Any calculation that requires banks to project the future exposures of an exotic derivative over its expected lifetime is bound to be complex – but the computation of margin valuation adjustments (MVA) is perhaps the most intensive of all of them.
It’s been a couple of years since banks started paying attention to MVA, an adjustment made to prices to reflect the cost of funding the initial margin requirement of the trade over its lifetime. For bilateral over-the-counter trades, initial margin is set using the standard initial margin model (Simm) – a detailed, sensitivity-based approach developed by the industry to meet regulatory requirements around margining for non-cleared derivatives.
The most straightforward way to do this is to brute-force the calculation – that is, spend resources and plenty of time running them using one of several computationally intensive techniques. For many products, this is easier said than done, however.
Applying a brute force method for callable exotics – Bermudan swaptions, for instance, where exercise is allowed at certain specified dates before expiry – is extremely tricky, since their MVA must be valued by differentiating the least squares Monte Carlo simulation (LSMC) technique. This combines a Monte Carlo simulation and a regression analysis, wherein the instrument’s exercise conditions are calculated in a backward run, before Monte Carlo is used to aggregate the final result: the price.
Calculating a Simm-based MVA would mean the LSMC would need to be differentiated – possible, but not easy to do.
“It takes hours to calculate a single MVA for one [exotic] instrument,” says Alexandre Antonov, head of core modelling in the financial markets risk models team at Standard Chartered. “If I have 100 time steps and several thousands of Monte Carlo paths, and maybe 20 sensitivities – or even 10 sensitivities – this would be a huge amount of work. It’s feasible, but it’s too slow.”
This is where leveraging algorithms used for fast sensitivities calculation could come in handy.
In this month’s technical, Efficient Simm-MVA calculations for callable exotics, Antonov, and his co-authors, Andrew McClelland, a director in the quantitative research team at Numerix in New York, and Serguei Issakov, a San Francisco-based senior vice-president in the quantitative research group at Numerix, propose a much quicker algorithmic technique to calculate MVA based on the Simm model, the so-called Simm-MVA.
We demonstrate there is a perfect match with brute force, but there is enormous acceleration with respect to brute force – several hundred timesAlexandre Antonov, Standard Chartered
One key trick the quants use in their technique is converting the sensitivities of future values with respect to risk factors such as a swap rate or implied volatility to sensitivities, with respect to just the model parameters. This allows the use of algorithms that can compute the sensitivities a lot quicker.
The quants use a combination of algorithmic differentiation (AD) and tangent differentiation (TD) for this purpose. AD, a technique most risk managers today use to calculate their risk sensitivities, is used during model simulation to calculate intermediate values; TD is used in the backward pricing step, where sensitivities with respect to model parameters are calculated. These sensitivities of future values with respect to model parameters are then converted using certain mathematical transformations to sensitivities of portfolio exposures with respect to market variables.
This can save a lot of time compared with the brute force approach – by more than a factor of a hundred, the quants claim.
“The algorithm follows the pricing and runs at the same time as the pricing goes backward. We demonstrate there is a perfect match with brute force, but there is enormous acceleration with respect to brute force – several hundred times,” says StanChart’s Antonov.
Because of the sheer computational effort of using the brute force method, most techniques in existence in the industry today to calculate MVA only approximate the value by making a projection of Simm-based initial margin over its lifetime instead of doing a full computation of the Simm. One way to do this is to calculate Simm initial margin as of today and then scale it over the life of the trade based on future exposure of the trade.
Some studies carried out by market participants show the approximated MVA could differ from MVA based on a full Simm computation quite significantly. It doesn’t help that over the past year or so, the push to appropriately calculate and account for MVA has taken a backseat, given other, more pressing issues such as non-cleared margin rules compliance and benchmark reform.
As with many other modelling issues, quants have continued to independently work in the background to fix and improve existing techniques. Antonov and his co-authors’ paper is an example of that, and should be a step in the right direction for calculating MVA more accurately in the future.
Editing by Tom Osborn
Listen to Alexandre Antonov discussing his work in our podcast here.