Stochastic Processes Used to Model Financial Prices and Rates
Summary
This overview lists stochastic processes used in quantitative finance and gives their representative stochastic differential equations. Brownian motion provides the random driver; geometric Brownian motion models an asset with proportional drift and constant volatility, while constant elasticity of variance lets volatility depend on the asset level. The Ornstein–Uhlenbeck process introduces mean reversion, and the Cox–Ingersoll–Ross process applies square-root dynamics to interest rates.
The list also covers jump diffusion, where a Poisson process adds random price jumps, and stochastic-volatility models. In Heston, variance follows CIR dynamics and is correlated with the asset’s Brownian motion; Hull–White and Stein–Stein are mentioned as alternatives for volatility dynamics. The examples connect processes to uses such as Black–Scholes pricing, options, and interest-rate modeling. This is a brief catalog rather than a comparative treatment: it does not discuss calibration, assumptions in depth, estimation, or how model choice affects pricing and risk.
Key ideas
- Brownian motion is a common random driver for continuous-time financial models.
- Geometric Brownian motion models proportional asset drift with constant volatility.
- Ornstein–Uhlenbeck dynamics describe mean-reverting behavior, while CIR dynamics use square-root diffusion.
- Jump diffusion adds Poisson-driven price jumps to continuous asset dynamics.
- Stochastic-volatility models make variance random and may correlate it with asset returns.
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# Processes used in quant finance
# Processes used in quant finance
What are the main stochastic processes (and their SDE) used in quant finance? For example to model currency prices, stock prices, etc.
## Answer by Basj (score 18, accepted)
https://quant.stackexchange.com/a/24466
Here is a short list (to be edited and improved - community wiki) :
- Standard brownian motion (also called Wiener process) for which: $d\, W_t \sim \mathcal N(0, \sqrt{d t})$
- Geometric brownian motion, used in the Black-Scholes model (1973): $d\,X_t = \mu X_t\,dt + \sigma X_t\,dW_t$
- Constant elasticity of variance ("CEV") model (1975): $d\,X_t=\mu X_t dt + \sigma X_t\,^\gamma\, d W_t$, with $\gamma \geq 0$
- Orstein-Uhlenbeck process, with mean reversion property, used e.g. in Vasicek model (1977): $d\, X_t = \theta(\mu - X_t) dt + \sigma\,dW_t$
- Merton jump diffusion process (1976), used for options pricing: $d\, X_t = \mu X_t\, dt + \sigma X_t\, d W_t + y_t\, d N_t$, with $N_t$ a Poisson process, and $y_t$ the jump size as a random process
- Cox–Ingersoll–Ross ("CIR") process (1985), used for interest rates model: $d\, X_t = \kappa (\theta - X_t) dt + \sigma \sqrt{X_t} dW_t$
- Heston model (1993), in which the volatility of the asset is not constant but follows a random process: $d\, X_t = \mu X_t dt + \sqrt{\nu_t}\, X_t\, dW_t^X$ $d\, \nu_t = \kappa (\theta - \nu_t) dt + \xi \sqrt{\nu_t} dW_t^\nu$ (i.e. $\nu_t$ is a CIR process), with $W_t^X$, $W_t^\nu$ two Wiener processes with correlation $\rho$ Other processes that use a random process for volatility: $\nu_t$ follows a geometric brownian motion (Hull and White, 1987), $\nu_t$ follows a Orstein-Uhlenbeck process (Stein and Stein, 1991).
See also Modern Pricing Models.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.