### Stochastic Credit Analysis: AA Credit to SP500

#### As of:

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

#### AA credit is now at fair value in relationship to SP500

##### 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 at fair value,considering equity volatility at current levels. This supports a view that the equity market while recently at all time highs, has underpinnings of strength as this fair value of debt indicates that firms are not overly leveraged.

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 path was considered:

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

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 of 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). 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:

AA credit is approximately 10 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, if a trade was out on, is short the credit spread, long ATM or OTM puts on the SP500.

If volatility does not change, then the credit spread will close 10 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.

However, when considering the high levels and low vol if SP500, perhaps a trade can be put on where the lower convexity/gamma of the credit spread is considered versus a higher convexity/gamma of a short dated put on the SP500, resulting in a net long gamma. Thereby if the SP500 stays in this area or even trades upward, little loss will occur if not small profit as the credit spread closes in even further. Yet if the SP500 gaps down the explosion in vega for the put and the underlying downtrade would more than offset the credit spread widening. This results in a near costless put on the SP500.

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 analysis 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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