### Stochastic Credit Analysis: AA Credit to SP500

#### AA credit will tighten by 20 basis points relative to SP500 vega.

#### As of:

`## [1] "February 13 2017"`

##### Discussion:

Assuming that market efficiency have priced in ex post (backward looking) variables, such as capital structure, interest coverage, profitability and corporate action, an ex ante forward looking view can be formed from these ex post inputs via an analysis of the stochastic (option) valuation of the firm or entity.

By transposing the equity option pricing using implied or realized volatility to deriving the voatility of the firm,and analysis of rich/cheap of the credit spread can be arrived upon.

Equity volatility is forward looking and the persistence of the volatility regime is the basis for option trading and analysis like GARCH and ARCH, using the “auto-regressive” qualities - for forecasts for volatility in the future. We transform this forward looking equity volatility to credit and then end with a forward looking view on credit.

AA credit is about 20 basis points wide and will close towards 60, relative to equity volatility at currrent levels. While 20 basis points is not enough spread tightening to trade upon, it does make a good case study and gives forward insight into the “quality” of the equity market, in this case SP500, as shown by the pricing of credit.

Asset volatility is derived by the Merton transformation, simultaneous solving for the equity value of the firm as a call value with the known equity volatility to the the firm asset volatility. Then the “distance to default” (D2D) is derived which provides a variable that can qualify credit spreads with the forward looking property of equity volatility.

This process can be applied to any name or index that is traded and has listed options and a credit spread data array, such as CDS prising. If you wish, I would be pleased to provide this analysis on any liquid traded name and if the credit spread data is data is provided - preferably CDS.

The ML AA Index and the SP500, which when just using the SP500 level (equity level) to the credit spread does not show useful trade signals:

Annual volatility for the SP500 is derived using 6 months (60 day) realized volatility:

When the vol of SP500 is compared to the credit spread some idea of credit spread value is reached but only in the extremes of very high volatility or low volatility:

Equity volatility is transformed first into the asset volatility of the index, and then into a normalized “Distance to Default” or “D2D” (D2D being the “d2” in the Black Scholes formula) which gives a normalized value of how far the underlying SPY is to the strike in terms of asset volatility, time and the current SPY price level. Intuitively understand D2D by seeing it as how long in time, given volatility, would default be reached if only bad events occurred, that only the lower branch on the binomial tree would result:

Black Scholes Merton, and Merton(74) for credit, take the D2 and D1 to solve for the price of the option.

Yet in investment grade the price as given by the “hazard rate” or the probability of default is not a useful variable for trading but at the extremes. The Probability of default is indicative of inputs of d1 and d2 with time gives the odds of hitting the strike (default) - in credit, the probability of default (pdef) is being investment grade always small. The pdef is not useful as pdef odds rarely provide useful input for investment grade credit trading, but for the odd times of extreme stress like in 2009.

But D2, of distance to default (D2D), the step in derivation to get to the pdef answer does give a robust forward looking qualifying of current credit spreads. By not seeking a final pricing of credit, but stopping for the solving for d2, or D2D, insight is provided as to rich cheap of the credit spread compared to the underlying, This shows that AA credit is 10 to 16 bass points wide. The point being made is not to trade this AA as perhaps better trades can be found with different names, but that now we can take the “all or none” indication fo the credit spread and the probability of default, and instead use the nuanced D2D:

A close up using only the last 6 months of the SP500 levels (x axis):

AA credit is approximately 10 to 16 basis points cheap on a relative basis to the current D2D level.

Since D2D is solved using equity volatility, it is common to both equity and the credit, the entire firm or in this case the SP500 index.

The trade setup is short the credit spread, long ATM or OTM puts on the SP500.

If volatility does not change, then the credit spread will close 15 to 20 basis points realizing profit. And if the SP500 trades off the credit spread will likely be unchanged as the D2D shortens and as the volatility perhaps increases, making profit in the long SP500 put position.

Most credit models are based upon ex post (backward looking) company valuation of the Graham Dodd or Altman type. A rough idea of this ex post analysis is what most do, simply compare credit spread to the equity level:

And a “close up”" of an ex post analyis which can be represented by comparing SP500 to AA credit, showing no useful trade signals:

Please seek me out to discuss this report.

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