# A Guide to Stochastic Models for Finance

In this article I am going to show you how stochastic models for portfolio balance can be used to calculate the risk of investment. A stochastic model is one which uses the random variables to generate expected returns from a steady state model. Most models employed in financial analysis are stochastic in nature, as they attempt to estimate the risk and the return of an investment. An example of such a model is Portfolio Distributional Analysis, otherwise known as APR. This model employed in many investment management research and risk management programmes.

Most investment managers and fund managers will accept that stochastic models are used in risk management programs as they allow for better decisions regarding investment portfolios. These models are considered ideal for many investment purposes as they are simple to use and are time sensitive. The best way to describe the probability of risk in any investment context is “riskiness” as it has both high levels and low probability.

As a manager or a fund manager looking to implement a portfolio risk management approach into your trading strategy you should firstly consider using stochastic models. They are simple to use as you only require one variable, the parameters of which can be changed with a few mouse clicks. You would then use the model to estimate the likely returns of the selected portfolio. These estimates can then be used to help you set risk targets and compare multiple investment opportunities.

One of the main reasons why stochastic models for portfolio balancing are so widely used is due to their simplicity and thus low cost and accuracy. Models like APR are designed so that the long-term objective function (LTF) can be obtained and used to find the optimal level of risk. This objective function can be solved with finite difference methods (FDIC-like models). This allows the trader to obtain the most accurate estimates of returns over time. It is not however, enough to simply use these models as they must be fitted to the relevant investment scenario and the manager must also know how much of the portfolio should be risked in order to meet the objectives. Thus the model will need to be evaluated to make sure that it is offering the desired return.

As mentioned earlier, stochastic models for portfolio balance can be fitted using finite difference methods (FDIC-like methods). With this method, the parameters of interest are time and price (trend) terms, with parameters varying linearly over time. This form of model can provide excellent sensitivity but suffers from significant errors in price and quantity data. These errors lead to the over- or under-estimation of the expected returns and also generate a range of systematic volatility. They are therefore not suitable for applications where the uncertainty of results is large, such as in the stock or bond market.

Most traders use stochastic models for finance that are based on binomial models. Binomial models assume that the random variables are independent, which can be a disadvantage as they are based on the assumption that returns are independent. Therefore stochastic models that use binomial parameters can be more robust than stochastic ones and can provide good estimates over short or longer periods. Another drawback associated with binomial models is that they assume that the random variables can be studied separately. The model can then be estimated from the results obtained, which may lead to an invalid comparison with other models.

A better choice when fitting stochastic models for portfolio risk management is to fit a random effects model (REM). A REM can be fitted using a normal logistic function to take advantage of the non-periodicity of stochastic processes. It provides a higher level of predictive accuracy and therefore is usually preferred. The model is fitted by the logistic function to the portfolio mean and by the past history of each variable as it influences the value of each period. This ensures that the parameters are truly random and not necessarily Poisson distributions. This can be a significant cost savings compared with traditional stochastic models.

Other forms of stochastic models for portfolio risk management include the logistic, cubic, and range bounds. A logistic model can be fitted in cases where there is a high level of uncertainty in the data used to form the risk function. A cubic model can be fitted for cases where the distribution of risk is very irregular. A range bound can be fitted using a range function to estimate the maximum likelihood that a value of the risk function will occur in any interval of time. These various models all have significant limitations and are suited for specific applications and are recommended as suitable applications for financial risk management.

A Guide to Stochastic Models for Finance
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