Previously, we have talked in a very generic sense about the issues with Gamma and how to identify the problems with current calculation methodologies. This research showed just how important a complete quantification of second order effects can be. The feedback has been tremendous, but one running question was "how do you really know the drivers are at the second order and not the fourth order, and how can you be sure your 'metric' isn't just some hidden risk at a different order you are estimating?" One of the amazing things of presenting this work, is the very insightful questions that follow, and here we will address exactly that question.
Rather than looking at the PnL accuracy of our suggested model vs traditional metrics, it will help to revisit the original model, but attempt to model the error and look for the order of risk that best explains the PnL errors.
First we should start with the simplest model to test if our implementation is valid. If the volatility of A = volatility of B, and the correlation is 100%, then the market will move in a perfectly parallel manner. In this case, we should expect no improvement from using a full quantification of second order risk, and we should expect a near perfect regression statistic to the 4th order. We should also predict a far less good second order R-Squared value, simply because the 4th and 2nd order are similar distributions locally. Because it is an at-the-money straddle, we predict a 0 R-squared for the first and third order.
It's nice when things work out exactly as theory suggests, because with interest rate derivatives, it never feels like that is the case.
Now let's progress to a more reasonable model. First let's keep a flat vol surface but decorrelate A and B, to 90%.
Even with such a small magnitude decorrelation, you can no longer say the errors under the traditional metric are driven by the 4th order. In fact, it looks like there is a question of 2nd and 4th order risks, and neither can properly quantify the error. Our complete metric though, continues to be modeled as 100% 4th order risk.
But of course, we haven't stressed the model in any way yet. By preserving a flat volatility surface, the market will still mostly approximate parallel moves. What if the vol surface shape changes, such that the ratio of volatilities is 1.5 and correlations are at 90%? This may seem extreme, but in fact, it is regularly encountered when central bank policy is expected to be on hold for the near future. In fact, ratios as high as 3 were seen during the quantitative easing period in the US. In Australia at the beginning of the year, the 1m2y/1m1y vol ratio (a good approximation for the ratio of volatility of the 1y and 2y spot swap) was 1.77.
Sure enough, the complete model continues to see 100% of the unexplained risk in the 4th order. Traditional metrics now look even worse, with significant unexplained PnL being of the 2nd order.
We can also see that some of the stress is showing up as 1st or 3rd order explanatory power (at an irrelevant amount, but there).
Of course, we can keep going. One way to see how impactful this is would be to consider a more highly levereaged asset C. Consider a relatively flat volatility ratio, 1.2, with a good correlation, 95%, but where C = 4B-3A. Our experience says very few banks will ever risk manage on a higher correlation for these types of non-linear derivatives because it is selling a wing (the decorrelation risk) extremely cheaply.
This is an extremely simplified model for an option on a 3y1y midcurve. Traders inherently know these highly leveraged rate points are extremely dangerous to write options on, and this is why. Even when the vol surface is extremely flat and well correlated, the smallest deviation means the a priori assumption currently used of a parallel shift causes severe risk management errors. These highly leveraged options are dangerous precisely because the non-linear PnL is also a very significant number. Consider the absolute actual non-linear PnL ratio between a 2:1 leverage with a vol ratio of 1.5 and a 4:3 leverage with a ratio of 1.2. We calculate it as 1.5x. And the misattribution of PnL (and therefore, by extension, delta bucketing risk) is 2.5x. That is a stunning level of error increase.
What about when you hedge? Let's consider a leverage ratio of 3:4, a correlation of 95%, and first order hedges (hedged to 100% rho). This is the natural dispersion portfolio, or the first order hedge to the above trade.
Functionally, the entire error is 2nd order under traditional metrics: All the variation is driven by 2nd order considerations. In fact, in our original paper, we showed how these hedged portfolios are significant worse, with traditional metrics misattributing 100% of PnL, making all traditional second order risk relatively meaningless for quantifying portfolio dynamics. We see here suddenly why that is, without fully computing the second order risk, these portfolios in particular end up far harder to manage than naively unhedged portfolios.
Worse yet, it could all be avoided with a method to approximate full second order risk. An entire world of customer trades would become tractable to market. The underlying risks, and volatility, would be the same, but by finally quantifying these risks, banks would be able to market the derivatives with far more certainty. We can both accept why these limitations were originally accepted (computational power with no tractable solution) and enjoy the benefits of modern methods to estimate this risk without a server farm worthy of Google dedicated to calculations.
We hope this was instructive, and PLATSON will be releasing a simple excel spreadsheet so market participants can fiddle, we know how important this can be for understanding.
Interest rate models are much more complicated. Forward swaps are not static linear combinations of spot swaps. In fact, they are very complicated convolutions of the yield curve, because the underlying assets are also non-linear. The PLATSON solution is a generic approach to deal with these complexities in tractable computational time. Contact us today to see how much of an improvement PLATSON can offer for derivatives risk modeling without changes in infrastructure requirements for your options portfolios.
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