# Particle Filter for structural credit risk model

Kwon (2012)* proposes a structural credit risk model where the asset value process and the noise are estimated based on the observed equity prices:

• $$S$$ - equity prices
• $$V$$ - value of the assets
• $$Z$$ - noise which contaminates the equity prices.

The assumed model for the log equity price is: $$\ln S_t = g_S(V_t,t,\Theta_V) + Z_t$$ with $$g_S$$ being some function of $$V_t$$ (value of firm's assets at time $$t$$), $$t$$ (time) and $$\Theta_V$$ (parameters governing the asset dynamics)

$$V_t$$ and $$Z_t$$ are modeled as follows: \begin{align} dV_t &= \mu V_t dt + \sigma_V d W_t^{\mathbb{P}V}\\ dZ_t &= -\theta_Z Z_t dt + \sigma_Z dW_t^{\mathbb{P}Z} \end{align} $$W_t^{\mathbb{P}V}$$ is assumed to be independent of $$W_t^{\mathbb{P}Z}$$ ($$\mathbb{P}$$ denotes the physical or the real-world probability measure).

The author writes:

In the process of following the particle filter algorithm steps, we confront the problem that re-sampling weight $$w^j \approx f_{t+1}\left(S_{t+1}\mid V_{t+1}^{(*j)},Z_{t+1}^{(*j)}\right)$$ is a $$\delta$$-function, which takes only 0 and 1 values. In order to solve this problem, the algorithm is adjusted as follows:

where:

• $$J$$ is the Jacobian transformation of $$Z_t=\ln(S_t)-g(V_t,t,\Theta)$$
• $$q_{t+1}$$ is the density function of $$V_{t+1} \mid V_t$$
• $$k_{t+1}$$ is the density function of $$Z_{t+1} \mid Z_t$$

I don't quite understand how this algorithm solves the "problem" and in particular I don't see why the Jacobian appears in step (3).

My intuition is that since, given $$V_{t+1}^{(*j)}$$, there is a one-to-one relationship between $$S_{t+1}$$ and $$Z_{t+1}^{(*j)}$$, if we were to sample $$Z_{t+1}^{(*j)}$$ along with $$V_{t+1}^{(*j)}$$ and evaluate the density function $$f_{t+1}\left(S_{t+1}\mid V_{t+1}^{(*j)},Z_{t+1}^{(*j)}\right)$$ we would simply get a lot of $$0$$s (maybe by chance and/or computer rounding error some $$1$$s). To sidestep this, we make it look like we've actually sampled the $$S_{t+1}$$ alongside $$V_{t+1}^{(*j)}$$ and then evaluate the density of $$Z_{t+1}^{(*j)} \mid S_{t+1}, V_{t+1}^{(*j)}$$. However, $$Z_{t+1}^{(*j)}$$ depends on $$S_{t+1}$$ only through the auto-correlation with $$Z_{t}^j$$. Therefore we are left with $$k_{t+1}(Z_{t+1}^{(*j)}\mid Z_t^j)$$ which is a normal distribution according to the properties of the Ornstein-Uhlenbeck process. Not 100% sure of this.

And I'm still confused about step (3). It looks very much like a transformation of variables sorcery.

* Kwon, T. Y. (2012). Three essays on credit risk models and their Bayesian estimation (Doctoral dissertation)

• I would recommend asking this question at the SE Quantitative Finance forum. – user32398 Mar 18 at 18:36
• The question is more statistical in nature than "quantic". My guess is that there are more people familiar with Particle Filters here than on quant.stackexchange.com. – Sandu Ursu Mar 18 at 18:58
• What is f? It is not clear from your post. – Forgottenscience Mar 18 at 20:43
• @Forgottenscience I've updated the question with the bit where $f$ appears. So it's just the conditional probability, which would normally (e.g. in the bootstrap filter) serve as the weights. – Sandu Ursu Mar 18 at 21:03
• The question is indeed a statistical question, I agree that this the correct forum. – Jonas Mar 19 at 0:10