 lynn
Topic Author
Posts: 35
Joined: September 13th, 2002, 8:02 am

### Value at Risk for portfolios with linear and non-linear products

I have a portfolio containing shares, equity futures and equity options. I calculate the VAR for shares and equity futures using the simple variance covariance VaR method but calculate the VAR for options using delta-gamma VaR. Would it be wrong to then sum up the VaR metric for each category to obtain the portfolio VaR? ppauper
Posts: 70239
Joined: November 15th, 2001, 1:29 pm

### Value at Risk for portfolios with linear and non-linear products

Last edited by ppauper on December 19th, 2004, 11:00 pm, edited 1 time in total. lynn
Topic Author
Posts: 35
Joined: September 13th, 2002, 8:02 am

### Value at Risk for portfolios with linear and non-linear products

Must the volatility used in both the variance covariance method and delta gamma approximation method be the same? Gmike2000
Posts: 801
Joined: September 25th, 2003, 9:49 pm

### Value at Risk for portfolios with linear and non-linear products

why cant you just set the gamma for stocks and futures to zero and shove them all into the delta gamma model? battant
Posts: 11
Joined: April 10th, 2004, 12:07 pm

### Value at Risk for portfolios with linear and non-linear products

if you sum up the different Var, you are under the conditions that the correlations are equal to one and so it's a conservatism approach.So you can use different methods in order to construct your total VaR : - use the historical correlation between your assets- use the theory of correlation breakdown (in this case, you can use the stress tests and so cf Bale 2 or some econometrics models
Last edited by battant on August 12th, 2004, 10:00 pm, edited 1 time in total. lynn
Topic Author
Posts: 35
Joined: September 13th, 2002, 8:02 am

### Value at Risk for portfolios with linear and non-linear products

Can i continue to use historical volatility for linear and non-linear products. Ziggy
Posts: 149
Joined: January 27th, 2002, 10:59 pm

### Value at Risk for portfolios with linear and non-linear products

I assume you are calculating VaR to get the best estimate of true risk, using a confidence interval.1) One idea is to use implied volatilities from the market to calculate VaR as usually this is superiour to historical vol. This might is a good way to go, since you get consistant volatilities on both the underlying and the options in this case which reflect the future risk much better than historical estimates2) Gamma is a greek which measures sensitivity to infinetly small market movements. When calculating VaR you are really looking at LARGE movements. For a static porfolio, the Delta-Gamma approach only works for VaR with a infinetly small confidence interval. Obviously if you have active delta rebalancing the Delta-Gamma approch works better, but you have to keep your strategy in mind when calculating VaR figures for a portfolio being rebalanced.If the portfolio is static (no relabancing), avoid delta-gamma, calculate option values for the [underlying =max 95%] and [underlying = min95%] and report VaR as Up-VaR and Down-VaR, since the measure will not be symmetric 3) I would not underestimate the power of higher order greeks in any risk calculations. If you are truely looking for a 95% VaR of a portfolio that includes options you should at least estimate the effects of changes in implied volatility, particularly if you are short a lot of wings. There you might have to look at the correlation between implied vol surface and spot levels, as well as what happens when/if this correlation breaks down.4) If the options do have full vol-smile information from the market, a smart idea is to estimate the "implied distribution" of the underlying which fits the options prices and using that distribution on both the options and underlying to get a consistant estimate between asset classes. This also has the added benefit of conforming to market expectations to some extent, and include some of the stochastic vol premiums embedded into the smile.PS: I'm not a VaR expert and not happy with VaR as a risk measure. Never use it on non-linear products unless you understand and all the shortfalls of VaR and know exactly what you are doing Best regards,Z fearless
Posts: 94
Joined: October 9th, 2003, 6:45 pm

### Value at Risk for portfolios with linear and non-linear products

Quote Can i continue to use historical volatility for linear and non-linear products.I am not sure what exactly you are referring to when you say linear but you only need volatility for options and non-linear terms. By linear I mean straight equity, i.e. stocks, futures on stocks, tracking stocks, ETFs, and indices. Nonetheless, since do have some options in your portfolio, you need to use volatility to value this portion of your portfolio. As stated earlier, for a given senario, i.e. values of underlying and vols, you have to determine the value of all the securities in your portfolio and add them up to figure the portfolio value for that given senario. Come up with 100 such values and take the 5th worse case and that is your 95% VaR.I am not VaR expert either, but this how I have seen it done. riskguru
Posts: 79
Joined: August 11th, 2004, 4:24 pm

### Value at Risk for portfolios with linear and non-linear products

A more fundamental approach to the question (I hope):The risks in your portfolio include the following:1. Delta risk (impacts all the instruments you described)2. Gamma/Non linearity risk: impacts the options3. Vega/implied volatility risk: Impacts the options4. Rho/theta: Impacts the options (if you really want to include them as well, they typically tend to be small numbers).To do a VaR calculation, you basically need the underlying stochastic characteristics of equity prices and implied vols (ignoring rates etc). This can be historical time series (historical simulation) or a covariance matrix. The draws from this distribution need to be used to reprice your portfolio. This is easy for futures and stocks, but (typically) requires a Taylor series approxmation (using the option greeks) for options. So the change in the value of the option would be approximately delta * Stock price * stock return + (1/2) * gamma * Stock price ^2 * stock return ^2 + vega * implied vol * vol return etc. As pointed out earlier in this thread, everything except the first term is zero for the 'linear instruments'. Having said that, one obviously needs to use a consistent set of properties for the stock return volatility across all instruments (i.e I cannot use different volatility assumptions for the stock return series influencing the impact of the delta on my intel option vs. my short delta stock hedge!). The use of historical vs. "better" measures of predicted volatility (such as implied vols) is a different subject altogether and not directly related to the original question on using different vols for the normal vs delta-gamma VaR.Finally, all that this gives you is a 'local approach' based risk measure, which can always be misleading for a non-linear book. Hence all the caveats about the added benefits of stress testing the portfolio etc. obviously apply.Hope this helps.  