Apr 3, 2014

Derivatives Valuation: A Quantitative Approach to Modelling Optionality within Collaterals


In this video blog CMO, Jim Jockle speaks with Dr. Alexander Antonov, Senior VP of Quantitative Research, co-author of the recent Risk Collateral Management Cutting Edge feature "Options for Collateral Options."

Current credit support annexes (CSAs) are complex, specifying rules for posting collateral in different currencies. When multiple currencies are allowed, this choice leads to optionality that needs to be accounted for when valuing even the most basic of derivatives. Leveraging a deterministic approach, interest rates in different currencies can be projected at certain points in the future, where the cheapest option at each point in time is used to discount the derivative.

However, the question on everyone's mind is what about optionality and how can it be modelled? By leveraging smart approximations with certain assumptions Dr. Antonov and Barclay's Dr. Vladimir Piterbarg propose a new approach to modelling the optionality of collateral currency choices continuously throughout time, potentially leading to more sophisticated interest rates that could be applied for discounting versus simply selecting the future cheapest rate.
Weigh in and continue the conversation on Twitter @nxanalyticsLinkedIn, or in the comments section.


Video Transcript: A Quantitative Approach to Modelling Optionality within Collaterals

Jim Jockle (host): Hi, welcome to the Numerix video Blog, I'm your host Jim Jockle. Continuing on the theme that we see in the market of vanilla instruments getting more and more complex especially around derivatives pricing and all of the inputs not as it relates to profitability, join us today is Alexandre Antonov, SVP of Quantitative Research  at Numerix – Alexandre how are you?

Alexandre Antonov (guest):  I'm fine thank you Jim.

Jockle: Thank you for joining us. The think I want to talk to you about – if you haven't seen it already in Risk Magazine in the March edition it is a must read, a co-authored paper by Alexandre Antonov and Barclay's Vladimir Piterbarg titled "Options for Collateral Options." I just want to give a very quick summary from the report – then get into a couple questions.

So the summary is: When collateral can be posted in multiple currencies, pricing even the simplest derivatives involves optionality, which is often tackled numerically. But by conditioning on a risk factor to make variables independent, this can be simplified. This paper shows that this approach is both quicker and more accurate than more obvious methods.

So the first question I have for you Alexandre is – what are the obvious methods in the marketplace and what are some of the challenges by past approaches?

Antonov: Let's probably say a couple 6f words about the subject itself. Imagine we have a portfolio – a big portfolio with our counterparty, containing vanillas, exotic- or semi-exotic instruments. If in the CSA we have a right to choose a collateral currency–we have an associated option. Thus, to evaluate the portfolio, we should calculate this option for any time horizon in the future in order to come up with an efficient discount factor. In the article we consider a case of two currencies – a domestic and a foreign one. Our efficient rate is a maximum of a domestic and foreign (currency adjusted) short rates. Suppose that both follow the Hull White (HW) process. 

Then, mathematically, this problem is equivalent to the calculation a zero coupon bond (as of today) of a zero-floored HW short rate.

Such a model known as "the Black Scholes one" has appeared in the early 90s, very long ago. It was a natural modification of the HW process (a normal mean reverting one) with a zero floor– these times people were afraid of the negative rates.
We know that for the Hull White standard model an analytical solution for zero bonds is very simple – however for this Black Shadow model – the exact analytical solution is impossible. The simplest trick here is to approximate the bond as expectation of the exponent of the short rate integral by exponent of the expectation. Such approach is the first order one with respect to volatility of the rates.
In the small volatility regime this approximation is simple and efficient. We can also take a second order approach; however, this method appears to be quite slow. The accuracy is relatively good for moderate maturities and volatilities but the speed in not sufficient. That is why we've proposed our advanced method.

Jockle: So Alexandre my question for you is: perhaps you could explain a little bit of the work you and Vladimir Piterbarg did who's head of quantitative research at Barclay's did to improve the simplified mathematical approach taken by many institutions today to solve this problem.

Antonov: As said in first part of this blog, the work is related to the calculation of the zero bonds for the Hull White model with a zero floor: simple setup but not so simple to solve mathematically. To improve the accuracy and the speed of the simple approximations we have developed Conditional Independence methods. We have several flavors of these methods:  what is why our paper is called "Options for Collateral Options" – you have the option of what method to use.

The main idea is to freeze an essential stochastic factor of the underlying process which makes it (approximately) conditionally independent for different times. In this case we can calculate the zero bond integral easily using Bachelier like closed formula and integrate over this frozen factor to get the final answer. How to choose this conditional factor? It is, probably, a question of "art" and we have proposed several solutions, different in the accuracy and the speed. All these methods, of course, over-perform the previous approximations and give a calculation tools for the effective discount factor for the collateral currency option.

Jockle: In your conclusion you state some elements of this – is the traditional approach of using analytic methods might be faster and not necessarily efficient there's an over emphasis in the approach as it relates to choice of numerical method whether it be finite differences etc. What is the best practice or best approach that individuals should be thinking about as it relates to validation of new processes for the selection of analytics methods as it relates to implementation within a portfolio?

Antonov: As a validation method we have used the finite differences described in details in the underlying SSRN paper. Couple words to conclude the presentation. We have studied analytical approximations to measure an impact of the collateral currency choice with respect to the trivial answer based of deterministic curves. The volatility of the movement of two currencies can be relatively big, so people should be careful to account for that volatility.  If the adjustment is tiny with respect to deterministic rates, one can obviously come up with the linear approximation or even ignore it. On the other hand, if the volatility effect is big, people should take into account our advanced adjustments.

Jockle: Alexandre thank you so much I wanted to thank you so much for joining us today, the paper "Options for Collateral Options" published in risk magazine in March is available on Numerix.com you can request it right there, and we'll make sure it's readily available on our Linkedin Page, please feel free to follow up on Linkedin or on twitter @nxanalytics.com. Alexandre Thanks you so much for your time today.

Antonov: Thank you very much Jim.

Jockle: And I hope you'll join us again soon. 


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