### Stochastic Credit Analysis: AA Credit to SP500

#### As of:

`## [1] "May 19 2017"`

#### AA credit is at fair to rich value in relationship to SP500, yet has asymptoted out stochastically along this report’s causal axis (x) which means it is not useful to consider credit spread richness as no reasonable risk control is available to manage a trade. 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 volatility of the equity to the volatility of the entire firm (or in this case the SP500 index), ending with a forward looking view on the credit spread pricing.

AA credit relative to SP500 is now at rich to fair value,considering equity volatility at current levels.

Yet the volatility is at recent historic lows so likley credit is rich. No timing can be provided with this stochastic analysis now as the relationship has asymptoted out to large causal value of the distance to default.

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 (in this report it is SP500) that liquid and traded and has similar liquid listed options and credit spread data, such as CDS pricing.

If you wish, I would be pleased to provide this analysis on any such liquid traded name and if the credit spread data is data is provided - preferably CDS. Since the transformation of the equity volatility to asset volatility ends with an “apples to apples” relationship dynamically managed hedge ratios can be established and monitored for risk control. In this report the trades would be expressed by being long or short the CDX and with offsetting options. Not only can the appropriate hedge ratio be defined and dynamically managed, the convexity and gamma can also be defined.

This report will now go through the steps of deriving the transformation of equity volatility to an asset volatility and examine each step’s utility in ascertaining rich or cheap or the AA credit spread.

The ML AA Index and the SP500, which when just using the SP500 level (equity level) mapped 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. Using SP500 volatility does provide insight to the AA credit pricing but is not sufficient to define trades:

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 (standardized to one year unit) 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 along the causal axis (X). With the asymptoted condition zero gamma in the change of spread occurs which means there is no timing available for a trade. Still this is a superiour and to our knolwedge the only effective way to consider pricing of credit spread versus looking at only the index levels, the volatility of the index, or the probaility of default:

A close up using only the last 6 months of the SP500 levels (x axis) which shows how rich credit is currently.But the ’0" gamma makes trade risk management impossible now:

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, which at heart is simply comparing the credit spread to the equity level. The credit spread is by definition ex ante and the equity level analysis is usually ex post.

And a “close up”" of the above ex post analyis, using the index level, shows no useful trade signals:

Please seek me out to discuss this report.