It was not until the last consultative paper was issued in December 2014 that the Basel Committee took a final decision on the new standardised approach for market risk. It decided to implement the “enhanced delta plus method”, which is sensitivity-based, differentiating between three different risk components: delta risk as a foundation for capturing linear risks, and vega and curvature risk as two additional components which apply to products with optionality. Vega risk assesses the risk of price changes based on market expectation on future volatility. Curvature risk captures the non-linear risk, which is not accounted for by delta risk.

Here we will illustrate the concept of curvature risk in an intuitive way.

To be precise, let us consider a sample portfolio consisting of a long stock and two short European vanilla calls, which are almost at the money.[1]

Figure 1 shows the value of the portfolio and its delta approximation depending on the underlying risk factor. Stress scenarios were introduced for the purpose of regulatory capital charge calculations. The scenarios are created by shifting defined risk factors. For equities, the risk factors are spot prices and the shifts are in range from 30% to 70% in spot prices depending on the specific bucket. Let us assume the risk factor of this stock is stressed by 50%.

The portfolio is non-linear because of the call options, so it contributes to the curvature risk.

For the curvature risk framework, we have to consider the value of the delta-hedged position. This is the difference between the **fully revaluated** stressed portfolio scenario and the delta approximation (the difference between the blue line and the red line in figure 2). Compared to the delta risk framework the difference lies in whether the stress scenario consists of an increase (positive shift) or a decrease (negative shift) in the underlying risk factor (see the different values of the two green circles in figure 2).

When the value of the delta-hedged portfolio is positive for a given stress scenario, this means the delta approximation is conservative and underestimates the potential gain or overestimates the potential loss. On the other hand, if the value of the delta-hedged portfolio is negative (as in figure 2) the delta approximation overestimates gains and underestimates losses.

Thus, only negative values of the delta-hedged portfolio add risk. So, the negative minimum of the values in both scenarios (e.g. in figure 2 the more negative of the two circled values) is used to determine the curvature risk charge. Then, contributions from different delta-hedged portfolios corresponding to the same curvature risk factor are netted. Across different risk factors and risk buckets, correlation matrices apply for diversification. As positive values of the delta-hedged portfolio do not add further risk, they neither count towards the aggregation on risk factor or bucket level nor contribute to any increase in the capital charge through correlation.

As a rule of thumb long gamma strategies do not raise the curvature risk charge whereas short gamma strategies increase the curvature risk charge – but the curvature risk approach adds more complexity than the gamma risk approach.

There are three important things to notice when putting this concept of curvature risk into practice.

First, in contrast to gamma risk, curvature risk is no second order approximation but rather a **full revaluation** which is needed for every instrument affected. This means that banks have to be ready with regard to infrastructure, data availability and (IT) capacity to run the revaluation for all products with optionality.

Secondly, separate treatment of curvature and delta risk can lead to overly exaggerated capital charges. This can happen especially, but not only, with short gamma strategies. From figure 3 it can be seen that the delta risk charge (red circle) comes from the negative shift, whereas the curvature risk charge (green circle) comes from the positive shift. Thus, two values from mutually exclusive events are added for the aggregated risk charge. This leads to the problem of their sum being significantly higher than the worst case loss (blue circle)!

Thirdly, as mentioned before, in long gamma strategies a positive value in the delta-hedged portfolio cannot offset the delta capital charge. This leads to high capital charges for an imperfectly hedged long gamma strategy even though a long gamma strategy should have a very low worst-case loss. In figure 4 we have depicted the P&L of such a sample long gamma portfolio consisting of two long European vanilla calls almost at the money and a short stock. As can be seen, the delta capital charge is not offset and thus the capital charge is too high.

In summarising, it can be said that computing curvature risk is a complex task which needs **full revaluation** of all financial products. Because of the separate treatment of curvature and delta risk, this approach can lead to higher aggregated capital charges than expected in some cases.

If you have any questions, do not hesitate to contact us; we would be happy to help you. Why not check out the past blog entry on the effect of overcapitalisation due to data insufficiency and the opening post on the FRTB including the flyer.

[1] This example is inspired by a very similar example given by KBC Bank in its comments to the consultative document “FRTB: outstanding issues”.