Some people are concerned about stochastic models for risk and ask what their relevance is to the study of finance. Stochastic volatility is a way of modeling the risk of a portfolio by evaluating expected future values of the parameters of a model, conditional on current conditions. They are extremely useful in all types of model analysis, but especially when they are used in a finite-date context like portfolio projections. In fact, stochastic volatility is one of the main concepts behind finite-date option pricing where future prices (the value of options) are conditioned on past prices (the value of the underlying instruments).
There are three main types of stochastic models for risk and investment. The first type is the binomial tree. This is a very convenient model for calculating expected returns given historical data on investment and historical volatility. It is also good for solving the problem of time-trend reversals in option pricing.
The second type of stochastic models for finance i consider more of a mathematical convenience. They are not directly comparable to the binomial model in that they treat time as a random variable and do not use expected future values. Instead, there are Monte Carlo stochastic models that can approximate the expected value and time trend with some success. The main advantage of this type of model is that it can provide reasonable estimates for certain economic scenarios while being unbiased in other situations. Unfortunately, they are not very accurate.
The third main class of stochastic models for finance i focus on more traditional statistical concepts. In particular, we will look at the binomial, Black and Sharpe method, logistic, and random walk models. Binomial tree stochastic models are based on a theoretical framework called the binomial probability. The theory is that random variables have a certain probability of occurring, and by knowing the binomial tree (a distribution of probability terms, where the probability of a value appearing as a real number within a finite range of outcomes is a binomial constant), you can approximate an underlying mean or standard deviation. Black and Sharpe random walk models, which also take into account the binomial tree, are the two most widely used and accurate stochastic models for finance.
As one can gather from the name, random stochastic models for finance utilize random variables to describe potential financial situations. They can be used in any model, but the more realistic stochastic models often combine binomial models with a random initial condition in order to deal with range problems. These can also be combined with other models such as the Fibonacci ratio in order to deal with the range problem.
Finally, the last main class of stochastic models for finance I will discuss those stochastic models that attempt to use as many different analytical techniques as possible in order to capture as much of the uncertainty in the economic data set as possible. Some examples of these include Monte Carlo methods and finite difference methods. Although these methods have been around for a long time, the addition of an element of stochasticity by using various lagging parameters has only recently made them particularly attractive to the mainstream. This allows these models to deal with both initial conditions well and to deal with varying results once the initial conditions are stabilized. These models are extremely powerful and often form the basis for a significant portion of economic models across all industries.
Hopefully this brief overview gives some insights into the most common types of stochastic models for finance. There are many more which can be introduced into the discussion once all these key categories are understood. The model with the most predictive power will typically fit the needs of the user the best, but it’s important to understand which form is best for your purposes. Good luck and have fun!