Adding Liquidity Effects to Intensity-Based CDS Pricing

Summary

The document frames how liquidity might be incorporated into a credit default swap model built around a stochastic default intensity. Survival probabilities are derived from the integrated hazard rate, while the CDS value is the difference between protection and premium legs, discounted using given discount factors. The question asks whether a constant illiquidity spread could be placed in discounting while leaving survival probabilities unchanged, whether it should affect both legs, and how a stochastic liquidity process could be calibrated alongside the hazard process.

No answer or calibration method is included, so the document does not establish a preferred model or quantify liquidity effects. It highlights a modeling identification problem: market CDS quotes are already used to calibrate hazard rates, and the text gives no separate observations or constraints for distinguishing default compensation from liquidity. Any implementation would therefore require additional assumptions or data beyond what is presented.

Key ideas

• The baseline setup models default time through a Cox process with survival probabilities driven by a stochastic hazard rate.
• CDS value is expressed as the protection leg less the premium leg, using discount factors and survival probabilities.
• The document asks whether a constant liquidity spread belongs in discounting and whether it affects both legs.
• It raises the calibration challenge of separating liquidity from hazard rates inferred from CDS quotes.
• No solution or empirical evidence is provided, so the proposed liquidity extensions remain open questions.

Tags

# CDS pricing using intensity models incorporating liquidity


# CDS pricing using intensity models incorporating liquidity












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)?

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.