1. **Background**

Debt-equity trades are hard to analyze. Traditionally it is not trivial to hedge out the common factors driving both credit and equity.

In general buy-side firms have sophisticated models to analyze the sensitivity of each leg of the trade separately, but they lack a robust way to propagate common risks. For example, how a trade involving a CDS and a put option might behave due to movements in the underlying stock? The put option would be trivial: the trader might have a volatility surface to compute various option prices for different underlying prices. But the CDS is more complex: how changes in stock price propagate to spreads and probabilities of default? Simple regressions of stocks and credit spreads will yield incorrect results.

This is where Everysk can complement buy-side models by simulating the cross-asset propagation, so that the trader or risk manager can stress-test her/his trade assumptions.

In what follows we will illustrate how it is done. Let's use an illustrative capital structure arbitrage trade involving Macy’s with 2 OTC legs: a short 5year Macy’s credit and short put option:

**CDS:M 20230924 P1.5204****M 20200118 P32 3.81**

The 2 legs above follow our symbology: the first is a 5 year credit default swap on Macy’s with 2023 maturity and paying 152.04 bps spread over Libor (short credit). The second leg is a put option on Macy’s expiring on January 2020 struck at $32 and with a mid-price of $3.81.

The notional for the 5yr payor CDS is $1M and the trade is setup short 300 put contracts. Additionally we are assuming that the portfolio equity is $1M. Presumably this trade was established by the client with a jump-to-default scenario in mind, who now wants to stress-test its behavior within a holding period. The plots below simulate a 6 months holding period and a wide range of stock shocks: [-40% , +40%].

Starting with the put option by itself:

The x-axis are shocks in Macy's stock, ranging from -40% to +40%. The y-axis plots the expected profit and loss (PL) in black and 2 envelopes with the best and worst 5-percentile PLs in red.

This plot contains a lot more information than simply applying shocks to the stock and repricing the put option. In that case, we would obtain a single PL for each shock that would be comparable to the black line. Everysk will produce the full dispersion around that expectation. It can be seen that for a +40% move in Macy's stock, the expectation is asymptotically converging to the best 5-percentile, which reflects the trade making a positive PL equivalent to the full premium received from the short put (gain of approximately 11%), For a -40% move, the expectation is converging to the intrinsic value of the option.

Then, looking at the CDS independently:

In the plot above Everysk is propagating shocks on Macy's stock to the CDS via a structural model. Additionally the negative convexity (positive for a payor swap) is captured as shown above. The expected PL for a -40% move in the stock (expected PL of +10%) is much higher than the expected loss from a symmetrical 40% up move in the stock (expected PL of -1%), due to a higher probability of default in the down move. The following table shows the calculations for various stock prices:

Stock price ($) | Mkt Cap ($M) | Firm Value ($M) | PD | spread (%)/L |

95.75 | 29,394 | 34,226 | 0.46% | 0.05 |

46.76 (+40%) | 9.39% |
1.21 |
||

33.4 | 10,252 | 15,788 | 11.76% | 1.52 |

20.04 (-40%) | 34.88% |
5.86 |
||

10.90 | 3,347 | 8,179 | 50.69% | 8.87 |

The central row reflects current conditions whereas the upper and bottom rows reflect the conditions for extreme simulated stock prices. Calculations use a level of total debt of $5.5 Bi, a stock volatility of 40%, an implied firm volatility of 28% and a 40% recovery value.

Finally putting both legs together:

The trade might experience significant PL dispersion for a 6month horizon, despite being properly dimensioned for jump-to-default.

Other configurations that are more balanced for a 6 month horizon can be easily calculated via our API, by varying each leg quantity.

2. **Conclusion**

Everysk's transitive risk engine can be effectively used to stress test complex multi-asset trades by propagating shocks (Macy's stock in the example above) , regardless if the shock is directly used in the securities pricing or not.