### Stochastic Credit Analysis: AA Credit to SP500

#### As of:

`## [1] "April 03 2017"`

#### AA credit is likely at fair value in relationship to SP500, yet has asymptoted out stochastically and is likely at max richness. No useful trade can be setup now.

##### 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 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 volatility 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 of volatility for forecasts for volatility in the future. We transform this forward looking equity volatility of the to the debt of the firm, and then end with a forward looking view on credit.

AA credit is relative to SP500 is at fair value,considering equity volatility at current levels. Yet the volatility input is at near historic lows so likley credit is rich. No timing can be provided with a stichastic analysis now.

Asset volatility is derived by the Merton transformation of equity volatility, 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:

Annualized realized 6 month (60N) SP500 volatility:

When the 6 month realized vol of SP500 is compared to the AA credit spread, greater nuance of credit spread valuation is provided than just comparing spread to market index level, 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 default 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 the current asset volatility, would default be reached if only bad events occurred, that only the lower branch on the binomial tree path would result:

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

Another measure of credit risk is the probability of default. In investment grade the probability of default is not a useful variable for trading as it does not provide timing. 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), being investment grade, is 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.

D2, or distance to default (D2D) does give a robust forward looking qualifying of current credit spreads.Insight is provided as to rich cheap of the credit spread compared to the underlying using D2D. The below shows that AA credit is not as rich as when the D2D was this large pre-2007, yet has asymptoted. With asymptoted with now zero gamma in the change of spread, there is no timing available to consider a trade. Still this is the better way to consider pricing of credit over spread v the index levels or the volatility of the ondex or the probaility of default:

A close up using only the last 6 months of the SP500 levels (x axis). While it does suggest that credit could close another 10 basis points to the SP500, the cost of putting on and taking off the trade would eat all the profit as well as there are no signals for timing:

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 this ex post analyis using the index level shows no useful trade signals:

Please seek me out to discuss this report.

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