Some of the Stochastic Processes used in Finance:
There are many different types of stochastic processes that are used in finance, but I will focus on few of them.
Brownian Motion:
Brownian motion is a type of stochastic process that describes the random movement of particles suspended in a fluid. In finance, it is often used to model the random fluctuations of stock prices. The movement of the stock price is assumed to be continuous and follows a normal distribution. The parameters of the distribution, such as the mean and variance, can be estimated using historical data. An example of this would be the daily fluctuations in the price of a stock like Google or Microsoft.
Geometric Brownian Motion:
Geometric Brownian motion is a variation of Brownian motion that is used to model the exponential growth of financial assets over time. In this model, the asset price is assumed to grow at a constant rate (the drift) and experience random fluctuations (the volatility). Geometric Brownian motion is commonly used to model stock prices, commodity prices, and foreign exchange rates. An example of this would be the long-term growth of Apple's stock price over several years.
Jump Process:
Jump processes are a type of stochastic process that includes sudden, discontinuous movements in the value of an asset. This type of process is useful for modelling events such as market crashes or sudden changes in the price of a commodity due to a supply shock. The jumps in the process can be modelled using a Poisson distribution, and the size of the jumps can be estimated using historical data. An example of this would be the sudden drop in the value of Bitcoin in 2018.
Ornstein-Uhlenbeck Process:
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The Ornstein-Uhlenbeck process is a type of stochastic process that is used to model mean-reverting behaviour in financial markets. This means that the process has a tendency to return to a long-term average value over time. The parameters of the process can be estimated using historical data, and the model can be used to predict future market behaviour. An example of this would be the behaviour of interest rates in response to changes in economic conditions.
GARCH Model:
GARCH (Generalized Autoregressive Conditional Heteroscedasticity) models are a type of stochastic process that is used to model the volatility of financial assets over time. This model is based on the idea that the volatility of an asset is not constant over time, but rather varies in response to changes in market conditions. GARCH models can be used to predict the likelihood of extreme events, such as a market crash, and can help investors manage risk. An example of this would be the use of GARCH models to manage risk in a portfolio of stocks.
Jump-diffusion model:
The jump-diffusion model is a stochastic model that is used to describe the behaviour of financial assets that experience sudden and unpredictable jumps in their prices or values. This model combines a stochastic process (such as Brownian motion or a Poisson process) with a jump process to capture both the continuous and discontinuous components of asset price movements. An example of this would be the use of jump-diffusion models to estimate the price movements of stock options.
Cox-Ingersoll-Ross Model:
The Cox-Ingersoll-Ross (CIR) model is a stochastic model that is used to describe the behaviour of interest rates over time. This model assumes that interest rates are mean-reverting, with the speed of mean reversion being dependent on the level of interest rates. The CIR model can be used to estimate the probability of interest rates hitting certain levels in the future and is commonly used in the pricing of interest rate derivatives.
Levy Processes:
Levy processes are stochastic models that are used to describe the behaviour of financial assets that exhibit fat-tailed or heavy-tailed price distributions. This model is named after French mathematician Paul Levy and is based on the idea that asset prices can jump by large amounts over short periods of time. An example of this would be the use of Levy processes to model the price movements of commodities such as gold, oil, or natural gas.
