Stochastic process are collections of random variables which can be used to describe the evolution of random systems over time. Stochastic processes are used in a wide variety of fields including engineering, genetics, computer graphics, and quantitative finance. Despite their usefulness, stochastic processes can be quite difficult to learn about and understand. For this reason we teamed up to produce this interactive educational web-application for learning about stochastic processes. In this web-application we feature four stochastic processes commonly used for simulating asset returns and prices and two mean-reverting stochastic processes used for modelling interest rates. This is our first quantitative web-application and would very much appreciate your feedback via either of the supporting articles linked to at the top. Enjoy!

The first use of a Wiener Process, also called Brownian Motion after Robert Brown, for simulating returns on financial assets was in 1900 when in Louis Bachelier wrote a paper entitled The Theory of Speculation which used a Wiener process to describe the returns on stock options. A Wiener process is described by three properties:

- $W_0 = 0$,
- The function $t → W_t$ is almost surely everywhere continuous, and
- $W_t$ has independent increments where $W_t−W_s \sim N(0,t−s) \text{ for}\ 0 ≤ s < t$ is approximately normally distributed with expected value $μ$ and variance $σ^2$.

This model was popularized by Fisher Black and Myron Scholes in their 1973 paper, The Pricing of Options and Corporate Liabilities. In this paper Black and Scholes famously derived the Black-Scholes closed form formula for pricing European Options. Their proof assumed that returns on assets evolve according to a Geometric Brownian Motion stochastic process. Geometric Brownian Motion applies drift and volatility factors to an underlying Wiener process. The stochastic differential equation for Geometric Brownian Motion is given by, $$dS_t = μS_tdt + σdS_tW_t$$ where $dS_t$ is the change in the asset price, $S$, at time $t$; $μ$ is the percentage drift expected per annum; $dt$ represents time, $σ$ is the daily volatility expected in the asset price, and $W_t$ is a Wiener process (Brownian Motion).

One limitation of Geometric Brownian Motion is that it does not exhibit the erratic upward and downward jumps in price as seen in the real-world. This is because Geometric Brownian Motion is continuous and upward and downward jumps can be seen as discontinuities in the price. In order to address this limitation, Robert Merton proposed adding a jump process to an underlying Geometric Brownian Motion which follows a compound Poisson distribution in his 1976 paper, Options Pricing when Underlying Stock Returns are Discontinuous. The stochastic differential equation of the Merton Jump-Diffusion model is given in two parts by, $$dS_t = μS_tdt + σdS_tW_t + dJ_t \text{ where}\\ dJ_t = S_td(\sum\nolimits_{i=0}N_t (Y_i - 1))$$ where Nt is the compound Poisson process with rate $λ$ and $Y_i$ is a log-normally distributed random variable.

Another limitation of Geometric Brownian Motion is that it assumes volatility is constant over time. In the early 1990's Steve Heston extended the Black Scholes model to incorporate stochastic volatility. Volatility in the Heston model evolves according to another stochastic process called the Cox Ingersoll Ross (CIR) process. In the Heston model the stochastic volatility CIR process and the Geometric Brownian Motion which describes the evolution of the asset's price are correlated with some correlation, ρ. The stochastic differential equation for the Heston model is given in two parts by, $$dS_t = μS_tdt + \sqrt{v_t}S_tdW_t^S \text{ where}\\ dv_t = a(b - v_t)dt + σ \sqrt{v_t}dW_t^v$$ where $W^s$ and $W^v$ are two correlated Wiener processes, $a$ is the rate of mean-reversion of the Cox Ingersoll Ross process $(dv_t)$, $b$ is the mean value of the CIR process over time, and the term $a(b - vt)$ is the drift factor of the CIR process.

The Cox Ingersoll Ross (CIR) stochastic process is used to describe the evolution of interest rates over time. The CIR stochastic process was first introduced in 1985 by John Cox, Johnathan Ingersoll, and Stephen Ross as an extension of the Ornstein Uhlenbeck process. The stochastic process has also been used in the Heston model to describe the evolution of volatility over time. One interesting characteristic of the CIR stochastic process is that it is mean-reverting over time. The stochastic differential equation for the Cox Ingersoll Ross is given by, $$dr_t = a(b-r_t)dt + σ\sqrt{r_t}dW_t$$ where $W_t$ is a Wiener process, $a$ is the rate at which the process mean reverts (a larger number results in a faster mean reverting process), $b$ is the long run average interest rate, and $σ$ is the volatility of the process.

The Ornstein Uhlebneck stochastic process is another mean-reverting process which is sometimes used for modelling interest rates. The Ornstein Uhlenbeck process is named after Leonard Ornstein and George Eugene Uhlenbeck. The difference between the Ornstein Uhlenbeck stochastic process and the CIR process is that the CIR processes multiplies the stochastic component by the square root of the previous value for the interest rate. The stochastic differential equation for the Ornstein Uhlenbeck process is given by, $$dr_t = a(b-r_t)dt + σdW_t$$ where $W_t$ is a Wiener process, $a$ is the rate at which the process mean reverts (a larger number results in a faster mean reverting process), $b$ is the long run average interest rate, and $σ$ is the volatility of the process.