use delta-gamma approxamation to calculate the VAR
- Thread starter ajsa
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Hi asja,
Did you see this is the video?
starts @ 17:00 at http://www.bionicturtle.com/premium/screencast/2009_4.a.ii_valuation_and_risk_models_var/
or here is a 9 min video:
http://www.bionicturtle.com/learn/article/taylor_series_approximation_illustrated_9_minute_screencast/
and I have an illustrated example with stock option at 4.a.4:
http://www.bionicturtle.com/premium/screencast/2009_4.a.ii_valuation_and_risk_models_var/
this is a key building block (it's my #1 building block in risk...most of our applications that are not full simulation are taylor approximations)
here is an application for the bond;
i.e., using duration + convexity adjustment is to apply truncated taylory
http://www.bionicturtle.com/learn/article/duration_convexity_of_zero_practice_market/
and here is a recent question posted re: an option:
http://forum.bionicturtle.com/viewthread/1840/
the headline is that duration/delta and convexity/gamma are both application of, respectively, 1st and 2nd order taylor series...
it is good to see:
delta = dollar duration = 1st derivative
gamma = dollar convexity = 2nd derivative
Taylor = f[1st] + f[2nd] + ...
after you take a look at the resources we've got, let me know if i can clarify further...Thanks, David
Did you see this is the video?
starts @ 17:00 at http://www.bionicturtle.com/premium/screencast/2009_4.a.ii_valuation_and_risk_models_var/
or here is a 9 min video:
http://www.bionicturtle.com/learn/article/taylor_series_approximation_illustrated_9_minute_screencast/
and I have an illustrated example with stock option at 4.a.4:
http://www.bionicturtle.com/premium/screencast/2009_4.a.ii_valuation_and_risk_models_var/
this is a key building block (it's my #1 building block in risk...most of our applications that are not full simulation are taylor approximations)
here is an application for the bond;
i.e., using duration + convexity adjustment is to apply truncated taylory
http://www.bionicturtle.com/learn/article/duration_convexity_of_zero_practice_market/
and here is a recent question posted re: an option:
http://forum.bionicturtle.com/viewthread/1840/
the headline is that duration/delta and convexity/gamma are both application of, respectively, 1st and 2nd order taylor series...
it is good to see:
delta = dollar duration = 1st derivative
gamma = dollar convexity = 2nd derivative
Taylor = f[1st] + f[2nd] + ...
after you take a look at the resources we've got, let me know if i can clarify further...Thanks, David
Hi David,
I think I understand Taylor expansion.. my question is how to use it to evaluate the VAR of a derivative? maybe an example is helpful.
Also for the full revalution method for derivitive's VAR, it says to revalute the derivative by decreasing underlying's value by x%VAR.. But the 'delta' may not be positive, so the derivative may increase value in that case, and so this method would not work. Do I misunderstand anything here?
Thanks.
I think I understand Taylor expansion.. my question is how to use it to evaluate the VAR of a derivative? maybe an example is helpful.
Also for the full revalution method for derivitive's VAR, it says to revalute the derivative by decreasing underlying's value by x%VAR.. But the 'delta' may not be positive, so the derivative may increase value in that case, and so this method would not work. Do I misunderstand anything here?
Thanks.
Hi asja,
if you look at the duration +convexity example:
http://public.sheet.zoho.com/public/btzoho/sep24-taylorbond
the yield shock (+20 basis points) is the risk factor; so you'd take the VaR of the yield change. In this case, assume +20 basis is the 95% %ile VaR. Or if we assume (unrealisitically) yields are normally distributed, we can replace that cell with = NORMSINV(95%) * yield volatility = 1.645 * yield volatility (e.g.) Then, for the 95% %ile worst shock to the risk factor, this TS gives as an approximation of the instrument's/portfolio's wost expected loss (because it follows for the worst exp loss on the underlying risk factor)
similarly, Taylor truncated to just one term is : option change = delta * change in stock price.
So to apply, we take the VaR of stock price change and multiply by delta
Regarding full revaluation, i don't understand the directional point, sorry, but i full revaluation is full repricing so i don't think it's important...
David
if you look at the duration +convexity example:
http://public.sheet.zoho.com/public/btzoho/sep24-taylorbond
the yield shock (+20 basis points) is the risk factor; so you'd take the VaR of the yield change. In this case, assume +20 basis is the 95% %ile VaR. Or if we assume (unrealisitically) yields are normally distributed, we can replace that cell with = NORMSINV(95%) * yield volatility = 1.645 * yield volatility (e.g.) Then, for the 95% %ile worst shock to the risk factor, this TS gives as an approximation of the instrument's/portfolio's wost expected loss (because it follows for the worst exp loss on the underlying risk factor)
similarly, Taylor truncated to just one term is : option change = delta * change in stock price.
So to apply, we take the VaR of stock price change and multiply by delta
Regarding full revaluation, i don't understand the directional point, sorry, but i full revaluation is full repricing so i don't think it's important...
David
Hi David,
1. How does one know how many basis points will be needed to represent 95% %ile VaR?
2. are you using bond as an analogy of derivative like option, so bond=derivative and yield=stock?
3. Is VAR(bond) = Bond Price - New Bond Price? But you also said "take the VaR of stock price change and multiply by delta", it sounds like VAR(opt)=VAR(stock)*delta?
4. BTW it seems you calcuate convexity using (maturity +1)*maturity /(1+yield)^2, this does not look familar, could you explain?
sorry, i am confused..
Thanks!
1. How does one know how many basis points will be needed to represent 95% %ile VaR?
2. are you using bond as an analogy of derivative like option, so bond=derivative and yield=stock?
3. Is VAR(bond) = Bond Price - New Bond Price? But you also said "take the VaR of stock price change and multiply by delta", it sounds like VAR(opt)=VAR(stock)*delta?
4. BTW it seems you calcuate convexity using (maturity +1)*maturity /(1+yield)^2, this does not look familar, could you explain?
sorry, i am confused..
Thanks!
Hi asja,
1. based on a distributional quantile. e.g., if yield volatility = 100 basis points, then 95% VaR of yield (under unrealistic normality) = 1.645 * 100 basis points
2. yes
3. VaR (bond) = Bond price - new bond price is full revaluation. The other is approximation
4. It's what you get if you take the 2nd derivative of the price formula for a zero under continuous compounding. It is a worthwhile exercise to try this (I think Jorion does it in the handbook). if that's too much trouble, notice that it's sort of near to maturity^2 which isn't a terrible approx. of convexity for a zero
David
1. based on a distributional quantile. e.g., if yield volatility = 100 basis points, then 95% VaR of yield (under unrealistic normality) = 1.645 * 100 basis points
2. yes
3. VaR (bond) = Bond price - new bond price is full revaluation. The other is approximation
4. It's what you get if you take the 2nd derivative of the price formula for a zero under continuous compounding. It is a worthwhile exercise to try this (I think Jorion does it in the handbook). if that's too much trouble, notice that it's sort of near to maturity^2 which isn't a terrible approx. of convexity for a zero
David
Hi David,
3. So your example is actually an example of 'full" revaluation, and the delta-gamma approxamation is to calcuate new derivative (or bond) price not to directly calculate the VAR, right? is it what this AIM means?
Or is there a way to directly calculate derivative's VAR using delta-gamma approxamation? what are the delta and the gamma of VAR(opt)? Is VAR(opt)=VAR(stock)*delta the delta-gamma approxamation? Here are we assuming opt and stk have the same vol?
thanks.
3. So your example is actually an example of 'full" revaluation, and the delta-gamma approxamation is to calcuate new derivative (or bond) price not to directly calculate the VAR, right? is it what this AIM means?
Or is there a way to directly calculate derivative's VAR using delta-gamma approxamation? what are the delta and the gamma of VAR(opt)? Is VAR(opt)=VAR(stock)*delta the delta-gamma approxamation? Here are we assuming opt and stk have the same vol?
thanks.
asja,
i'll have to come back to this if/when time permits (i need to record 7e...meanwhile several customers are upset i spend too much time on the forum)....so many compound questions suggest maybe we need to start at the beginning....sorry, i just don't have infinite time....David
i'll have to come back to this if/when time permits (i need to record 7e...meanwhile several customers are upset i spend too much time on the forum)....so many compound questions suggest maybe we need to start at the beginning....sorry, i just don't have infinite time....David
Hi David,
I found this: http://forum.bionicturtle.com/viewthread/1566/
Now I beleive the delta-gamma approxamation is to approximate the price of the derivative, than its VAR directly.
Thanks.
I found this: http://forum.bionicturtle.com/viewthread/1566/
Now I beleive the delta-gamma approxamation is to approximate the price of the derivative, than its VAR directly.
Thanks.
Hi ajsa,
Yes, absolutely true...delta/gamma and duration/convexity are asset-class specitic (option, bond) descriptions of Taylor and the Taylor approximation is "merely" a mathematic way to treat a non trivial function...
if the derivative can be prices as a function, symbolically, e.g., price[derivative] = function[factor1, factor2, factor3]
then the Taylor is merely giving us a way to estimate the change in price for a change in factor(s)
Taylor approximation: estimated price change = function[factor shock, factor shock, etc], or...
Full revaluation: price change = price as function [new set of factors] - price as function [current set of factors]
Both produce a loss (change in derivative price) given a set of "shocks" on underlying factor...
VaR enters by way of calibrating the magnitude of the underling shock; e.g., for bond, yield change in risk factor; for option, stock price is risk factor. So, we say, 95% VaR is the estimated price change given a 95th %ile change in the underlying yield. If +1% is the 95th %ile change in yield, then use that (for either Taylor approximation or full revaluation) to either (i) approximate or (ii) calculate specifically the loss to the instrument. I am nonchalant about derivative/security because it's not the issue; the Taylor is for nonlinear instruments...David
Yes, absolutely true...delta/gamma and duration/convexity are asset-class specitic (option, bond) descriptions of Taylor and the Taylor approximation is "merely" a mathematic way to treat a non trivial function...
if the derivative can be prices as a function, symbolically, e.g., price[derivative] = function[factor1, factor2, factor3]
then the Taylor is merely giving us a way to estimate the change in price for a change in factor(s)
Taylor approximation: estimated price change = function[factor shock, factor shock, etc], or...
Full revaluation: price change = price as function [new set of factors] - price as function [current set of factors]
Both produce a loss (change in derivative price) given a set of "shocks" on underlying factor...
VaR enters by way of calibrating the magnitude of the underling shock; e.g., for bond, yield change in risk factor; for option, stock price is risk factor. So, we say, 95% VaR is the estimated price change given a 95th %ile change in the underlying yield. If +1% is the 95th %ile change in yield, then use that (for either Taylor approximation or full revaluation) to either (i) approximate or (ii) calculate specifically the loss to the instrument. I am nonchalant about derivative/security because it's not the issue; the Taylor is for nonlinear instruments...David
WhizzKidd
Member
Hi David,
Can VaR be zero on a delta-gamma neutral portfolio? Or a delta neutral portfolio? Because we need delta is calculating it, and if it is zero does it mean VaR is zero too?
Can VaR be zero on a delta-gamma neutral portfolio? Or a delta neutral portfolio? Because we need delta is calculating it, and if it is zero does it mean VaR is zero too?
WhizzKidd
Member
Hi @David Harper CFA FRM / @ShaktiRathore
Have you had a chance to look the above?
Hi @S666, I see you posted some of these questions a while back, could you guide me on them?
Also, some final ones:
-Can VaR be zero on a delta-gamma neutral portfolio? Or a delta neutral portfolio? Because we need delta is calculating it, and if it is zero does it mean VaR is zero too?
-At a 99% VaR, VaR exceeding on 5 out of 150 days where we had to find out the probability of it exceeding on 6 out of 250 days? do you calculate it like this: 250C6*(0.01)^6*(0.99)^244
-A common trait between AIB and Barings? In the notes it's fake trades, but could it be back office loop holes?
-the replication of the 3% 1 year treasury bond using positions in a 5% 6 month maturity bond and a 4% 1 year maturity bond (all semi annual coupons) - How do you get this (below)?
The answer to that was short 0.005% of the 6 month 5% coupon and long 99.5% of the 1 year 4% coupon bond.
-Is the systematic risk in APT all the weighted sum of the Beta's? - can it ever be zero
-Between the IR, Sharp and Treynor? Which one should take the most weight when making a portfolio decision?
Have you had a chance to look the above?
Hi @S666, I see you posted some of these questions a while back, could you guide me on them?
Also, some final ones:
-Can VaR be zero on a delta-gamma neutral portfolio? Or a delta neutral portfolio? Because we need delta is calculating it, and if it is zero does it mean VaR is zero too?
-At a 99% VaR, VaR exceeding on 5 out of 150 days where we had to find out the probability of it exceeding on 6 out of 250 days? do you calculate it like this: 250C6*(0.01)^6*(0.99)^244
-A common trait between AIB and Barings? In the notes it's fake trades, but could it be back office loop holes?
-the replication of the 3% 1 year treasury bond using positions in a 5% 6 month maturity bond and a 4% 1 year maturity bond (all semi annual coupons) - How do you get this (below)?
The answer to that was short 0.005% of the 6 month 5% coupon and long 99.5% of the 1 year 4% coupon bond.
-Is the systematic risk in APT all the weighted sum of the Beta's? - can it ever be zero
-Between the IR, Sharp and Treynor? Which one should take the most weight when making a portfolio decision?
Regarding Bond Question we just need to match the cash flows.
Let weight be w on the 4% 1 year maturity bond so that Cash flow in after year 1=102*w=101.5(cash flow from 3% 1 year treasury bond after 1year) => w=101.5/102=.995
Cash flow after 6 month=.995*2+w'*102.5=1.5=>w'=(1.5-.995*2)/102.5=-0.005 thus long .995(99.5%) portion of the 4% 1 year maturity bond and short 0.005(0.5%) of the 6 month 5% coupon.
thanks
Let weight be w on the 4% 1 year maturity bond so that Cash flow in after year 1=102*w=101.5(cash flow from 3% 1 year treasury bond after 1year) => w=101.5/102=.995
Cash flow after 6 month=.995*2+w'*102.5=1.5=>w'=(1.5-.995*2)/102.5=-0.005 thus long .995(99.5%) portion of the 4% 1 year maturity bond and short 0.005(0.5%) of the 6 month 5% coupon.
thanks
Hi @HTHNIK
Re: Can VaR be zero on a delta-gamma neutral portfolio? Or a delta neutral portfolio? Because we need delta is calculating it, and if it is zero does it mean VaR is zero too?
Yes, but with important caveats. If we neutralize delta and gamma, then a simple analytical (parametric) VaR that measures only delta and gamma would return a zero VaR but it would only be an instantaneous estimate. So it has a very narrow application. We know about the greeks that when the asset price changes, the greek values (delta, gamma) change, such that the perfect neutralization cannot hold for long. Therefore, if the portfolio is static (does not rebalance), a risk metric that assumed constant delta and gamma over any realistically long horizon would fail to appreciate the risk. In practice, over a longer time horizon, we should consider the dynamic changes to the risk factors and we can further (or not) assume rebalancing. This is the point of dynamic VaR estimates. Here is a good resource on this big topic (but please do not refer to this before tomorrow's exam, this is outside the exam scope) https://forum.bionicturtle.com/resources/market-risk-analysis-value-at-risk-models-volume-iv.93/ Thanks,
Re: Can VaR be zero on a delta-gamma neutral portfolio? Or a delta neutral portfolio? Because we need delta is calculating it, and if it is zero does it mean VaR is zero too?
Yes, but with important caveats. If we neutralize delta and gamma, then a simple analytical (parametric) VaR that measures only delta and gamma would return a zero VaR but it would only be an instantaneous estimate. So it has a very narrow application. We know about the greeks that when the asset price changes, the greek values (delta, gamma) change, such that the perfect neutralization cannot hold for long. Therefore, if the portfolio is static (does not rebalance), a risk metric that assumed constant delta and gamma over any realistically long horizon would fail to appreciate the risk. In practice, over a longer time horizon, we should consider the dynamic changes to the risk factors and we can further (or not) assume rebalancing. This is the point of dynamic VaR estimates. Here is a good resource on this big topic (but please do not refer to this before tomorrow's exam, this is outside the exam scope) https://forum.bionicturtle.com/resources/market-risk-analysis-value-at-risk-models-volume-iv.93/ Thanks,
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