# How does delta-gamma VaR work in practice and when can it be preferable to Monte-Carlo VaR?

So I will start off by just stating my understanding of the two methods through some examples and lead that into my question. Hopefully it is correct but if not then perhaps the answer to my question will become clear still.

Example 1. Portfolio with a single option.

This is the example I always see worked out for delta-gamma VaR and where it makes perfect sense to me, you approximate the change in value of the option by it's delta and gamma and assume the return of the underlying is normal and get the VaR simply by taking the return of the stock in the desired quantile and multiply by your greeks appropriately.

Example 2. Portfolio consisting of many stocks and vanilla options.

Now to use the delta-gamma approach you form a correlation matrix, C, from the stock returns and get your change in portfolio by $$W^T*C*W$$ where W is the weights of your position in each stock, being your actual stocks you have + the sum of the deltas from each option in the stocks. Add your gamma term to this and you get your change in portfolio (bonus question: I have read that the gamma term will produce "cross gammas", i.e $$\Gamma_{i,j} = \frac{\delta^2 V}{\delta S_i \delta S_j}$$. Is the only way this can be non-zero if you have an option dependent on two different stocks, or are there other situations?)

Now my first question is how do you make use of this in practice? In example 1 we knew which stock return would produce the 5% worst outcome, simply the 5% worst stock return. Here when we have a complex portfolio that may have lots of different puts and calls, how can we know which outcome will produce 5% worst outcome? What I imagine is that you would simulate the stock returns 10,000 times and find the outcome from that, but at that point what was the point of the delta-gamma approximation? Couldn't you just as well plug the stock returns into your Black-Schooles formula and get the actual price if you're simulating the price either way? Is there a way to get around having to do a Monte-Carlo simulation of the stock prices?

Example 3. Portfolio consisting of exotic options.

Here I can appreciate the need if the actual evaluation of the option is time consuming, so you would perform your Monte carlo simulation like in example 2. But is the delta-gamma approximation still accurate for exotic options like barriers or other path dependent options?

Summing up:

Am I on the right track with this thinking? Would you ever want to use Delta-Gamma in example 2? From my experience the delta-gamma method seems very popular in practice, and I'm also imagining the situation in example 2 is the most common one.

As you have already noted, the Delta-Gamma (DG) approximation, and its 'brother', the Delta-Gamma-Normal (DGN) are used to approximate the distribution of future portfolio returns, e.g. for the value at risk.

At this point, let us have a look at the DG-approximation to the theoretical PnL of a portfolio of financial products as a function of a move of market factors, only (i.e. no time decay, or $$\Theta$$ effect). For simplicity, we will suppress notional and position amounts and write:

\begin{align} \tag{1} \label{1} PnL&\equiv PV(\mathbf{S}+\Delta\mathbf{S})-PV(\mathbf{S})\\ &\approx \nabla\left(\Delta\mathbf{S}\right)+\frac{1}{2}\left(\Delta\mathbf{S}\right)^T\Gamma\left(\Delta\mathbf{S} \right)\\ &=\sum_j\nabla_{j}\left(\Delta \mathbf{S}\right)_j+\frac{1}{2}\sum_j\sum_k\left(\Delta \mathbf{S}\right)_j\left(\Delta \mathbf{S}\right)_k\Gamma_{j,k} \end{align} where $$\Delta\mathbf{S}$$ is the vector of market factor movements, $$\nabla$$ is the vector of first order derivatives of all financial instruments in your portfolio with respect to (all) valuation factors (the Delta vector), and $$\Gamma$$ is the Hessian matrix of second order derivatives, also called the Gamma matrix of your portfolio. You can always separate these into instrument level contributions, but that is not useful here. Finally, $$j$$ and $$k$$ are indices that help us navigate across the FO and SO 'contributions' to the PnL approximation.

Let us simplify this to a setting with one valuation factor. For simplicity, let $$a$$ denote the delta sensitivity (FO derivative) and $$b$$ be the gamma (SO), and $$x$$ be the change in the underlying risk factor: $$\label{2} \tag{2} PnL\approx ax + \frac{1}{2}bx^2$$

Using \eqref{1} and \eqref{2}, we are now able to tackle your various questions.

Regarding your first example and question: This is only partially correct. Only if your first and second order contributions are of the same sign can you simply 'read off' the desired quantile from your return distribution (which can be from a normal variate, the historical return series, or any other distribution).

Regarding your second example and question: I think that \eqref{1} gives us a basis for this discussion. If we assume the valuation factors in $$\Delta\mathbf{S}$$ to be multivariate normally distributed with covariance matrix $$\Sigma$$ and mean return of zero, we unfortunately cannot simply aggregate delta and gamma and multiply by some correlation matrix. The reason for this is twofold: First, as we discussed in your example 1 above, we need to keep track of signs, and second, as you already noted in your question, there is this cross gamma piece.

At this point, let me sketch the way to a DGN distribution. We start with the second line in \eqref{1} and successively, apply the following steps:

1. Decorrelate the covariance matrix
2. Diagonalise the (transformed) Gamma matrix
3. Split the result into constants, normal variates, and quadratic normal variates

As a result, we have transformed the sum of squares of (correlated) random normals into a sum of independent normals and squared normals. Then we

1. apply a suitable transformation method, e.g. the characteristic function transform, moment generating function transform or others to either come up with characteristics of the portfolio distribution or with an approximation to its quantile.

Now to your questions around application and applicability: Yes, the method is a simple approximation that happily survived the 90ies; your comments are technically on the right track. Personally, I see it applied in the industry at various points: Either for explanatory purposes with in risk controlling functions, where the user has no resource to the valuation machinery, or for íntraday / short term risk approximations in the front office systems, where the risk factor shock is usually of small magnitude and constantly revaluation would be too expensive.

Also, the DGN method is the 'spiritual ancestor' to the risk measurement methods introduced with the new CRR (i.e. the Fundamental Review of the Trading Book piece, FRTB) and also some other risk models.

• I am happy to elaborate on follow up questions. Mar 31, 2020 at 7:32
• Hey, thank you very much for the response, I'll happily take you up on that. I still don't quite understand the cross gamma term, would it not always be zero apart from rainbow options? Regarding your sketch 1. when you decorrelate the matrix, do you mean extracting a multivariate normal vector from the matrix by e.g. using a Cholezky decomposition? I'm not too familiar with 2 and 3 so perhaps I need to read up on that some more. 4. Does doing this then mean that you dont have to do a monte-carlo simulation? The transformed function will tell us the correct quantile? Mar 31, 2020 at 11:17
• After thinking about it a little more I realized that these cross gammas can probably be derivatives of more than just the underlyings, e.g cross derivative of the underlying and an exchange rate? In that case I see how there would be cross gammas even for plain vanilla options. Mar 31, 2020 at 13:23
• Yes: you will have cross gammas for, eg, underlying and implied volatility. But for exotic options, these could of course be quite higher; and sometimes practitioners simply assume zero cross gammas -> this simplifies the calculation, although we then re-introduce another error. Mar 31, 2020 at 20:34