The state of practice of assessing the financial impact of jumps in market variables on derivative positions is far from ideal: (1) the mechanics of theoretical perfect replication that are the foundation of pricing models for derivatives are challenged in the face of jumps of random magnitude and uncertain timing, let alone practical difficulties with replication; (2) many pricing models in practice are continuous-diffusion process based and do not entertain jumps (see Cont & Tankov [2004] for an overview). Controlling the risk profile of derivative trading, however, requires understanding P&L impacts due to realistic changes in pricing input variables, which can involve sudden moves not captured by diffusive processes. Furthermore, managing a derivative trading book requires understanding and anticipating the impact of jumps in basic market variables on more exotic pricing model inputs.
In the face of jumps in basic market parameters, significant segments of market participants can become risk-aware and risk-averse, and that can manifest as a correction in implied parameters of pricing models. For example: (1) the 1987 equity market crash and its impact on volatility skew resulting from a greater recognition of fat tails and heteroskedasticity of return distributions, and (2) the May 2005 investment grade CDO equity tranche correlation correction resulting from a recognition of un-priced cost of hedging idiosyncratic spread jumps within the standard model, as analyzed by Petrelli et al [2006]. All these challenges get compounded when the derivative references multiple issuers, and its payoff is triggered by jumps alone in the case of CDOs, triggered by an issuer state variable switching from no-default to default. This work examines the basic synthetic CDO contract and how the impact of defaults on a tranche investment might be offset by taking a position in the reference pool assets.
Not all jumps of issuer state from no-default to default come as surprises. The credit spread revealed in the CDS market will often advertise distress. For a CDS position, marked to market daily with prescient knowledge of recovery, the impact of default on the day of default does not have to result in a significant P&L event if default occurs after the credit spread of that name has already widened significantly. Of course recovery is not perfectly known beforehand and the credit spread of an issuer itself can have sharp moves en-route to default which can cause jumps in P&L en-route to default. Whether default arrives as a shock to market participants or as a gradual deterioration of a credit, the standard synthetic CDO contract payout occurs only after a default occurs hence it would be inconceivable to attempt to devise a vanilla synthetic CDO pricing model without jumps to default.
The standard synthetic CDO model (e.g., Li [1999]) considers jumps to default, given the reference asset spread term structures, and does not consider diffusion or jumps in the credit spreads. In that model, a risk-neutral description of time-to-default is effected by fitting marginal default probabilities to observable credit spreads. The joint distribution of issuer default-time is described by a Copula approach with the implied asset correlation found by fitting a modeled CDO tranche price to market. While that model assesses risk-neutral expectations of tranche losses due to jumps to default to value the tranche - it is silent on replication-hedging in the face of defaults or diffusion or jumps in the spread. This paper assesses the costs and irreducible errors of static hedging the financial impact of defaults in Synthetic CDO contracts.
