泊松分布在金融领域的应用.docx

Random (Poisson distribution) in the field of financial applications【Abstract】 mathematical finance as a subject. Using a great deal of teaching theory and method study and solve major theories in financial issues, practical problems, and some, such as the pricing of financial innovation. Due to financial problems the complexity of the mathematical knowledge, in addition to the base of knowledge, there are plenty of theories and methods of modern mathematics. In this article we introduce the volume fluctuations in stock price model. Application of Poisson process theory describes the volatility of stock prices, and based on option pricing theory, European call option pricing formula is derived. In the course of financial investment, investors typically shy away from risks, and control the risks in the first place, so we further risk aversion in the market of European call options price range. In order to give investors a more specific reference.【Key words】 stochastic process of compound Poisson process shares traded options pricingAlong with rapid economic development, a variety of financial tools continue to produce. The correct valuation of financial instruments is a necessary condition for effective management of risk, we used the prices of securities described in geometric Brownian motion process is continuous. With fair prices and financial instruments is that they are reasonable and the key. Mathematical finance is 20 centuries later developed a new cross discipline. It is observed with a unique way to meet financial problems, which combine mathematical tools and financial problems. Provide a basis for creative research, solving financial problems and guidance. Through mathematics built die, and theory analysis, and theory is derived, and numerical calculation, quantitative analysis, research and analysis financial trading in the of various problem, to precise to description out financial trading process in the of some behavior and may of results, while research its corresponding of forecast theory, reached avoided financial risk, and achieved financial trading returns maximize of purpose, to makes about financial trading of decision more simple and accurate. Because of financial phenomena studied in mathematical finance strong uncertainty, stochastic process theory as an important branch of probability theory, and are widely used in the financial research. Stochastic process theory include: theory of probability spaces. Poisson process, the updating process, discrete Markov chains and continuous parameters of the Markov chain, theBrown campaign, martingales theory and stochastic integration, stochastic differential equations, and so on. In recent decades, theory and applications of stochastic processes has been developing rapidly. Physics, automation, communication sciences, economics and Management Sciences and many other fields are active figure of the theory of stochastic processes.This stochastic process theory of option pricing using Poisson process theory to the study of regularity of stock price fluctuation in the stock market, consider the impact of transactions on stock prices, stock price process model is constructed. And gives the option of avoiding risks in the investment process.And thePoisson process conceptsDefinitions 1. 1 random processNt,T0 is called the counting process, if theIn time intervals (0,t occurs in a certain event ( due to a point on the timeline of events, so people called the event ) number. Therefore, a counting process must meet:(1) Nt Take non-negative integer values;(2)If st, thenNsNt (3) Nt InR+=0，）There are continuous and piecewise fetch constants,(4)For st, Ns,t=NS-NtIs equal to the time (s,t the number of events occurring in,Said the counting processNt,T0 has independent increments. If its in any finite number of disjoint events that occur in the time interval of a few independent of each other, said the counting processNt,T0 with stationary increments, if at any time the probability distribution of the number of events that occurred in the interval depends only on the length of the interval, and has nothing to do with its location. That for any0t1t2And s0 IncrementalNt1,t2 AndNt1+s,t2+sHave the same probability distribution.Definitions 1. 2 counting processNt,T0 is called intensity ( or speed )The homogeneous Poisson process if it meets the following conditions:(1) P(N0=0) =1，(2)Has independent increments.(3)For any 0s0 If(1) N0=0；(2)Processes with stationary independent increments.IfYou can proveThat is, Ns+t-NtHas mean m (t+s) m (t) of the Poisson distribution.Non-homogeneous Poisson process is important because no longer requires a stationary increments, allowing the possibility of events at certain times than others.Dang strength(t) Territories can be non-homogeneous Poisson process is regarded as a homogeneous Poisson random sampling. Established specifically to meet(t) And, for all t0 and considered a strength forPoisson process. Set up the process at time t with probability(t) /Count, was count of events is the process of with intensity function(t) Non-homogeneous Poisson process. Second, based on complex Poisson process model of stock prices1. model constructionAssumptions in the stock market, a yin and the strength of each transaction is a sequence of independent identically distributed random variables. We useI trade intensity, then for any iO,Have the same distribution. Set the stock tradesIs a parameter for(o) Poisson process, its trading volume for the compound Poisson process. We believe that the trading volume in the stock market will have an impact on stock prices, established the following model to simulate the volatility. Setting the time parameter is set to T=o,)。 Stock price process is s (t)ands (0) =S 0 time stock prices. The following definition(t)(Sufficiently small ) changes in stock price within a time interval:uS probabilitiesIIs probabilitydS probabilityU,d (uld0) is a constant,o (t) 2) meet . Thus we get the stock price process, in a very small time (t) 2 , price change has three States, each State is associated with the trading volume was expected, stock prices rise or fall by parameters of each， (0，O) lation ( also referred to as market depth parameter ). Compound Poisson process, the expected value can be obtained , and the modeluS probabilitiesIIs probabilitydS probabilityTable 1 volume impact on the share priceVolume comparison in order to make it clear that the model of stock price levels and market parameters，Extent of the impact on the stock price, we give specific examples to illustrate this. Assuming stock strengthObey the interval a,b of the uniform distribution on ( unit: shares ), intervalst= 0 . 1, the parameter=0 . 08, =O . 02。 Shares s up to uS the probability p rise, fell to dS probability p down,p unchanged stock prices did not change the probability. After probabilities calculated from table 1 it can be seen that when the stock transactions number parameters=1 Shi, trading intensity increased probability of increases pushed up share prices, which led to rising share prices. Similarly, when stock trading intensity is constant. The increase in number of transactions can also cause stock prices to change, for example: in this example, because theAnd so the number of transactions increased probability of rising stock prices. Thus it can be seen, deal strengthDepends on the distribution of the number of volumes and market parameters，The size of the rise and fall in stock prices. On the stock market of which the practical problems, we can obtain a wealth of historical data, by reasonable analysis, processing these data and apply effective statistical analysis, we can pending, and approximate volume of distribution and the corresponding parameters of the market.Three, model analysisIn the model assumptions above, we consider the European call option pricing problem. Set there is a bond and a stock market, bonds are risk-free asset, the risk-free interest rate is r, stocks are riskier assets, the price process as described in the model. Set associated with the stocks of European call options expire as t, the strike price for the k. In t=0 times, bond prices for b, then (t) time bond value ( principal and interest ) . Assuming risk neutral probability, volatility in stock prices are as follows:uS probabilitiesIIs probabilitydS probabilityType in then0,n0,n0Market depth corresponding to the risk-neutral probability coefficient index n said neutral (neutral). Because the risk-neutral, so the expectations of stock yields equal to the yields on bonds, namely: Weve got a European buying option pricing for:Four, risk avoidance and risk controlOn the stock market, investors often Avoid Risk and risk control as a top priority, so consider how to avoid investment risks is of great theoretical and practical significance. Assume the expectations of stock returns than yields on bonds, whichu-1E-（1-d）ErProbability is assumed by the model shows that a stocks price fluctuations are as follows:Under the risk-neutral probability, according to the model assumptions, fluctuations in stock price probability is:Five, model the specific practical problems of applicationSelect 2009 years 6 months 22 days until 2010 yea