real data of the exchange rate between the US dollar and the Japanese yen, we make three simple models to explain the distribution of price change and the interaction of traders.<br /> In the first model, we consider change (difference, returns) in stock index prices and exchange rates for currencies. These are said, from empirical studies, to be distributed by a stable law with a characteristic exponent α< 2 for short sampling intervals and by a Gaussian distribution for long sampling intervals. To explain this phenomenon, we introduce an Ehrenfest model with large jumps (ELJ), which explains the empirical density function of price changes for short time intervals as well as for long time intervals. In chapter 3, we discuss mathematical details and related problems of ELJ.<br /> The second model is a majority orienting model which we introduce to show the majority orienting behavior of the traders in a market. It seems that the interaction among the traders must exist not only at the time of crashes and bubbles but also at the time of usual trading. And the interaction makes the time series of the market prices such as an exchange rate between the US dollar and the Japanese yen a typical trajectory.<br /> The third model is the majority orienting model with feed back process which we introduce to understand the oscillation of the market price. We study a simplified market in which the dealers' behavior changes by the influence of the price. We show that in such a market, the price oscillates perpetually by applying the van del Pol equation which is obtained from a deterministic approximation of our model.<br /> The advantage of these models is it is easy to understand the connection with real market such that there are N agents, each of whom is in one of two possible microeconomic states. To explain volatihty clustering, trend of market and non-symmetrical trading is left for our further study. To combine the above models may be also an interesting next problem.<br /> We also study a default probability of companies by applying the Iogit model to the data from the database as a starting point of making stochastic model on the default of a company. It is important in credit risk management to determine the probability of bankruptcy. Few reliable analyses of bankruptcy have been developed for small and medium-sized enterprises because of the delay in developing of databases to capture credit risks for these enterprises. Recently, a large-scale database for estimating credit risks for such enterprises has become available as "Credit Risk Database". We use the Wald statistic to evaluate the significance of the model's parameters. We discuss the differences in explanatory factors of credit risk depending on the enterprise scale. In general, financial data for small and medium-sized companies contain many missing data, and many statistical difficulties are caused by it. To avoid these difficulties, 0-1 dummy variables were incorporated into the Iogit model. This method can be also interpreted that the condition of a certain company's missing data in its financial indices is valuable for predicting a company's default. To make a simple stochastic model for this problem is our next problem.