I want to price a CDS using an intensity based model, but I want to account for liquidity as well.
General model: The default time $\tau$ is the first jump time of a cox process, and the survival probabilities are given by: $$SP_{t,T} = \mathbb{E}\left[\exp \left(-\int_t^T h(s)ds \right) | \mathcal{F}_t \right]$$ Where: $$dh_t = a(t,h_t)dt + \sigma(t,h_t)dW_t$$
The price of the CDS is given by: $$V_t^{CDS} = V_t^{Protection} - V_t^{Premium}$$
$V_t^{Protection}$ and $V_t^{Premium}$ are functions of discount factors (assumed given) and survival probabilities. Also, interest rates and hazard rates are assumed independent.
Is there a simple way to introduce liquidity into the above model? Specifically:
Can we introduce a constant CDS illiquidity spread to the model? How will this spread be computed? Is it ok to introduce this constant spread into the discount factor, and not alter the Survival Probability computations? Will both the protection and premium leg contain this constant liquidity spread?
If we want liquidity to be stochastic, and introduce a new stochastic process for liquidity: $$dl_t = a^1(t,h_t)dt + \sigma^1(t,h_t)dW_t^1$$ then how will this process be calibrated (given that the hazard rate SDE is being calibrated from market CDS spread quotes)?
