## Static pricing-hedging duality for credit default swaps and the negative basis arbitrage

Assuming the absence of arbitrage in a single-name credit risk model, it is shown how to replicate the risk-free bank account until a credit event by a static portfolio of a bond and infinitely many credit default swaps (CDS). From the viewpoint of classical arbitrage pricing theory this static portfolio can be viewed as the solution of a credit risk hedging problem whose dual problem is to price the bond consistently with the CDS. This duality is maintained when the risk-free rate is shifted parallelly. In practice, there is a unique parallel shift that is consistent with observed market prices for bond and credit default swaps. The resulting, risk-free trading strategy in case of a positive shift earns more than the risk-free rate, is referred to as negative basis arbitrage in the market, and the parallel shift defined in this way is a scientifically well-justified definition for what the market calls negative basis. In economic terms, it is a premium for taking the un-modeled residual risks of a bond investment after interest rate risk and credit risk are hedged away, predominantly these are liquidity risk and legal risk.

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## The negative basis conditioned on default

The negative basis is an annualized earnings figure that measures the expected excess return of a bond investment over a reference discounting rate, after credit risk has been hedged away via credit default swap (CDS) protection. Unfortunately, the most appropriate measurement of the negative basis is in terms of a root of a non-trivial decreasing function. The goal of the present article is to derive closed proxy formulas for the negative basis, at least in special situations. Under the assumption that a credit event before maturity of bond (and CDS) is certain, i.e. conditioned on default before maturity, we are able to derive a very simple formula. Furthermore, a small enhancement of the formula provides a proxy formula for the (unconditional) negative basis. The availability of such closed formulas allows to study qualitative properties of the negative basis, such as dependence on the recovery rate assumption or leg prices

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## Empirical evidence for the existence and temporal stability of the negative basis

We describe an analysis which underpins the existence of the negative basis as a temporally stable source of income, based on historical price data of the assets in a negative basis fund. The analysis essentially relies on the computation of a correlation between primary and hedging assets, which can easily be conducted on an xls-sheet.

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## Index CDS options: a survey of pricing approaches

This is a survey of methods proposed in the literature and the marketplace regarding the pricing of index CDS options. The challenges of the topic are highlighted, and the heavy assumptions on which common formulas rely are pointed out.

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## Negative basis measurement for callable bonds

The negative basis is an annualized earnings figure. It is supposed to measure the annualized excess return of a bond investment after having hedged away issuer-specific default risk completely via credit default swap (CDS) protection. For callable bonds a mathematically rigorous definition of the negative basis is particularly challenging because the future cash flows of the bond depend on the issuer’s call decision and are random. We propose a definition based on a specific default intensity model. It is demonstrated by means of an example that the resulting negative basis measurement is smaller than it would be under the assumption of knowledge about the call time point. This is a desirable property because eliminating the credit risk associated with a bond issuer via a CDS hedge is more expensive in the presence of call rights for the issuer.

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