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Say we have the following two SDEs driven by the same single Brownian:

$$ dx_t = -0.5\sigma^2g(\psi)^2dt + \sigma g(\psi)dW_t \quad\quad d\psi_t = -(H\psi_t+0.5\sigma^2)dt + \sigma dW_t$$

where $x_t$ is the asset log-price process with variable volatility and $\psi_t$ is an OU process connected to that same driving noise. $H>0$ is the mean-reversion speed of $\psi$, $\sigma>0$ is a constant and $$ g(\psi) = 1_{\psi \in (a,b)}\quad\quad a<0<b$$

I would like to calculate the univariate affine characteristic function of $x_t$ i.e. $$\phi(u,\tau) = \mathbb{E}(e^{iux_T}|x_t,\psi_t) = e^{iux_t+\beta(u,\tau)\psi_t+\gamma(\tau)} $$

I think I have multiple issues:

  • $g(\psi)$ needs to be continuous - but we can use a continuous approximation such as $$g_n(\psi) = \frac{1}{1+(2\frac{\psi-a}{b-a}-1)^{2n}}$$
  • If $g$ is anything but a linear function of $\psi$ (and therefore unable to exhibit the properties that I am looking for), I cannot get the system in affine form
    • I end up with the characteristic function looking like $\phi(u,\tau) = e^{iux_t+\beta(u,\tau,\psi)} $ which is leaving me with a system of ODEs that don't suggest a closed-form solution
    • I have tried ansatz approaches such as $\beta(u,\tau,\psi) = \alpha(u,\tau)\phi(u,\psi)$ but to no avail

Bottom line I am now getting stuck - I can use PDE methods on the above system to price options on $x_t$ but it requires three spatial dimensions and requires excessive calculation time.

Does anyone have any ideas on how to deal with coupled systems with non-constant coefficients such as this?

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  • $\begingroup$ Why would you expect a closed form solution? Even in affine models (which I am unconvinced this is...) one often needs to numerically solve ODEs $\endgroup$ – user9403 Apr 23 '17 at 12:30

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