Cutting edge introduction: Expanding collateral options
Pricing the optionality in derivatives trades where counterparties have the right to post multiple currencies as margin is a notoriously complex problem – so many ignore it. Research by quants at RBC Capital Markets might help put it back on the agenda
Over the past decade, derivatives pricing has become more and more complex, as quants have discovered ways to value everything from the funding costs in uncollateralised derivatives positions to the lifetime capital costs of new trades.
Quants, however, have always struggled to price collateral posting options. Many credit support annexes (CSAs) give counterparties the right to post a range of currencies as variation margin, but modelling the value of this option has traditionally been tricky.
"I don't think there is a standard method right now. Talking to various banks in the industry, it seems at least several have chosen not to implement this at all because of the perceived complexity of the model and its parameters," says Vladimir Sankovich, a managing director in the fixed-income and credit quantitative research team at RBC Capital Markets.
Those who do want to model collateral optionality have two choices: brute force Monte Carlo simulations; or an analytic solution proposed jointly by Alexander Antonov, a senior vice-president in the quantitative research team at technology vendor Numerix, and Vladimir Piterbarg, the head of quantitative analytics at Rokos Capital Management. Antonov and Piterbarg's method was published in Risk last year (Risk March 2014), however it starts to lose accuracy when there are more than two currencies in the CSA.
In our first technical, Collateral option valuation made easy, Sankovich and his co-author Qinghua Zhu, a vice-president in the fixed-income and credit quantitative research team at Royal Bank of Canada (RBC) Capital Markets, propose an analytical method to value collateral optionality that can also extend to multiple currencies.
"For the multi-currency case, there were no analytical approximations before - they are the first," says Numerix's Antonov.
In their model, the authors extend the method to cover five currencies, which they claim aligns with banks' practical needs. "In practice, we have seen CSAs stipulating baskets of up to five currencies while the most common ones give a choice of up to three – euro, sterling and US dollar – which makes modelling baskets of only two currencies unrealistic," says Sankovich.
Dealers would prefer to deliver collateral in a currency that would earn a higher interest rate for them, when adjusted for the exchange rate. Since maximising the interest rate earned is the objective, the authors approximate the maximum of the adjusted interest rate process by trying to find an analytically tractable distribution that can capture the higher moments of the distribution of the time integral of the maximum – the authors use the quadratic Gaussian process to achieve this. The coefficients of this process are then estimated using an expansion technique, called Gram-Charlier, for better accuracy.
The paper also presents a technique to estimate the parameters used in the model – competing methods typically assume the values of the parameters instead of estimating them, since it is hard to find data to calibrate them to.
"One of the things that make this complicated is that there is no market for estimating any of these parameters. You cannot calibrate the model to anything that is liquidly tradable. The model can be as efficient as you want it to be, but if you don't have a way of estimating the parameters, it is useless," says Sankovich.
As a solution, the authors observe the historical time series of traded instruments. For each of the dates of the time series, they construct a yield curve model that allows them to imply unobservable parameters from observable market quotes by using the yield curve model as a transformer.
"For instance, to get volatilities of and correlations between short rates, the latter being not observable in the market, one can build the yield curve from observable data from futures and swaps and query the curve to get the instantaneous forward rates for the given data and for a given tenor," says Sankovich.
The resulted is a method that is not only calibrated to real data but is also easy to implement.
In our second technical, A non-linear PDE for XVA by forward Monte Carlo, Rokos's Piterbarg proposes a solution to a non-linear partial differential equation (PDE), using the minimum of solutions of related linear PDEs, which in turn can be used to develop a forward simulation algorithm for calculating derivatives valuation adjustments. The alternative solutions are far more computationally demanding and difficult to handle for higher dimensions.
Only users who have a paid subscription or are part of a corporate subscription are able to print or copy content.
To access these options, along with all other subscription benefits, please contact info@risk.net or view our subscription options here: http://subscriptions.risk.net/subscribe
You are currently unable to print this content. Please contact info@risk.net to find out more.
You are currently unable to copy this content. Please contact info@risk.net to find out more.
Copyright Infopro Digital Limited. All rights reserved.
As outlined in our terms and conditions, https://www.infopro-digital.com/terms-and-conditions/subscriptions/ (point 2.4), printing is limited to a single copy.
If you would like to purchase additional rights please email info@risk.net
Copyright Infopro Digital Limited. All rights reserved.
You may share this content using our article tools. As outlined in our terms and conditions, https://www.infopro-digital.com/terms-and-conditions/subscriptions/ (clause 2.4), an Authorised User may only make one copy of the materials for their own personal use. You must also comply with the restrictions in clause 2.5.
If you would like to purchase additional rights please email info@risk.net
More on Markets
BofA’s e-FX rebuild pulls it closer to rivals
Deploying its equities tech stack, bank seeks to get ahead of the pack with its algo and e-FX offerings
Corporates look to tackle unhedgeable inflation indexes
As inflation risks mount for corporates, some are finding their exposures are linked to niche indexes
Talking Heads 2024: All eyes on US equities
How the tech-driven S&P 500 surge has impacted thinking at five market participants
Hedge funds pile into short volatility QIS options
New twist on capturing vol premium remains popular despite mixed performance in August vol spike
Exotic FX derivatives bets in play as US election vol jumps
Forward volatility agreements see profits for funds; new trades include vol knockouts
CGB steepener trade gains traction amid PBoC actions
Stimulus measures and warnings on long-dated yields have seen basis more than double since March
New benchmark to give Philippine peso swaps a fillip, post-Isda add
Isda to include new PHP overnight rate and Indonesia’s Indonia in its next definitions update
Boeing’s descent to junk doesn’t scare investors
Analysts and managers say market can absorb any selling pressure from potential downgrade