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tw
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Joined: May 10th, 2002, 3:30 pm

Asymptotic VAR?

May 17th, 2002, 1:03 pm

[I'm caveating this by stating that I'm not a risk manager: myinterest in the calculation of VAR is mostly academic]I was playing around with formulae for VAR recently and wonderedif it was either an accomplished calculation or an area of active researchto calculate an approximate Value at Risk using asymptotic analysis?It would seem that a full calculation of VAR for multi-asset portfolioswith derivatives valued using Black-Scholes formulae, is basically a multidimensional integral over a simply connected range. Whilst thisisn't a trivial task, there are results in the asymptotic analysis literaturefor doing this, which would circumvent the need for Monte Carlo calculation (always a good thing in my opinion), and give much quicker VARresults.Before I devote any more hours to it, I just thought I ask and see if I'm reinventing the wheel, or it there are commonly known trapsthat make the whole thing unworkable.cheers,Tom
 
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Aaron
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Asymptotic VAR?

May 17th, 2002, 1:55 pm

I'm not sure at what point you are proposing to apply asymptotic methods.One approach to estimating VaR is to simulate a one-day price movement for every asset in the portfolio and add them up to a firm P&L. Repeat that, say, 10,000 times, and take the 100th biggest loss. This is your 99%, one-day VaR.Another approach says with lots of assets, and enough independence among them, only mean and standard deviation matter. So all I care about is the mean return on each asset and its Beta with the total portfolio P&L. So I can estimate the mean and standard deviation of the one-day portfolio P&L and set VaR at 2.33*SD - Mean. I think this is what you have in mind as an asymptotic approximation, although no doubt you are interested in more sophisticated methods than alpha and Beta.Neither one of these pure approaches is adequate for practical VaR computation in large portfolios (say 1,000,000 positions in 100,000 different assets of 1,000 different types). There's no practical way to simulate 1,000,000 different assets, you don't know what all the statistical relationshps are. On the other hand, there's no way to determine Beta (or other price relationships) for 1,000 different asset types on one central factor.Therefore, the portfolio is broken up into sectors such that within each sector all assets can be valued based on a small number of factors, possibly with some idiosyncratic risk (this is significant, even a very large multinational bank will have significant VaR due to exposure to a few companies or market factors). Asymptotic approximations are used if appropriate (such as for large-cap US equities), but do not work in many markets with lumpier risk.Once you know the distribution of P&L within market sector, you can simulate each sector using the first method and estimate an overall VaR. Generally you will have a sparse covariance matrix for this step. Some sectors (like FX and Interest Rates, or Equities and Equity Implied Volatility) will have stable correlations, but for most pairs of sectors you assume zero (not because it is, but because it isn't stable enough to measure). Asymptotic methods are not appropriate because the number of sectors, 10 to 100, is not large.
Last edited by Aaron on May 16th, 2002, 10:00 pm, edited 1 time in total.
 
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zq
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Asymptotic VAR?

May 17th, 2002, 3:07 pm

Hi Tom,I remember seeing once a paper by Luis Seco (U Toronto) and someone else about harmonic analysis applied to the calculation of VAR. I think they get approximate expressions by careful integration. I heard him lecture about it and I remember he mentionned spherical symmetries being important.I hope this helps.zq
 
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tw
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Joined: May 10th, 2002, 3:30 pm

Asymptotic VAR?

May 18th, 2002, 8:22 am

Aaron,Thanks for the reply. The nature of asymptotic approximation I had in mind was something along the lines of the following:Suppose you have a portfolio consisting of several underlyingsand derivatives priced off of those underlyings, the change in P&L would be something likeD(P&L) = a1*Dp1 +a2 *Dp2 + .... + b1 *(delta1*Dp1 + 0.5*Gamma1*Dp1^2 + ...) + b2 *(delta2*Dp2 + 0.5*Gamma2*Dp2^2 + ...) +...where you a1, a2.. are the positions in thge underlyings, b1, b2...are the equivalent positions in the derivatives Dp1, Dp2, ..are the changes in underlings prices and the Greeks should be prettyobvious.As I understand it, the object of the exercise is to take an assumedjoint distribution for Dp1, Dp2, ... over the particular time scale of interest and derive a distribution for D(P&L) and read of the particular percentile of the distribution required.The way to get the D(P&L) from the joint Dp distribution is to do a multipleintegral over it, using the range where the Dp's are constrained to a particular D(P&L). It's this integral (a multiple integral of Laplace type;the smoothness and monotonicity of many common option pricing formulaemake the range of integration simple) I was thinking of using an asymptotic approximation for. The theory of these approximations isn't pretty, but I don't know if it's unworkable or not.Thanks,Tom
 
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Athletico
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Asymptotic VAR?

May 18th, 2002, 4:24 pm

Tom,The distribution of D(P&L) is non-normal when you incorporate gamma -- this does indeed complicate the problem. The approximate distribution you are after involves the sum of normals and chi-squareds -- an excellent reference is:VAR: One Step BeyondAlso check out JPMorgan's website -- there are several papers that deal with generalizing VaR for derivatives (I remember one called "Delta-Gamma VaR Four Ways", or something like that).
 
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Aaron
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Joined: July 23rd, 2001, 3:46 pm

Asymptotic VAR?

May 20th, 2002, 12:13 am

Most VaR computations do not use a gamma*Dp1^2 term, or other higher order terms. Instead they use a piecewise linear input of derivative price versus underlying price (really, market factor price, since there are usually fewer market factors than underlyings). The problem is that many exotic and far-from-the-money positions are not well-captured unless you use an unreasonably high-order approximation. A Taylor expansion at the current price is not good enough, although it would save a lot of effort if it were.Your idea might work if you could input the value of derivative positions as functions of underlyings. Then you could compute an asymptotic P&L distribution. That might be practical for a hedge fund or desk that traded a limited number of types of products. I don't think it would work for a multinational bank trading lots of different types of derivatives and introducing new products frequently.