The views expressed here are those of the authors, and do not necessarily represent those of their employers. Optimal Static Hedging of Defaults in CDOs Andrea Petrelli, Olivia Siu, Jun Zhang, Vivek Kapoor April 2006 Abstract The optimal static hedging of a CDO tranche position with a portfolio of bonds that constitute the CDO reference pool is addressed here. The hedge ratio and tranche pricing that result in a fair bet on the average and minimum hedge error measures are found for synthetic CDO tranches, employing two default models (1) Reduced form Normal Copula; (2) Structural Variance-Gamma. The sensitivities of the break-even spread, optimal hedge ratios, and un-hedged risks to the underlying credits and the CDO structure are illustrated. The relationship between the no-default carry and residual default risks of hedged CDO tranches are illustrated. In the same framework hedging a bond with a CDS is also examined. The residual hedge error dependence on recovery uncertainty and deviation of bond price from par are shown. Keywords: CDO, Hedging, Default, Carry, Expected Shortfall I. Introduction 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