## Markets Risks

#### Topical Issues on Risks in Financial Markets.

Last updated on 6 September 2016

Curvature risk charge (CVR) is the third and final component of the sensitivity based approach for calculating market risk capital using standardised method. Aim of this component is to calculate capital charge for the second order risk inherent in the non-linear instruments. It is to account for the fact that the price of an option in general moves more than what can be explained by delta of an option only.  In summary, curvature risk charge is calculated by applying upward and downward stress stress shock to each risk factor and then using the worst loss scenario in aggregating within buckets and across buckets. Risk weights are used as a stress shock.

Step 1 – Net curvature risk charge $CVR_k$ is calculated for each risk factor as per following equation:

$CVR_k = -min (Up Shock CVR, Down Shock CVR)$

$Up Shock CVR = \sum_i[V_i(x_k^{RW(Up)})-V_i(x_k)-RW_k^{(Base)}.s_{ik}]$
$Down Shock CVR = \sum_i[V_i(x_k^{RW(Down)}) - V_i(x_k) + RW_k^{(Base)}.s_{ik}]$

Effect of delta is taken away from the above equations to isolate curvature risk. Here,

• represents an instrument for which CVR is calculated for the risk factor k;
$x_k$ is the current level of risk factor. For example, current spot price of the underlying equity;
• $V_i(x_k)$ represents value of the instrument i at the current level of the risk factor $x_k$. Note that if the price of the instrument depends on several risk factors, the CVR is determined individually for each risk factor;
• $V_i(x_k^{RW(Up)})$ represents value of the instrument i when the risk factor $x_k$ is applied upward shock;
• $V_i(x_k^{RW(Down)})$ represents value of the instrument i when the risk factor $x_k$ is applied downward shock;
• $RW_k^{(Base)}$ is the risk weight of the instrument i for the risk factor k;
• For FX and Equity risk classes, $s_{ik}$ is the delta sensitivity of the instrument i for the risk factor k;
• For other risk classes, $s_{ik}$ is the sum of delta sensitivities to all the tenors of the relevant curve of the instrument i for the risk factor k.

Step 2 -Ignore all the negative curvature risk exposures except for the ones which particularly hedge positive curvature exposure. Negative net curvature risk exposure does not attract any CVR.

Step 3 – Next step is to aggregate curvature risk exposure within each bucket using a formula below –

$\mathbf{K_b = \sqrt{max(0,\sum_kmax(CVR_k,0)^2+\sum_k\sum_{k\neq l}\rho_{kl}^2 .(CVR_k.CVR_l).\psi_{k,l})}}$

Here $\rho_{kl}$ represents same bucket correlation and is same as used for calculating Delta risk position. Details are in a blog post on delta risk charge.

$\psi_{k,l}$ is a function that takes the value 0 or 1. It is set at 0 if both $CVR_k$ and $CVR_l$ are negative. In all other cases it is set at 1.

Step 4  – Final step is to calculate curvature risk by aggregating curvature risk positions across buckets within each risk class.

$\mathbf{CurvatureRiskCharge = \sqrt{\sum_bK_b^2 + \sum_b\sum_{b\neq c}\gamma_{bc}^2.S_b.S_c.\psi_{b,c}}}$

$S_b=\sum_kCVR_k$ for all the risk factors in bucket b;
$S_b=max[min(\sum_kCVR_k, K_b), -K_b]$ when a number under square root is negative using first approximation of $S_b$;
$\gamma_{bc}$ represents correlation across buckets and is same as used for calculating Delta risk charge. This correlation is different for each risk class;
$\psi_{b,c}$ is a function that takes the value 0 or 1. It is set at 0 if both $S_b$ and $S_c$ are negative. In all other cases it is set at 1.

## Example for FX Risk Class

 CCY V(-30%) V(+30%) WSK CVRk USD/AUD -100 150 80 20 USD/EUR -150 65 -45 195 USD/CNY 290 75 65 -10

Table above shows a portfolio of options in three currency pairs. Option portfolio will be exposed to forward curve and the term structure of at the money (ATM) volatilities (as for the standardised approach, option vol smile is not to be considered). A few points to note about the numbers in the table:-

• All the numbers in tables in this section are in millions;
• Each currency pair line shows the result for only one risk factor i.e. FX rate.   V(-30%) and V(+30%) figures have been obtained by shocking the FX rate but keeping volatility and interest rate risk factors constant;
• Shock of 30% is applicable only to USD/CNY currency pair as it is not included in the specified currency pairs whose Risk Weight can be reduced by square root of 2.

In a table below $K_b$ is calculated by aggregating CVR within a bucket across all the risk factors -FX Rate, FX Volatilities and Interest Rates. This is accomplished by using a formula provided in Step 3 above. $S_b$ is calculated using a formula in Step 4.

 CCY Kb Sb USD/AUD 35 25 USD/EUR 200 160 USD/CNY 0 -45

Curvature risk charge for the above portfolio will come at 210 using a formula noted in Step 4 above. In this example, curvature risk charge calculation does not consider curvature risk exposure to option positions in USD/CNY due to this exposure being negative. This will in general be true for long gamma positions. On the contrary short gamma position will always increase the curvature risk charge. It can be seen that sensitivity based approach does not consider theta and other second order risks like cross-gamma in the calculations. Even without these missing pieces, complexity in calculating market risk capital charge using standardised approach has gone up significantly under the new approach.

1. #### sunny

can you post excel sheet with the calcs

2. #### HX

Can you please show how to compute the risk charge for low, medium, high correlation scenarios?