# 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.

## 1 Answer

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.

• CGMY (math.nyu.edu/research/carrp/papers/pdf/jbarticle.pdf), time changed Levy (generalized Heston) (faculty.baruch.cuny.edu/lwu/papers/timechangeLevy_JFE2004.pdf) – user9403 Feb 23 '16 at 16:45
• @user9403 Is there a way to describe the idea behind these models in a few words? Can they be described as solution of a SDE like the other processes listed here? What are the relevant SDE for this processes? – Basj Feb 25 '16 at 23:57
• These are Levy models which are defined by their characteristic function. It is possible to write the infinitesimal generator as well. However as far as I know there is no good way to write the SDE as the jump process may contain infinite jumps in finite time. – user9403 Feb 26 '16 at 10:34
• Ok @user9403. For the purpose of Monte Carlo simulation, do we have something that offers the same simplicity of a SDE, allowing to compute X[i+1] from X[i], X[i-1], etc. ? – Basj Feb 26 '16 at 16:22