The third stage of Monte Carlo VaR is so time-consuming that it is impossible to perform thousands of revaluations of the portfolio within a realistic time frame

3. Apply the pricing model to the simulated correlated h-day risk factor returns. This will give one simulated / -day portfolio return, and multiplying this by the current price gives a single / -day profit or loss figure.

These three stages are repeated several thousand times, to obtain many P&L figures and thereby build an / -day P&L distribution for the portfolio. How many times will depend on the number of risk factors, that is, the dimension of the hypercube. Figure 9.6 illustrates why. If only a few hundred random points are taken from the cube it is quite possible that they will, by chance, be concentrated in one part of the cube ... and the next time, by chance, the points could be concentrated in another area of the cube, in which case the Monte Carlo VaR measures will vary enormously each time they are recalculated.

To increase robustness of Monte Carlo VaR estimates one could, of course, take many thousands of points in the cube each time, so that sampling errors are reduced. But then the time taken for all these portfolio revaluations would be completely impractical. Jamshidian and Zhu (1997) show how principal components analysis can be used to gain enormous increases in computational efficiency in Monte Carlo VaR calculations. It is also standard to employ advanced sampling methods, such as deterministic sequences, so that the cube will be covered in a representative manner by fewer samples. But even with the aid of advanced sampling techniques or principal component analysis it will usually be necessary to employ several thousands of points in the sample.

Stages 1 and 2 take hardly any time, and if the third stage is also not too time-consuming there will be no impediment to using thousands of simulations to obtain accurate and robust VaR estimates. However, it is often the case that the third stage of Monte Carlo VaR is so time-consuming that it is impossible to perform thousands of revaluations of the portfolio within a realistic time frame. In that case approximate pricing functions will need to be employed, including analytic forms for complex options and Taylor approximations to portfolio value changes (§9.4.3).

Figure 9.6 Sampling the hypercube with too few points.

The main advantages of Monte Carlo simulation are that it is widely applicable, it is able to capture path-dependent behaviour of complex products and, since simulation techniques are often already employed in the front office, it is operationally efficient to employ these models in VaR calculation also. The main disadvantages of Monte Carlo VaR are the need to use a covariance matrix, which introduces another well-known source of error, and the fact that there is considerable trade-off between speed and accuracy. Often accuracy will need to be compromised in order to complete the VaR calculations within a realistic time frame.

9.4.3 Delta-Gamma Approximations

The Monte Carlo VaR spreadsheet calculates the VaR for an option portfolio using full (Black-Scholes) valuation and also using a number of different Taylor approximations based on the option Greeks.

Delta approximations for options and complex instruments are, of course, very convenient. With a delta approximation the portfolio value changes are just a linear function of the underlying asset price changes. Delta approximations are simple, but the VaR obtained in this way tends to be very inaccurate, in any portfolio with gamma or convexity effects. The delta approximation is a local approximation, holding good only for small changes in the underlying price. For VaR measures we need to consider the tail of the P&L distribution, that is, the effect of large price changes. Figure 9.7 explains why the VaR approximation errors from a delta-only representation tend to be rather too large.

The delta-gamma-theta representation is a second-order Taylor expansion of the portfolio value change with respect to the changes in the underlying prices (denoted by the vector AS). That is,

AP « QAt + 8AS + \ ASTAS,

(9.10)

Delta Approximation

P(S)

Portfolio Value Function

Error at Sr

Figure 9.7 Error from delta-only approximation.

where 0 is the partial derivative of the portfolio value with respect to time, 8 is a vector of first partial derivatives of the portfolio value with respect to the components of S, and is the Hessian matrix of second partial derivatives with respect to S. Without the first term, (9.10) is the delta-gamma representation.

Finite differences are normally employed to compute the option sensitivities. For example, the option delta and gamma may be computed using central differences by valuing the portfolio at the current price S, giving the current value P(S) and for small perturbations above and below this price, at S + e and 5-e for each underlying S in the vector S. Then the delta and gamma corresponding to S are approximated by:

[P(S + e)-P(S- e)]/2e

(9.11)

[P(S + 8) - 2P(S) + P(S - 8)]/82.

(9.12)

The accuracy of the delta-gamma approximation can be tested in the usual way, by comparing the delta-gamma value changes with the full valuation value changes over a suitable historic period. If the approximation is not working sufficiently well it is normal to include volatility as another risk factor, in the delta-gamma-vega-theta representation

AP % QAt + 8AS + { AS AS + vAcr,

(9.13)

where v is a vector of first partial derivatives of the portfolio value with respect to the volatility of each component of S, this vector of volatilities being denoted o\18 Theta and the components of the vega vector are also computed using first finite differences. The procedure is the same as with the delta components, only this time using small perturbations on time and the current value of the volatility rather than the underlying price. More details can be seen on the CD.

The delta-gamma, delta-vega and delta-gamma-vega approximations are commonly employed as an approximate pricing function when full revaluation is considered too time-consuming

The delta-gamma, delta-vega and delta-gamma-vega approximations to value changes in a portfolio have an important role to play in simulation VaR models. They are commonly employed as an approximate pricing function when full revaluation is considered too time-consuming. Glasserman et al. (2001) show how to use them in Monte Carlo VaR, not as a pricing approximation but as a guide to sampling, with a combination of importance sampling and stratified sampling methods. Rouvinez (1997) shows how the delta-gamma approximation can be used for an analytic approximation to the VaR for an options portfolio.

This assumes that volatility changes are independent of price changes, which is only a rough approximation (§23).