Value at Risk Monte-Carlo using Generalized Pareto Distribution(GPD)

Question

I have created a VBA program to calculate VaR by using Monte Carlo, I have simulated Brownian Motion. This method might be ok for 100% equity portfolio, but let's say this portfolio may have fixed income/alternative investment/derivative etc and that composition percentage may or may not be known to me. In that case as a generic model can I use Generalized Pareto Distribution(GPD) for Monte Carlo ? Usage will not be for accurate Market Risk valuation purpose but more of a Investment Performance Risk Analytics report. I got a link for GPD in excel at below link. http://www.quantitativeskills.com/sisa/rojo/pareto.xls But in that case what/how will be my stochastic equation for price/return modelling ? Simply randomizing x0 or alpha will not do I believe. My current VBA code is as below for easy reference:

    Function ValueAtRiskMC(confidence, horizon, RiskFree, StDv, StockValue)

Dim i As Integer

Dim stockReturn(1 To 10000) As Double
    'start of monte carlo loop
    For i = 1 To 10000

    'According to the Black Scholes model, the price path of stocks is defined by
    'the stochastic partial differential equation dS = (r - q -1/2sigma^2)dt + sigma dz
    'where dz is a standard Brownian motion, defined by dz = epsilon * sqrt(dt)
    'where epsilon is a standard normal random variable; dS is the change in stock price
    'r is the risk-free interest rate, q is the dividend of the stock,
    'sigma is the volatility of the stock.

    'The model implies that dS/S follows a normal distribution with mean
    'r - q -1/2sigma^2, and standard deviation sigma * epsilon * sqrt(dt))
    'As such the price at time 0 < t <= T is given by
    'St = S0 * exp( (r – q - ½ sigma^2) dt + sigma * epsilon * sqrt(dt))

    'As we are ignoring dividends etc here so
    'below line is for geometric brownian motion
    stockReturn(i) = Exp((RiskFree - 0.5 * StDv ^ 2) + StDv * Application.NormInv(Rnd(), 0, 1)) - 1
    Next i
    'end of monte carlo loop

    ValueAtRiskMC = StockValue * (-(horizon) ^ 0.5) * Application.Percentile(stockReturn, 1 - confidence)

End Function

Answer 1

the risk neutral drift is needed for pricing of derivatives. For a $100\%$ equity portfolio you can take the real world drift - sometimes a good guess is a drift of zero.

For fixed-income you could do the same and might need more sophistication for the variance term. If you have short-dated bonds then you will need a special model for the pull-to-par.

For derivatives: for options the best is modelling the spot price (and other parameters) and then do full valuation. Or as an approximation you do Delta-Gamma. But again: to cover many asset classes in one application is a big project (that's why there are those vendors of commercial solutions).

Why do you want to use GPD? If you want to use GPD to model fat tails then you could use it for all asset classes.

If you want to model fat tails you can use Levy models (t-distributed returns, normal-inverse Gaussian mixtures).

To find some key-words to search for you can have a look here or you brows through this book.

Edit: if you want to use the GPD anyways then looking at the wikipedia page you find an algorithm for random number generation of this distribution. E.g. in the special case where $\xi=0$ you draw a uniform $U$ and calculate $$ X = \mu - \sigma \log(U) $$ and $X$ has the desired distribution. The above formula comes from the random number generation by inversion of the CDF which is a general approach. Of course you have to fit the parameters of the GPD. Does this help?