Monte Carlo to calculate Risk - greeks or quantiles?

[factfinder]

Is it incorrect to run a monte carlo simulation on a position and then take (for example) the 95% quantile as your risk number as the dispersion of the different paths is actually the monte carlo error? Does anyone know of any papers that discuss this?



In terms of getting greeks out of a MCS the best idea I have come across is to hold all random numbers generated in an array then reapply the same numbers at points dS above and below the original price? Does anyone know of alternative methods to this?



Thanks

[aaron]

No it is not incorrect. The dispersion of paths is not your Monte Carlo error, it is your prediction of what the markets might actually do.



You have seen the correct method for getting greeks. There are some alternatives. One good one is to reweight the paths.



For example, suppose you want the partial derivative with respect to X (where X could be underlying price, volatility, interest rates; whatever). Each path has a realized X, the average of those realized values is the current value of X (that may not be exactly true, there may be some discounting involved). Call the price P0 and the value of X, X0. Suppose I reprice my derivative weighting paths according to, say, (1 + X). Now I have a new price and a new average X, call them P' and X'. My estimate of Delta is (P' - P0) / (X' - X0).



The one thing you cannot do is run a new simulation. Simulation noise will overwhelm your measurement.

[factfinder]

Thanks Aaron

[quantie]

There are other mc methods include path-wise method and the likelihood ratio method. A good treatment of this is in Glasserman's book



The idea is to either take the differential under the payoff function or to take the derivative of the density.