It is common for standard collateral agreements – known as credit support annexes (CSAs) – to include substitution rights, so that assets posted to mitigate the counterparty risk associated with an over-the-counter derivatives contract may be replaced with other eligible assets.
The choice of which collateral to post, and whether to switch one for another, is not trivial. Eligible assets have varying mark-to-market values which in standard approaches are not taken into account. But an optimal strategy may lead to significant cost savings.
Pierre Henry-Labordère, a senior quant in the global markets quantitative research team at Societe Generale in Paris and associate professor at the Ecole Polytechnique, in Optimal posting of collateral with recurrent neural networks, proposes a solution using two sets of tools that are increasingly popular in quantitative finance: stochastic control problems (SCPs) and recurrent neural networks.
Traditional methods for collateral substitution bear more resemblance to a rule of thumb than a rigorous quantitative procedure.
One is the full substitution rule, which holds that users should always opt for the cheapest-to-deliver asset – that is, the one with the lowest rate of return. While optimal in theory, this strategy may be impractical as most CSAs only permit a collateral payer to substitute an amount equivalent to the change in mark-to-market, rather than the entire mark-to-market.
The alternative approach, the so-called bang bang, is suboptimal by design and recommends that the asset with highest rate of return should be posted as collateral. While this is simple and practical, it does not account for the joint evolution of mark-to-market and rates, a simplification that can easily lead to inefficient collateral selection.
Therefore, the best method for collateral posting sits somewhere between the outputs of the bang bang and full substitution strategies, which represent the upper and lower bound of the possible choices. Unless they coincide, at least one of the two is suboptimal and the collateral poster has no way of knowing which.
Henry-Labordère’s paper tackles precisely this issue. Recurrent neural networks, a type of neural network that allows for time-dependent dynamics, are used to find the optimal solution, conditional to real-world constraints. This result is then compared with the two boundaries established by full substitution and bang bang strategies to test which of the two is closer to the optimal choice.
“One of the motivations for the paper was to assess the accuracy of the bang bang and the full substitution strategies,” says Henry-Labordère. “If at the end of the computation we find that the optimal is close to one of the two, then that one is already a good strategy.”
But the key to his solution is rewriting the collateral choice problem as a stochastic control problem.
SCPs are a family of mathematical problems in which a dynamic stochastic system has a control variable that can be changed arbitrarily. The aim is to find an optimal solution for that control variable to be minimised or maximised, conditional to the randomness underlying the system.
SCPs have found numerous applications in fields such as engineering, aerospace technologies, robotics and finance, among others. Financial applications are showing promising results and the literature on them is rapidly expanding. This is because the dynamic nature of SCPs makes them particularly suitable for path-dependent instruments, of which there are many in finance. The most common example is an American option, which the holder can exercise at any time during the life of the contract. While the least squares Monte Carlo method is a well-established technique for pricing American options and similar instruments, the SCP approach is also applicable.
SCPs can also be applied to the multi-period portfolio selection problem, where an investor seeks to optimise returns while rebalancing their portfolio. The optimisation involves inverting potentially large correlation matrices, which may be computationally demanding. A stochastic control approach can overcome such an obstacle.
Vladimir Piterbarg, now global head of quantitative analytics at NatWest Markets, was the first to frame the choice of collateral as a stochastic control problem in a 2013 paper published in Risk.net, Stuck with collateral. In it, he developed a solution suitable for an economy in which all assets are collateralised, a rather accurate theoretical representation of the OTC market.
Missing from that framework was a practical way to optimise for different choices of collateral. This is where Henry-Labordère’s idea of using recurrent neural networks comes into play.
“There have been some recent breakthroughs on the use of neural networks to solve optimal stochastic control. The novel contribution of this work is to introduce it to solve the problem of collateral choice,” says a senior counterparty risk quant at a large European bank.
The combination of stochastic control problems and variants of neural networks also featured in a recent paper by Hans Buehler, global head of equities analytics, automation and optimisation at JP Morgan. There, a similar framework is deployed to a very different application, the hedging of derivative books using deep learning, again with very promising results.