A random collection

QUANT: Important Stochastic Processes

Geometric Brownian Motion

  1. It’s SDE is:
      dY_t = \mu Y_t dt + \sigma Y_t dW_t

    or also as

      d \log (Y_t) = \left(\mu -\sigma^2/2\right) dt + \sigma dW_t
  2. Solution: Y_T = Y_0 e^{(\mu-\sigma^2/2)T+\sigma W_T}

Ornstein-Uhlenbeck Process (aka Vasicek Model)

  1. It is a Mean-Reverting Process, although random, shows a pronounced tendency toward an equilibrium value.
  2. It’s SDE is:
      dY_t = -\alpha(Y_t - \mu) dt + \sigma dW_t

    \mu – long term mean/equilibrium value/mean value
    \sigma – volatility to model random shocks
    \alpha – rate by which the shocks dissipate and variable returns towards equilibrium mean

  3. Solution for this SDE is:
      Y_t = Y_0 e^{-\alpha t} + \sigma e^{-\alpha t} \int_0^t e^{\alpha s} dW_s
  4. Useful for modeling interest rates, currency exchange rates, and commodity prices

Bessel Process

  1. Stochastic Differential Equation:
      dX_t = dW_t + \frac{a}{X_t}dt

Written by curious

February 8, 2011 at 12:59 pm

Posted in quant-finance

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