Three different but interrelated methods of managing financial risks

First, the historical simulation method

The core of historical simulation method is to simulate the future profit and loss distribution of securities portfolio according to the historical sample changes of market factors, and give the VAR estimation under a certain confidence level by quantile. Historical simulation method is a nonparametric method, which does not need to assume the statistical distribution of market factors, so it can better deal with non-normal distribution; This method is full-value simulation, which can effectively deal with nonlinear portfolio (such as portfolio with options). In addition, this method is simple, intuitive and easy to explain, and is often selected as the basic method with sufficient capital by regulatory agencies. In fact, this method is the basis of the Bank Capital Adequacy Agreement formulated by Basel Committee 1993 in August.

In the historical simulation method, the market factor model adopts the historical simulation method-the changes of market factors observed in a given historical period represent the future changes of market factors; In the estimation model, the historical simulation method adopts the full value estimation method, that is, the position is revalued according to the future price level of market factors and the change of position value is calculated; Finally, the profit and loss of the combination are sorted from small to large, and the profit and loss distribution is obtained, and the VAR is obtained through the quantile under a given confidence level. For example, there are 1000 possible profit and loss situations, and the quantile corresponding to 95% confidence level is the 50th maximum profit and loss value of the combination.

The calculation steps of historical simulation method are as follows:

1, mapping, that is, identify the basic market factors, collect historical data of market factors in appropriate periods (usually daily data of 3 to 5 years), and use market factors to represent the mark-to-market value of financial instruments in the portfolio (including options, which can be calculated by Black-Scholes or Garman-kohlhagen formula).

2. According to the time series of market factor prices in the past N+ 1 period, calculate the actual changes of market factor prices in the past n period. Assuming that the future price changes are completely similar to those in the past, that is, n changes in the past N+ 1 periods may occur in the future, so that the current price level of market factors may directly estimate the n possible price levels of future market factors.

3. Using the securities pricing formula, according to the future N possible price levels simulated by market factors, the N future mark-to-market values of the securities portfolio are obtained, and compared with the securities portfolio values corresponding to current market factors, the N future potential gains and losses of the securities portfolio are obtained, that is, the profit and loss distribution.

4. According to the profit and loss distribution, the VAR at a given confidence level is calculated by quantile.

Advantages and disadvantages of historical simulation method;

1, the advantages of historical simulation method

① Historical simulation method is intuitive in concept, simple in calculation, easy to implement and easy to be accepted by risk management authorities.

(2) The historical simulation method is a nonparametric method, which does not need to assume the statistical distribution of market factors, and can effectively deal with asymmetric and heavy-tailed problems.

③ There is no need to estimate various parameters such as volatility and correlation, so there is no risk of parameter estimation; In addition, it does not need market dynamic model, thus avoiding model risk.

④ It is a full-value estimation method, which can better cope with nonlinear and large market fluctuations and capture various risks.

2. Disadvantages of historical simulation method

① It is assumed that the future changes of market factors are completely consistent with the historical changes, and are independent and identically distributed, and the probability density function does not change with time (or obviously changes), which is inconsistent with the changes in the actual financial market. If historical samples are used according to the historical simulation method, it is impossible to predict and reflect future mutations and extreme events; However, when historical samples are included, there is a serious lag effect.

(2) need a lot of historical materials. It is generally believed that the historical simulation method needs no less than 1500 sample data, and if it is daily data, it is equivalent to 6 years (calculated according to 250 working days per year). On the one hand, the actual financial market is difficult to meet this requirement, for example, emerging market countries do not have so many necessary data; On the other hand, too long historical data can not reflect the future situation (outdated information), which may lead to the assumption of the same distribution. The so-called dilemma-if there are too few historical data, it will lead to fluctuations and inaccuracies in VAR estimation; However, long historical samples may increase the stability of VAR estimation, but may violate the assumption of independent and identically distributed.

③ The VAR calculated by historical simulation method is highly volatile. When the sample data is large, the historical simulation method has a serious lag effect, especially when it contains abnormal sample data, which will lead to a serious overestimation of VAR. At the same time, the abnormal data inside and outside the sample will cause the fluctuation of VAR value. Because the change of market factor only comes from the corresponding change of historical samples in the observation area, and the VAR estimation mainly uses tail probability, the number of historical observation values representing the tail of the real distribution may be very small, especially in the case of high confidence, the distribution of actual historical data is highly discrete, and the jump of VAR value is more obvious.

④ It is difficult to conduct sensitivity analysis. In practical application, it is usually necessary to consider the change of VAR under different market conditions, while the historical simulation method can only be limited to the given environmental conditions, and it is difficult to make corresponding adjustments.

⑤ The historical simulation method requires high computing power. Because the historical simulation method uses pricing formula instead of sensitivity, especially in the case of large combination and complex structure. In practical application, simplified methods can be used to reduce the calculation time. But oversimplification will weaken the advantages of full-value estimation method.

The empirical analysis results of the application effect of historical simulation method are inconsistent. In the study of spot foreign exchange portfolio, Hendricks found that when the return deviates from the normal distribution, the VAR estimated by historical simulation method with 99% confidence is more effective than the analysis method. Mahoney's research also supports this conclusion. Jackson et al. pointed out that the historical simulation method is superior to the analysis method in the case of thick tail, especially in the tail estimation event. However, Coupier's research conclusion is just the opposite. Using the simulation research of normal distribution and T distribution, he found that when the yield distribution is thick-tailed, the VAR estimated by historical simulation method has great changes and upward deviation.

Second, the analysis method

Analysis method is the most commonly used method in VAR calculation. It uses the approximate relationship between the value function of portfolio and market factors and the statistical distribution of market factors (variance-covariance matrix) to simplify the calculation of VAR. According to the different forms of portfolio value function, analysis methods can be divided into two categories: δ model and γ model. In the Delta model, the value function of portfolio is approximated by the first order, but the statistical distribution assumption of market factors in different models is different. For example, the Delta- normal model assumes that market factors obey multivariate normal distribution; Δ Weighted Normal Model uses the Weighted Normal Model (WTN) to estimate the covariance matrix of market factor returns; Delta-GARCH model uses GARCH model to describe market factors.

In gamma-class model, the value function of portfolio is approximated by second order, in which gamma-normal model assumes that the changes of market factors obey multivariate normal distribution, and Gamma-GARCH model uses GARCH model to describe market factors.

Third, Monte Carlo simulation method

This analysis method simplifies VAR by using sensitivity and statistical distribution characteristics. However, due to the special assumption of distribution form and the local characteristics of sensitivity, this analysis method is difficult to effectively deal with the nonlinear problems of heavy tail and large fluctuation in the actual financial market, which often leads to various errors and model risks. This simulation method can deal with nonlinear and non-normal problems well. The main idea is to repeatedly simulate the stochastic process of determining the estimated financial price, and each simulation can get a possible value of the portfolio at the end of holding. If a large number of simulations are carried out, the simulated distribution of portfolio value will converge to the real distribution of portfolio. In this way, the real distribution can be obtained by simulating the press conference, so as to find out the VAR.

Monte Carlo simulation method is also called stochastic simulation method. The basic idea is that in order to solve problems in science, engineering technology, economy and finance, firstly, a probabilistic model or stochastic process is established, so that its parameters are equal to the solution of the problem, then the statistical characteristics of the parameters are calculated by observing the model or process, and finally the approximate value of the problem is given. The accuracy of the solution can be expressed by the standard deviation of the estimated value.

Monte Carlo simulation method can solve many problems, and its application can be divided into two categories, depending on whether it involves the shape and results of random processes.

1, deterministic problem

The method of Monte Carlo simulation to solve this kind of problem is: first, establish a probability model related to the solution, so that the solution is the probability distribution or mathematical expectation of the model; Then the model is randomly sampled and observed, that is, random variables are generated; Finally, the arithmetic mean is used as an approximate estimate. Calculating multiple integrals, finding inverse matrix and solving linear equations all belong to this kind of problem.

2. Random questions

For this kind of problem, although it can sometimes be expressed as multiple integrals or some functional equations, and then random sampling can be considered to solve it, this indirect simulation method is generally not adopted, and the other is direct simulation method, that is, sampling inspection is carried out according to the probability law of actual situation. Inventory problem in operational research, queuing problem in stochastic service system and simulating the change of financial assets value all belong to this kind of problem.

The basic steps of Monte Carlo simulation method are as follows:

① Establish a simple probabilistic statistical model for practical problems, so that the solution is exactly the expected value of the model;

(2) Establish the sampling distribution of random variables in the model, conduct simulation experiments on the computer, extract enough random numbers, and make statistics on related events;

(3) Analyze the simulation test results and give the estimated solution and its precision (variance);

④ If necessary, improve the model to improve the estimation accuracy and simulation calculation efficiency.

Advantages and disadvantages of Monte Carlo simulation method;

The advantages of this method are:

(1) generating a large number of scenes, which is more accurate and reliable than the historical simulation method;

② Full-value estimation method, which can deal with nonlinear, large fluctuation and thick tail problems;

③ Different behaviors (such as white noise, autoregressive, bilinear) and different income distributions can be simulated.

Its main disadvantages are:

① The generated data sequence is pseudo-random number, which may lead to wrong results; There is clustering effect in random numbers, which wastes a lot of observations and reduces the simulation efficiency.

(2) It depends on the specific stochastic process and the selected historical data;

(3) The calculation amount is large and the calculation time is long, which is more complicated than the analytical method and the historical simulation method;

④ There are model risks, and some models (such as Geometric Brownian Hypothesis) do not need to limit the changing process of market factors, so there is no arbitrage. (For more information about the stock market, please go to the mysterious stock market zone ...)

Monte Carlo simulation method has been widely used in recent years because of its full value estimation, non-distribution assumption, strong ability to deal with nonlinear and non-normal problems and flexibility in practical application. Many researches are devoted to improving the traditional Monte Carlo simulation method and trying to improve its calculation speed and accuracy. (Zhang Jibao)