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

enter image description here

  • $S$ - 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:

enter image description here

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)

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  • $\begingroup$ I would recommend asking this question at the SE Quantitative Finance forum. $\endgroup$ – user32398 Mar 18 at 18:36
  • $\begingroup$ 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. $\endgroup$ – Sandu Ursu Mar 18 at 18:58
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    $\begingroup$ What is f? It is not clear from your post. $\endgroup$ – Forgottenscience Mar 18 at 20:43
  • $\begingroup$ @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. $\endgroup$ – Sandu Ursu Mar 18 at 21:03
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    $\begingroup$ The question is indeed a statistical question, I agree that this the correct forum. $\endgroup$ – Jonas Mar 19 at 0:10

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