Popular Stochastic Processes in Finance

This list contains the most common stochastic processes used in finance.

Ornstein-Uhlenbeck process

The Ornstein-Uhlenbeck process is the most common mean reverting stochastic process.

    \begin{align*} dS_t&=\lambda(\mu-S_t)dt+\sigma dW_t  \end{align*}

Geometric Brownian motion SDE

The Geometric Brownian describes the most widely used model in finance. It is used to simulate the stochastic behaviour of stocks, currencies, futures.

The value of this process is strick positive, St cannot get below zero.

    \begin{align*} dS_t &= \mu S_t dt + \sigma S_t dW_t \end{align*}

Vasicek stochastic process

    \begin{align*} dr_t &= (\alpha-\beta r_t) dt + \sigma dW_t\\ E[r_t]&=r_0e^{-\alpha t}+\frac{\beta}{\alpha}(1-e^{-\alpha t})\\ Var[r_t] &= \frac{\sigma^2}{2\alpha}(1-e^{-2\alpha t}) \end{align*}

Cox-Ingersoll-Ross interest rate model

    \begin{align*} dR_t &= \kappa(\theta- R_t) dt + \sigma \sqrt{R_t}dW_t\\ E[R_t]&= R_0e^{-\kappa t} + \theta\left(1-e^{-\kappa t} \right) \\ Var[R_t] &=R_0\frac{\sigma^2}{\kappa}\left(e^{-\kappa t}-e^{-2\kappa t}\right) + \theta\frac{\sigma^2}{2\kappa}\left(1-e^{-\kappa t}\right)^2 \end{align*}

Schwartz type 1 stochastic process

The Schwartz type 1 model is a log price Ornstein-Uhlenbeck stochastic process.

    \begin{align*} dS_t =&\; \kappa (\mu-\ln S_t) S_t dt + \sigma S_tdW \\ \end{align*}

Schwartz type 2 stochastic process

Swartz type 2 stochastic process is a two-factor process. The first factor is the spot price, the second factor a instantaneous convenience yield.

    \begin{align*} dS_t &= (\mu-\delta_t)S_tdt + \sigma_1S_tdW_t\\ d\delta_t &= \kappa(\alpha-\delta_t)dt+\sigma_2dW^\delta_t \\ dW_t W_t^\delta&=\rho dt \end{align*}

Heston stochastic volatility model

    \begin{align*} dS_t &= \mu S_t  dt + \sigma_t S_tdW^S_t\\ d\sigma^2_t&=\left(\alpha-\kappa \sigma^2\right)dt+\gamma\sigma_tdW^\sigma_t\\ dW^S_t W^\sigma_t &= \rho dt \end{align*}

GARCH(1,1) stochastic volatility model

Generalized Auto-Regression Conditional Heteroskedacity (GARCH) stochastic volatility model.

    \begin{align*} dS_t &= \mu S_t  dt + \sqrt{\nu_t}S_tdW_t\\ d\nu_t&=\left(\alpha-\kappa \nu_t\right)dt+\sigma\nu_t dW^\nu_t\\ dW_tW_t^\nu &= \rho dt \end{align*}

Constant elasticity of variance

    \begin{align*} dS_t=\mu S_t dt +\sigma S_t^\alpha dW_t \end{align*}

General stochastic differential equation

General stochastic differential equation

    \begin{align*} &\text{differential equation form:}\\ dS_t &=f(t,S)dt + \sigma(t,S)dW_t\\ &\text{integral form:}\\ S_T&=S_0+\int_{0}^{T}f(t,S)dt + \int_0^T  \sigma(t,S)dW_t \end{align*}

Merton jump-diffusion

The Merton jump-diffusion process is an extension to geometric Brownian motion.

    \begin{align*} \frac{dS}{S} = \mu dt + \sigma d W_t + (\eta-1)dQ_t \end{align*}

Clewlow and Strickland jump-diffusion mean-reversion

This is the Clewlow and Strickland jump-diffusion mean-reversion SDE. This process is used to describe processes in the energy markets.

    \begin{align*} \frac{dS}{S}=\lambda (\mu-\Theta K_\mu -\ln S)dt+\sigma dW + KdQ \end{align*}

Black-Karasinski short term interest rate

The Black Karasinski for the short term interest rate

    \begin{align*} d \ln R = \kappa(t) \left( \ln \Theta(t) - \ln R\right)dt + \sigma(t) d W_t \end{align*}

Hull-White short term interest rate model

The Hull-While model is an extended version of the Vasicek model. The short term interest rate is normal distributed, and is mean reverting.

    \begin{align*} d R = \kappa(t)\left(\Theta(t)-r \right)dt + \sigma(t) dW_t \end{align*}