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Algorithmic Differentiation: Shattering the Myths
By Russell Goyder PhD | August 13, 2015

Lately I’ve noticed some inaccurate ideas regarding Algorithmic Differentiation (AD) floating around in the press. And so, I feel obliged to set the record straight.

For those unfamiliar with AD, it is a mathematical technique that helps firms with derivatives on their books rapidly solve complex pricing and analytics problems. In fact, firms utilizing AD typically experience staggering improvements in speed and accuracy when compared to traditional risk methods such as “bumping.” 

One misunderstanding concerning AD pertains to its role in hedging. Greeks generated by AD are exact derivatives, not the differences that many firms rely on to hedge. As such, some believe that the Greeks produced by AD need to undergo additional transformations before they can be used for proper hedging. For this reason, their recommendation is to stick with the conventional bumping method that produces differences out of the gate. 

I have to counter this erroneous line of thought. In reality, there are no additional transformations needed to use AD sensitivities in hedging, over bumped ones.

What you really need to accurately hedge risk is the notional of the hedging trade (let’s use the example of a 5-year interest rate swap), which is neither an exact derivative nor a finite difference, but a rather simple calculation which I’ll call “equivalent notional.” This value is based on a sensitivity that you can calculate either by AD (very quickly), or bumping (often prohibitively slowly).

Now, if the relationship between your portfolio value and your quote, the 5-year swap rate, is linear (i.e., a straight line without much curvature, or gamma), then the hedge will be a good one for a wide range of market moves and it won’t matter much in terms of accuracy whether you use AD or bumping. Speed of course is another matter, as there is no possible way bumping could match AD in this arena. 

On the other hand, if the relationship between a portfolio and quote is significantly non-linear, and if, and only if,  the market moves by the same amount as your bump size, then the hedge based on bumping will, in fact, be better. But therein lies the rub – you don’t know how much the market will move until it actually happens. And because of gamma, the amount of the move is now very important. 
A far better approach to cope with gamma is to adjust your hedge if the market moves a great deal. For that, you need both the sensitivity and a measure of how the sensitivity changes, which allows you to approximate the true relationship between your portfolio and your quote.   

So, that in a nutshell, is an accurate portrayal of AD. With it, you obtain derivatives and sensitivities, and have to calculate equivalent notionals to hedge.  But this is a task you have to perform anyway with bumping. However, part of the beauty of FINCAD’s implementation of AD, known as Universal Algorithm Differentiation™ (UAD), is that equivalent notionals come pre-calculated in the risk report, saving you a tremendous amount of time and effort.

For more information on UAD, view our related blog post, Driving Better Trade Decisions with UAD.

About the author
Russell Goyder PhD
Russell Goyder PhD
Director of Quantitative Research and Development | FINCAD

Russell Goyder, PhD, is the Director of Quantitative Research and Development at FINCAD. Before joining FINCAD’s quant team in 2006, he worked as a consultant at The MathWorks, solving a wide range of problems in various industries, particularly in the financial industry. In his current role, Russell manages FINCAD’s quant team and oversees the delivery of analytics functionality in FINCAD’s products, from initial research to the deployment of production code. Russell holds a PhD in Physics from the University of Cambridge.