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.