XVA (valuation adjustments) arise when extending classical derivative pricing to account for credit risk, funding costs, capital, and margin. Since XVA calculation involves time integrals of stochastic quantities, and usually must be computed for a large portfolio, efficiently calculating it remains challenging.
In this article, we’ll remind you what the basic formulas are, before discuss techniques to efficiently compute them.
Formulae for XVA
The total XVA adjustment is typically decomposed into the following components:
\[
\mathrm{XVA} = \mathrm{CVA} + \mathrm{DVA} + \mathrm{FVA} + \mathrm{KVA} + \mathrm{MVA}
\]
The first is the Credit Valuation Adjustment (CVA), which represents the expected loss due to counterparty default:
\[
\mathrm{CVA} = (1 – R_c)\int_0^T \mathbb{E}^{\mathbb{Q}}\big[ D(0,t)\,\mathrm{E^{+}}(t) \big] \, dPD_c(t)
\]
where \(R_c\) is the counterparty recovery rate, \(D(0,t)\) is the discount factor, and \(\mathrm{E^{+}}(t) = \mathbb{E}[\max(V(t) – C(T),0)]\) is the expected positive exposure, with \(C(t)\) being the collateral.
The Debit Valuation Adjustment (DVA) is similar, but reflects the institution’s own default risk:
\[
\mathrm{DVA} = (1 – R_b)\int_0^T \mathbb{E}^{\mathbb{Q}}\big[ D(0,t)\,\mathrm{E^{-}}(t) \big] \, dPD_b(t)
\]
where \(\mathrm{E^{-}}(t)\) represents expected negative exposure.
Funding valuation adjustment (FVA):
\[
\mathrm{FVA} = \int_0^T \mathbb{E}^{\mathbb{Q}}\big[ D(0,t)\,(f(t)-r(t))\,E(t) \big] \, dt
\]
where \(f(t)\) denotes the institution’s funding rate, \(r(t)\) is the risk-free rate, and \(E(t)\) is the funding exposure or requirement, typically representing the amount of uncollateralised exposure that must be funded. Intuitively, FVA measures the discounted expected cost arising from funding at a rate above the risk-free benchmark.
Similarly, the Margin Valuation Adjustment (MVA) reflects the cost of funding initial margin, replacing exposure \(E(t)\) with the margin profile \(IM(t)\), where \(IM(t)\) denotes the initial margin posted at time \(t\).
Finally, the Capital Valuation Adjustment (KVA) accounts for the cost of holding regulatory capital:
\[
\mathrm{KVA} = \int_0^T \mathbb{E}^{\mathbb{Q}}\big[ D(0,t)\,\gamma\,K(t) \big] \, dt
\]
where \(K(t)\) is the regulatory capital requirement, and \(\gamma\) denotes the institution’s cost of capital, i.e. the required return demanded by shareholders for committing capital. KVA measures the discounted expected cost of holding capital over the lifetime of the transaction.
Techniques for efficiently calculating XVA
What makes XVA so hard to calculate is:
- The integrals for CVA and DVA must be calculated for a large number of risk factor paths. Since the exposure is floored (ceilinged) just like an option is, we can’t just replace the risk factor paths with their average (like we do when we value a swap).
- For each risk factor path, the trades need to be valued at every time step (typically once per day) to expiry. For a large portfolio, this is a huge number of trade valuations.
If we were using Monte Carlo to value trades, the above two bullet points would lead to a nested Monte Carlo, which beyond almost any amount of computing power.
For CVA and DVA, the trade valuations must be floored (or ceilinged) at 0. An astute reader may note that the payoff of an option is always non-negative for the holder, so that calculating \(E^+\) and \(E^-\) is trivial. However, it’s important to note that the netting is done per counterparty. This means that options must be combined with other trade types, and the net exposure to that counterparty cannot be assumed to be positive (or negative).
Another key consideration is called wrong way risk. It’s tempting to assume that exposure and probability of default are independent, but the reality is that as your exposure increases, the counterparty becomes more likely to default.
Common techniques used to speed up XVA calculation are:
- Using analytic or faster approximate models to value trades, instead of more accurate numerical models
- Use fast “proxy” models for trade valuation, such as Taylor series, linear regressions of pricing functions on (potentially non-linear functions of) risk factors, or neural nets fitted to model prices
- Reducing the number of trades by “bucketing” or grouping similar trades together
- Utilizing GPUs which are good at highly parallel calculations
- Using algorithmic differentiation to speed up calculation of XVA Greeks
Of course, with any approximation one needs to be able to quantify the error and make sure it is within some acceptable tolerance.
XVA consulting services
At Genius Mathematics Consultants we:
- Build production quality XVA engines
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- Implement algorithmic differentiation, GPU acceleration and proxy models
The major difficulty with XVA calculation in quantitative finance is that it is computationally intensive when calculated over a large portfolio. Are you interested in working with PhD quant consultants to research and develop more efficient methodologies for XVA calculation? Drop us a message today.
