Comparison of VaR methods

[Eveline]

Hi David,

Pls help me with this other one:

In early 2000, a risk manager calculates the VaR for a technology stock fund based on the last three years of data. The strategy of the fund is to buy stocks and write out-of-the-money puts. The manager needs to compute VaR. Which of the following methods would yield results that are least representative of the risks inherent in the portfolio?
a) historical simulation with full repricing
b) delta-normal VaR assuming zero drift
c) Monte Carlo style VaR assuming zero drift with full repricing
d) Historical simulation using delta-equivalents for all positions

Ans: D

Also, can you explain what is meant by delta-normal VaR assuming zero drift? And historical simulation using delta-equivalents?

Thanks a lot for your help!

 

[David Harper CFA FRM]

Hi Eveline

I like this question...a good meditation on the approaches...(although D is a difficult)

First, "full repricing" should generally connote something like "most accurate"

In regard to zero drift, our general analytical (parametric linear) VaR is given by:
Jorion's Absolute VaR (i.e., relative to initial value or zero): - return*Time + volatility*deviate*SQRT(T)
The first term is drift: positive expected return will offset the worst expected loss,
such that this Absolute VaR is less than:
Relative VaR (i.e., relative to final expected value) which omits the drift: volatility*deviate*SQRT(T)

...Absolute VaR is fine/preferred for longer periods *if* the drift is representative; if the drift overstates, you can see VaR will be understated
...both assumptions of "zero drift" are therefore *conservative* vis a vis losses and, therefore, cannot be bad assumptions (at worst, they upwardly bias the VaR but only by the drift, not deadly for short horizons)

HS using delta equivalents, in this problem, is:

1. take the historical series of stock returns, and
2. Rather than simulate the put options directly, simulate the put option values by using the delta approximation: change in option value = delta * change in underlying

...writing out of the money puts: interesting b/c many have used this metaphor for the crisis; e.g., writing CDS on senior CDO tranches ~ writing OTM puts
...the delta of a deep out of money put is near to (converges on) zero. Can you tell me why, intuitively?

so, the use of delta approximation for deep OTM put gives very little change in option (risk) for local changes in underlying; i.e., won't capture tail risk especially if historical dataset does not contain heavy tails.

So, i get some useful ideas out of this:

• Deep OTM puts have delta ~ 0. Why? This has implication on delta-normal VaR (or as Linda Allen says, Taylor Series not appropriate when extreme non-linearities involved)
• The relevance of historical dataset matters because it determines drift (e.g., was it during a bull runup so that absolute VaR is downwardly biased?)

David

 

[hsuwang]

Hello David,
so if d) includes an extra term "assuming zero drift", how would it compare to b) delta-normal VaR assuming zero drift?
I'm guessing they would be similar since delta normal VaR is using linear approximation as well where delta is close to 0.

Thanks!

b) delta-normal VaR assuming zero drift
d) Historical simulation using delta-equivalents for all positions

 

[ajsa]

Hi David,

A similar question.. Could you confim the answer of the below question should be b not a? Not very sure..

Thanks.

EXAMPLE 15.2: FRM EXAM 2002—QUESTION 38
If you use delta-VAR for a portfolio of options, which of the following statements
is always correct?
a. It necessarily understates the VaR because it uses a linear approximation.
b. It can sometimes overstate the VaR.
c. It performs most poorly for a portfolio of deep-in-the money options.
d. It performs most poorly for a portfolio of deep-out-of-the money
options.

 

[David Harper CFA FRM]

@Jack: that's a really good comparison, but after adding the "assume zero drift" they would not be similar. They still represent two different VaR approaches. For simplicity sake, assume we are using a hilariously short data window of five *sorted* returns: {r1, r2, r3, r4, r5}.

The delta-normal VaR assumes the risk factor is normally distributed, it is a "parametric" approach. So it uses the dataset of returns, {r-5, r-4....}, to compute a standard deviation, then estimates VaR based on the standard deviation. Maybe the standard deviation of our data series is +1%. So then the delta-normal VaR = normal deviate (e.g., 2.33) * 1% volatlity. Note how this "imposes" (unfairly!) the distributional assumption on the data; the data isn't normal but we treat it so.

For historical simulation, we'd use the raw dataset "as is" ... and "as is" the dataset has no parametric distribution ... e.g., the VaR is somewhere between r4 and r5, but we can still get a VaR quantile ... this is the key different between HS and parametric VaR. Both may get a 95th %ile (a quantile) but:

* HS: worst expected loss in underlying risk factor (stock, in this case) = percentile (array, 95%); i.e., no parametric distribution
* delta-normal: worst = deviate * volatility; parametric distribution generated by the data, but once we have it, we're done with the data.

@ajsa:
if it said a portfolio of "long options," then because long call/puts always have positive gamma, the delta-based VaR would always overstate (i.e., just like duration for a bond, the line implies a loss that is greater than the actual loss due to the curvature).
if it said a portfolio fo "short options" then b/c short always have negative gamma, delta-based VaR would always understate.
Ergo, it can either under or overstate depending on long/short

btw, why are (c) and (d) not true; i.e., why do deep OTM/ITM options fare better with delta-VaR?

David

 

[ajsa]

I think it is because deep out the money or in the money opt has very small gamma..

Thanks David!

 

[David Harper CFA FRM]

Exactly! Gamma ~ 0 by *definition* means delta isn't changing, and it's the changing delta that messes up the delta approximation...David

 

[hp97]

Hello David,

Could you please explain once again as to why option A is incorrect?

 

[David Harper CFA FRM]

Hi @hp97 The OP question above is from the FRM handbook and it is approximately 15 years old. In my opinion, it is not a good question for the "modern" FRM. I located it in the FRM handbook and below is a copy of the answer:

"d) Because the portfolio has options, methods a) or c) based on full repricing would be appropriate. Next, recall that technology stocks have had a big increase in price until March 2000. From 1996 to 1999, the Nasdaq index went from 1300 to 4000. This creates a positive drift in the series of returns. So, historical simulation without an adjustment for this drift would bias the simulated returns upward, thereby underestimating VAR."

... so you can see how the answer seems to be arguing that our first concern should be the written options. Written options have non-linear (i.e. short gamma) exposure which will not be measured by delta-only approaches, since they only look at the linear approximation (including delta-normal, which means that the risk factor is presumed to be normally distributed, and the option exposure is measured only in terms of its delta). So based on the short option position, we can immediately know that the answer should be (b) or (d) given we are looking for the least effective approach.

So choice (A) seems to be incorrect primarily because it utilizes full-repricing, aka full-valuation (rather than delta-based approximation such as delta-normal). But, as I look at the question freshly, I think it's a weak question: to me, it's not obvious why (b) is not the answer as "delta-normal VaR assuming zero drift" appears to be an inferior approach here! I hope that's helpful,

 

[hp97]

Thanks David.

I'm currently a student of the Associate Professional Risk Manager course, which essentially could be thought of as a foundation towards entering PRM. However I'm currently struggling a bit in analaysing how the questions might seem to appear in the examination sticking to the MCQ format, as there seem to be no resources online that I could strictly refer to only in terms of APRM. I would appreciate any suggestions and tips from your end in terms of the experience you've had in examining the past papers of PRM as a few topics are similar to both the courses. I would want to understand the nature of the questions that could be asked as well as if I needed to rely on external sources for preparing too.

I look forward to hearing from you soon as I intend to give off my examination in the next one month.

Regards