[Paper Review] A CDS Option Miscellany

R. J. Martin|arXiv (Cornell University)|Dec 30, 2011

Stochastic processes and financial applications8 references3 citations

TL;DR

This paper provides a consistent Black'76-based framework for pricing single-name and index CDS options with upfront strikes, resolving inconsistencies in prior models. It shows that upfront payments significantly impact option prices and introduces a closed-form solution for no-knockout options with embedded recovery options, avoiding the need to model the 'armageddon' event separately.

ABSTRACT

CDS options allow investors to express a view on spread volatility and obtain a wider range of payoffs than are possible with vanilla CDS. We give a detailed exposition of different types of single-name CDS option, including options with upfront protection payment, recovery options and recovery swaps, and also presents a new formula for the index option. The emphasis is on using the Black-76 formula where possible and ensuring consistency within asset classes. In the framework shown here the `armageddon event' does not require special attention.

Motivation & Objective

• To develop a consistent pricing framework for single-name CDS options with upfront strikes, reconciling with the standard Black'76 model used on trading desks.
• To address the embedded recovery option in no-knockout CDS options with upfront strikes, providing a closed-form solution.
• To correct mispricing in existing academic treatments of CDS index options by accurately modeling the physical settlement of index options into a CDS contract.
• To eliminate the need for ad hoc handling of the 'armageddon' event (all defaults) by ensuring the model remains well-defined under all scenarios.
• To bridge the gap between practitioner pricing and academic modeling by grounding the framework in real market conventions, especially for upfront payments and physical settlement.

Proposed method

• Adapts the Black'76 model to CDS options with upfront strikes by redefining the underlying as a combination of the index level, upfront payment, and running spread.
• Introduces a forward price formulation using the risky annuity (RPV01) as numeraire, ensuring consistency with standard market practice.
• Models the payoff of no-knockout options as a function of the realized recovery and default timing, deriving an explicit formula for the embedded recovery option.
• Treats CDS index options as physically settled into a CDS contract rather than as spread options, avoiding incorrect assumptions about payoff structure.
• Uses a multivariate factor model with a common risk factor to couple default and recovery dependence, enabling extension to multiple names.
• Employs a transformation of the recovery distribution conditional on a common risk factor to match observed marginal distributions while allowing for correlation.

Experimental results

Research questions

• RQ1 How can the Black'76 model be consistently extended to CDS options with upfront strikes, rather than all-running strikes?
• RQ2 What is the correct pricing mechanism for no-knockout CDS options with upfront strikes, given the embedded recovery option?
• RQ3 Why do many academic models of CDS index options produce incorrect results, and how can the payoff structure be correctly modeled?
• RQ4 Can the 'armageddon' event (simultaneous default of all names) be handled without special treatment or probability estimation?
• RQ5 How can recovery and default dependence be modeled consistently in a multivariate CDS context?

Key findings

• The inclusion of an upfront strike significantly alters option prices compared to the all-running case, even when the strike spread is identical, due to the mixed payoff structure in cash and annuity.
• No-knockout CDS options with upfront strikes contain an embedded option on realized recovery, which can be priced explicitly using a closed-form formula derived from the model.
• The 'armageddon' event does not require special treatment or probability estimation in this framework, as the model remains well-defined and consistent under all scenarios.
• CDS index options should be treated as physically settled into a CDS contract, not as spread options, and the correct payoff is derived from the net present value of the underlying CDS.
• The model preserves consistency with the Black'76 framework for all-running options, even under lognormal hazard rate dynamics, by using the risky annuity as numeraire.
• The proposed framework correctly handles the index option payoff, unlike prior academic models that either misstate the payoff or incorrectly model the numeraire, leading to internal inconsistencies.