## 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?

I will come back in next few days.

3. #### Andrewfaf

Seriously plenty of fantastic material!

viagra with atacand veterans administrations formulary viagra
viagra for women
phizer viagra – https://genericviagrarcp.com

Awaiting moderation
4. #### Andrewfaf

Incredible lots of superb tips!

2buy levitra viagra edinburgh viagra search find soft
generic viagra 100mg
wife hates viagra – https://genericviagramsl.com

Awaiting moderation
5. #### Andrewfaf

You actually said that exceptionally well!

viagra fraudsters viagra for hard erection
viagra online

Awaiting moderation
6. #### Andrewfaf

You expressed that effectively.

cheap viagra a alternative for viagra
generic viagra 100mg
viagra suppliers in stoke on trent – https://genericviagrarcp.com

Awaiting moderation
7. #### Andrewfaf

Lovely stuff. Cheers.

viagra for women patch senate bill about viagra commercials
generic viagra online
testosterone and viagra – https://genericviagrantx.com

Awaiting moderation
8. #### Andrewfaf

Factor effectively utilized.!

c-pill generic viagra viagra cause herpes outbreak
key bestellen viagra – https://genericviagrantx.com

Awaiting moderation
9. #### Andrewfaf

ap viagra hearing loss secretly give my fife viagra
cheap viagra
is generic viagra fda approved – https://genericviagrantx.com

Awaiting moderation
10. #### Andrewfaf

Really plenty of useful data.

viagra edinburgh find search free discount viagra online shop
viagra online
viagra 250mg – https://genericviagrarcp.com

Awaiting moderation
11. #### Andrewfaf

Wow many of excellent info!

cardiac bypass surgery and viagra – https://genericviagrarcp.com

Awaiting moderation
12. #### Andrewfaf

Many thanks. I value this.

viagra levitra cialis and face flushing super duper viagra stiffy
viagra for sale uk
viagra without buy prescription – https://genericviagrantx.com

Awaiting moderation
13. #### Andrewfaf

Thank you. Great stuff.

stories viagra porn canadian pharamacy viagra
generic viagra 100mg
non generic viagra – https://genericviagrantx.com

Awaiting moderation
14. #### Andrewfaf

You actually stated this perfectly!

herbal forms of viagra ages forn viagra use
cheap viagra
viagra interact with lisinopril – https://genericviagrantx.com

Awaiting moderation
15. #### Andrewfaf

Whoa a good deal of great data!

free sample of viagra or cialis viagra jokes funny pictures
viagra without a doctor prescription
viagra and emphsema – https://genericviagramsl.com

Awaiting moderation
16. #### Andrewfaf

research cost of viagra viagra sucess stories
generic viagra
discount viagra perscription drug – https://genericviagramsl.com

Awaiting moderation
17. #### Andrewfaf

Great info. Thanks a lot.

viagra tylenol trial i been using viagra everyday
viagra generic
generic viagra baku az rx-health – https://genericviagramsl.com

Awaiting moderation
18. #### Andrewfaf

Very good data. Many thanks!

viagra 100mg
buying viagra title object object – https://genericviagrantx.com

Awaiting moderation
19. #### Andrewfaf

Very good facts. Thanks a lot!

viagra online
cheapest price viagra deliverd uk – https://genericviagrarcp.com

Awaiting moderation
20. #### Andrewfaf

Nicely put. Many thanks.

what if viagra dosn’t work viagra meloxicam interaction
generic viagra online
otc alternitives for viagra – https://genericviagramsl.com

Awaiting moderation
21. #### Andrewfaf

Thanks, A lot of write ups!

levitra cialis and viagra viagra jetlag
viagra without a doctors prescription
viagra news edinburgh tid cfm moo – https://genericviagrantx.com

Awaiting moderation
22. #### Andrewfaf

buy viagra onli viagra’s corporate office photos
viagra without a doctor prescription
videos of boners with viagra – https://genericviagramsl.com

Awaiting moderation
23. #### Andrewfaf

Beneficial postings. With thanks!

viagra user expierences how much is a viagra overdose
generic viagra online
sites computer viagra edinburgh pages – https://genericviagrantx.com

Awaiting moderation
24. #### Andrewfaf

Terrific material, Thank you!

generic viagra canadian pharmacy cialis viagra compare
viagra for women
viagra effects on men – https://genericviagramsl.com

Awaiting moderation
25. #### Andrewfaf

viagra and performance enhancement compare levitra and viagra
generic viagra 100mg
women sex viagra – https://genericviagrantx.com

Awaiting moderation
26. #### Andrewfaf

Amazing tons of terrific tips.

amyl nitrate erection viagra call appointment doctor viagra
viagra without a doctors prescription
online drug stores viagra – https://genericviagramsl.com

Awaiting moderation
27. #### Andrewfaf

Whoa tons of good material.

viagra pills
viagra lymphoma – https://genericviagramsl.com

Awaiting moderation
28. #### Andrewfaf

tenuate vs viagra viagra effects
viagra for women
viagra purchase without perscription us pharmacy – https://genericviagramsl.com

Awaiting moderation
29. #### Andrewfaf

Reliable material. With thanks.

viagra circulation leg viagra trial
viagra without a doctor prescription
will viagra give you an erection – https://genericviagrantx.com

Awaiting moderation
30. #### Andrewfaf

With thanks! A lot of postings!

compare cialis and viagra viagra and nutrition
generic viagra 100mg
reasons why viagra would be ineffective – https://genericviagrantx.com

Awaiting moderation
31. #### Andrewfaf

Fantastic info. With thanks!

viagra in the philippines viagra women
generic viagra 100mg
investing in a herbal viagra scam – https://genericviagramsl.com

Awaiting moderation
32. #### Andrewfaf

Seriously a lot of great material.

viagra no doctor canada daily record scotland viagra
viagra pills
what works as well as viagra – https://genericviagrantx.com

Awaiting moderation
33. #### Andrewfaf

Cheers! Wonderful information!

reasons why viagra would be ineffective viagra effect on women
viagra without a doctor prescription
viagra alternative review – https://genericviagramsl.com

Awaiting moderation
34. #### Andrewfaf

You actually reported that effectively!

do men get hard with viagra pas viagra
viagra without a doctor prescription
viagra 2 day delivery – https://genericviagramsl.com

Awaiting moderation
35. #### Andrewfaf

Regards, I enjoy it.

calcium channel blocker and viagra women on viagra forum
generic viagra 100mg
viagra fail drug test – https://genericviagrarcp.com

Awaiting moderation
36. #### Andrewfaf

Thank you. A good amount of forum posts.

does viagra increaese size discount viagra or cialis
generic viagra 100mg
how does viagra affect blood pressure – https://genericviagrarcp.com

Awaiting moderation
37. #### Andrewfaf

You said it perfectly..

viagra australia information stories cheap female viagra
viagra generic
generic pill viagra – https://genericviagrantx.com

Awaiting moderation
38. #### Andrewfaf

purchase viagra uk penile insertion viagra
viagra pages edinburgh find generic girl – https://genericviagrarcp.com

Awaiting moderation
39. #### Andrewfaf

Nicely put, Thanks a lot.

viagra offical site viagra side effects long term
viagra without a doctor prescription
effects of viagra on child birth – https://genericviagrarcp.com

Awaiting moderation
40. #### Andrewfaf

Incredible quite a lot of wonderful knowledge.

3 cheap generic substitute viagra viagra clinic beijing
viagra for sale uk
viagra oral jelly – https://genericviagramsl.com

Awaiting moderation
41. #### Andrewfaf

viagra suppositories ivf asherman’s viagra dissent
buying viagra in philippines – https://genericviagrantx.com

Awaiting moderation
42. #### Andrewfaf

Appreciate it. Plenty of posts!

viagra dosage requirements mixing cocaine and viagra
viagra without a doctor prescription
viagra 100 mg canada pricing – https://genericviagrarcp.com

Awaiting moderation
43. #### Andrewfaf

Thanks a lot, Excellent information.

real viagra overnight viagra generic pharmacy iframe
cheap viagra
free samply viagra cialic levita – https://genericviagrantx.com

Awaiting moderation
44. #### Andrewfaf

Cheers! Very good stuff.

us sales of generic viagra cheap viagra generic
viagra for women
ed viagra online – https://genericviagrantx.com

Awaiting moderation
45. #### Andrewfaf

You actually explained it effectively!

viagra buy pharmacy iframe viagra burning pain right side
viagra for sale uk
what’s in viagra – https://genericviagrarcp.com

Awaiting moderation
46. #### Andrewfaf

You actually said this superbly!

pics of genuine pfizer viagra kamagra suhagra generic viagra
rush caught with viagra going to – https://genericviagrarcp.com

Awaiting moderation
47. #### Andrewfaf

metformin and viagra viagra euroa
generic viagra
nhs and viagra – https://genericviagrantx.com

Awaiting moderation
48. #### Andrewfaf

Cheers, I appreciate this.

charles linskaill search viagra find arutha is viagra affected by grapefruit juice
viagra for sale uk
cheap viagra from mexico – https://genericviagrantx.com

Awaiting moderation
49. #### Andrewfaf

Superb info. Thanks.

whats the generic name for viagra erectile dysfunction babies viagra
mariage viagra usa – https://genericviagrarcp.com

Awaiting moderation
50. #### Andrewfaf

Truly tons of wonderful knowledge!

5 citrate generic sildenafil viagra g postmessage viagra subject remember
viagra for sale uk
prices for viagra – https://genericviagrarcp.com

Awaiting moderation
51. #### Andrewfaf

You expressed this perfectly!

cost of viagra in the pi viagra tolerance
viagra generic
free fonts viagra – https://genericviagrantx.com

Awaiting moderation
52. #### Andrewfaf

You actually revealed it effectively!

where to buy viagra in cancun viagra soft tabs instructions
generic viagra
trusted generics viagra sildenafil – https://genericviagrarcp.com

Awaiting moderation
53. #### Andrewfaf

Kudos. A lot of forum posts.

if viagra does not work viagra vs cialis levitra
generic viagra 100mg
viagra heartburn – https://genericviagrantx.com

Awaiting moderation
54. #### Andrewfaf

Amazing forum posts. Kudos.

uaw union gets viagra viagra purchase by phone
viagra online
name for viagra – https://genericviagrarcp.com

Awaiting moderation
55. #### Andrewfaf

viagra without prescription india will hydrochlorothiazide affect viagra
viagra 100mg
toprol-xl and viagra – https://genericviagrarcp.com

Awaiting moderation
56. #### Andrewfaf

life with viagra using viagra in bodybuilding
viagra without a doctor prescription
will medicare pay for viagra – https://genericviagrantx.com

Awaiting moderation
57. #### Andrewfaf

viagra suppositories ivf ashermans compare viagra and cialis
generic viagra 100mg
women talk about viagra – https://genericviagrarcp.com

Awaiting moderation
58. #### Andrewfaf

us discount viagra cheap viagra from india
viagra pills
don’t mix viagra with steroids photo – https://genericviagrarcp.com

Awaiting moderation
59. #### Andrewfaf

Nicely put. Thank you!

reversing the effects of viagra uk viagra sales online
generic viagra 100mg
can viagra cause peyronie’s disease – https://genericviagramsl.com

Awaiting moderation
60. #### Andrewfaf

Thank you. I value it!

viagra overdose symptoms blue order pill viagra
generic viagra 100mg
why viagra is bad – https://genericviagrantx.com

Awaiting moderation
61. #### Andrewfaf

Useful content. Thanks a lot!

generic viagra buy form viagra natural herbs
generic viagra 100mg

Awaiting moderation
62. #### Andrewfaf

cialis genuinerx net viagra viagra viagra vaginal viagra suppositories side effects
generic viagra 100mg
generic viagra from india pages – https://genericviagramsl.com

Awaiting moderation
63. #### Andrewfaf

Reliable facts. Thanks.

viagra health concerns viagra lawsuits that have been settled
viagra online
viagra 100g – https://genericviagramsl.com

Awaiting moderation
64. #### Andrewfaf

You said it perfectly.!

viagra cialis levitra order online buying viagra in ireland
viagra online
viagra cupons – https://genericviagrarcp.com

Awaiting moderation

Theme by Anders Norén