Characteristic Functions for Coupled SDEs with State-Dependent Volatility
Summary
The document poses a modeling problem involving an asset log-price and a mean-reverting process driven by the same Brownian motion. The log-price volatility switches according to whether the auxiliary process lies inside a specified interval. The author seeks a univariate affine characteristic function for the asset log-price, conditional on both state variables, and asks whether the system can be handled analytically or more efficiently than a three-dimensional PDE calculation.
The question considers smoothing the interval indicator with a continuous approximation, but observes that nonlinearity in the volatility function appears to prevent the usual affine representation. Attempts to introduce a parameterized ansatz have not yielded a closed-form ODE system. No solution or evidence is supplied, so the document is useful chiefly as a formulation of challenges around affine tractability, shared noise, and state-dependent coefficients. It does not establish whether a closed form exists or compare alternative numerical methods.
Key ideas
- The asset log-price and auxiliary mean-reverting process share a Brownian driver.
- Volatility depends on whether the auxiliary state falls within a specified interval.
- Smoothing the interval indicator does not by itself guarantee an affine characteristic function.
- The document raises a computational concern about solving a three-dimensional pricing PDE but provides no resolution.
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Full text
# Characteristic function of SDE with coefficients depending upon second coupled SDE
# Characteristic function of SDE with coefficients depending upon second coupled SDE
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?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.