1.1 Introduction


1.1.1 The Practical Need

Financial institutions are facing the important task of estimating and controlling their exposure to market risk, which is caused by changes in prices of equities, commodities, exchange rates and interest rates. A new chapter of risk management was opened when the Basel Committee on Banking Supervision proposed that banks may use internal models for estimating their market risk (Basel Committee on Banking Supervision; 1995). Its implementation into national laws around 1998 allowed banks to not only compete in the innovation of financial products but also in the innovation of risk management methodology. Measurement of market risk has focused on a metric called Value at Risk (VaR). VaR quantifies the maximal amount that may be lost in a portfolio over a given period of time, at a certain confidence level. Statistically speaking, the VaR of a portfolio is the quantile of the distribution of that portfolio's loss over a specified time interval, at a given probability level.


The implementation of a firm-wide risk management system is a tremendous job. The biggest challenge for many institutions is to implement interfaces to all the different front-office systems, back-office systems and databases (potentially running on different operating systems and being distributed all over the world), in order to get the portfolio positions and historical market data into a centralized risk management framework. This is a software engineering problem. The second challenge is to use the computed VaR numbers to actually control risk and to build an atmosphere where the risk management system is accepted by all participants. This is an organizational and social problem. The methodological question how risk should be modeled and approximated is - in terms of the cost of implementation - a smaller one. In terms of importance, however, it is a crucial question. A non-adequate VaR-methodology can jeopardize all the other efforts to build a risk management system. See (Jorion; 2000) for more on the general aspects of risk management in financial institutions.


1.1.2 Statistical Modeling for VaR

VaR methodologies can be classified in terms of statistical modeling decisions and approximation decisions. Once the statistical model and the estimation procedure is specified, it is a purely numerical problem to compute or approximate the Value at Risk. The modeling decisions are:

  1. Which risk factors to include. This mainly depends on a banks' business (portfolio). But it may also depend on the availability of historical data. If data for a certain contract is not available or the quality is not sufficient, a related risk factor with better historical data may be used. For smaller stock portfolios it is customary to include each stock itself as a risk factor. For larger stock portfolios, only country or sector indexes are taken as the risk factors (Longerstaey; 1996). Bonds and interest rate derivatives are commonly assumed to depend on a fixed set of interest rates at key maturities. The value of options is usually assumed to depend on implied volatility (at certain key strikes and maturities) as well as on everything the underlying depends on.
  2. How to model security prices as functions of risk factors, which is usually called ``the mapping''. If $ X^{i}_{t}$ denotes the log return of stock $ i$ over the time interval $ [t-1,t]$, i.e., $ X^{i}_{t}=\log(S^{i}_{t})-\log(S^{i}_{t-1})$, then the change in the value of a portfolio containing one stock $ i$ is

    $\displaystyle \Delta S_{t}^{i} = S^{i}_{t-1}(e^{X^{i}_{t}}-1),
$

    where $ S^{i}_{t}$ denotes the price of stock $ i$ at time $ t$. Bonds are first decomposed into a portfolio of zero bonds. Zero bonds are assumed to depend on the two key interest rates with the closest maturities. How to do the interpolation is actually not as trivial as it may seem, as demonstrated by Mina and Ulmer (1999). Similar issues arise in the interpolation of implied volatilities.
  3. What stochastic properties to assume for the dynamics of the risk factors $ X_{t}$. The basic benchmark model for stocks is to assume that logarithmic stock returns are joint normal (cross-sectionally) and independent in time. Similar assumptions for other risk factors are that changes in the logarithm of zero-bond yields, changes in log exchange rates, and changes in the logarithm of implied volatilities are all independent in time and joint normally distributed.
  4. How to estimate the model parameters from the historical data. The usual statistical approach is to define the model and then look for estimators that have certain optimality criteria. In the basic benchmark model the minimal-variance unbiased estimator of the covariance matrix $ \Sigma$ of risk factors $ X_{t}$ is the ``rectangular moving average''

    $\displaystyle \hat{\Sigma}=\frac{1}{T-1} \sum_{t=1}^{T} (X_{t}-\mu)(X_{t}-\mu)^{\top}
$

    (with $ \mu \stackrel{\mathrm{def}}{=}E[X_{t}]$). An alternative route is to first specify an estimator and then look for a model in which that estimator has certain optimality properties. The exponential moving average

    $\displaystyle \hat{\Sigma}_{T}= (e^{\lambda}-1)\sum_{t=-\infty}^{T-1}
e^{-\lambda(T-t)}(X_{t}-\mu)(X_{t}-\mu)^{\top}
$

    can be interpreted as an efficient estimator of the conditional covariance matrix $ \Sigma_{T}$ of the vector of risk factors $ X_{T}$, given the information up to time $ T-1$ in a very specific GARCH model.

While there is a plethora of analyses of alternative statistical models for market risks (see Barry Schachter's Gloriamundi web site), mainly two classes of models for market risk have been used in practice:

  1. iid-models, i.e., the risk factors $ X_{t}$ are assumed to be independent in time, but the distribution of $ X_{t}$ is not necessarily Gaussian. Apart from some less common models involving hyperbolic distributions (Breckling et al.; 2000), most approaches either estimate the distribution of $ X_{t}$ completely non-parametrically and run under the name ``historical simulation'', or they estimate the tail using generalized Pareto distributions (Embrechts et al.; 1997, ``extreme value theory'').
  2. conditional Gaussian models, i.e., the risk factors $ X_{t}$ are assumed to be joint normal, conditional on the information up to time $ t-1$.
Both model classes can account for unconditional ``fat tails''.


1.1.3 VaR Approximations

In this paper we consider certain approximations of VaR in the conditional Gaussian class of models. We assume that the conditional expectation of $ X_{t}$, $ \mu_{t}$, is zero and its conditional covariance matrix $ \Sigma_{t}$ is estimated and given at time $ t-1$. The change in the portfolio value over the time interval $ [t-1,t]$ is then

$\displaystyle \Delta V_{t}(X_{t}) = \sum_{i=1}^{n} w_{i} \Delta S^{i}_{t}(X_{t}),
$

where the $ w_{i}$ are the portfolio weights and $ \Delta S^{i}_{t}$ is the function that ``maps'' the risk factor vector $ X_{t}$ to a change in the value of the $ i$-th security value over the time interval $ [t-1,t]$, given all the information at time $ t-1$. These functions are usually nonlinear, even for stocks (see above). In the following, we will drop the time index and denote by $ \Delta V$ the change in the portfolio's value over the next time interval and by $ X$ the corresponding vector of risk factors.

The only general method to compute quantiles of the distribution of $ \Delta V$ is Monte Carlo simulation. From discussion with practitioners ``full valuation Monte Carlo'' appears to be practically infeasible for portfolios with securities whose mapping functions are first, extremely costly to compute - like for certain path-dependent options whose valuation itself relies on Monte-Carlo simulation - and second, computed inside complex closed-source front-office systems, which cannot be easily substituted or adapted in their accuracy/speed trade-offs. Quadratic approximations to the portfolio's value as a function of the risk factors

$\displaystyle \Delta V(X) \approx \Delta^{\top} X + \ensuremath{\frac{1}{2}}X^{\top}\Gamma X,$ (1.1)

have become the industry standard since its use in RiskMetrics (Longerstaey; 1996). ($ \Delta$ and $ \Gamma$ are the aggregated first and second derivatives of the individual mapping functions $ \Delta S^{i}$ w.r.t. the risk factors $ X$. The first version of RiskMetrics in 1994 considered only the first derivative of the value function, the ``delta''. Without loss of generality, we assume that the constant term in the Taylor expansion (1.1), the ``theta'', is zero.)


1.1.4 Pros and Cons of Delta-Gamma Approximations

Both assumptions of the Delta-Gamma-Normal approach - Gaussian innovations and a reasonably good quadratic approximation of the value function $ V$ - have been questioned. Simple examples of portfolios with options can be constructed to show that quadratic approximations to the value function can lead to very large errors in the computation of VaR (Britton-Jones and Schaefer; 1999). The Taylor-approximation (1.1) holds only locally and is questionable from the outset for the purpose of modeling extreme events. Moreover, the conditional Gaussian framework does not allow to model joint extremal events, as described by Embrechts et al. (1999). The Gaussian dependence structure, the copula, assigns too small probabilities to joint extremal events compared to some empirical observations.

Despite these valid critiques of the Delta-Gamma-Normal model, there are good reasons for banks to implement it alongside other models. (1) The statistical assumption of conditional Gaussian risk factors can explain a wide range of ``stylized facts'' about asset returns like unconditional fat tails and autocorrelation in realized volatility. Parsimonious multivariate conditional Gaussian models for dimensions like 500-2000 are challenging enough to be the subject of ongoing statistical research, Engle (2000). (2) First and second derivatives of financial products w.r.t. underlying market variables ($ =$ deltas and gammas) and other ``sensitivities'' are widely implemented in front office systems and routinely used by traders. Derivatives w.r.t. possibly different risk factors used by central risk management are easily computed by applying the chain rule of differentiation. So it is tempting to stay in the framework and language of the trading desks and express portfolio value changes in terms of deltas and gammas. (3) For many actual portfolios the delta-gamma approximation may serve as a good control-variate within variance-reduced Monte-Carlo methods, if it is not a sufficiently good approximation itself. Finally (4), is it extremely risky for a senior risk manager to ignore delta-gamma models if his friendly consultant tells him that 99% of the competitors have it implemented.

Several methods have been proposed to compute a quantile of the distribution defined by the model (1.1), among them Monte Carlo simulation (Pritsker; 1996), Johnson transformations (Zangari; 1996a; Longerstaey; 1996), Cornish-Fisher expansions (Zangari; 1996b; Fallon; 1996), the Solomon-Stephens approximation (Britton-Jones and Schaefer; 1999), moment-based approximations motivated by the theory of estimating functions (Li; 1999), saddle-point approximations (Rogers and Zane; 1999), and Fourier-inversion (Albanese et al.; 2000; Rouvinez; 1997). Pichler and Selitsch (1999) compare five different VaR-methods: Johnson transformations, Delta-Normal, and Cornish-Fisher-approximations up to the second, fourth and sixth moment. The sixth-order Cornish-Fisher-approximation compares well against the other techniques and is the final recommendation. Mina and Ulmer (1999) also compare Johnson transformations, Fourier inversion, Cornish-Fisher approximations, and partial Monte Carlo. (If the true value function $ \Delta V(X)$ in Monte Carlo simulation is used, this is called ``full Monte Carlo''. If its quadratic approximation is used, this is called ``partial Monte Carlo''.) Johnson transformations are concluded to be ``not a robust choice''. Cornish-Fisher is ``extremely fast'' compared to partial Monte Carlo and Fourier inversion, but not as robust, as it gives ``unacceptable results'' in one of the four sample portfolios.

The main three methods used in practice seem to be Cornish-Fisher expansions, Fourier inversion, and partial Monte Carlo, whose implementation in XploRe will be presented in this paper. What makes the Normal-Delta-Gamma model especially tractable is that the characteristic function of the probability distribution, i.e. the Fourier transform of the probability density, of the quadratic form (1.1) is known analytically. Such general properties are presented in section 1.2. Sections 1.3, 1.4, and 1.5 discuss the Cornish-Fisher, Fourier inversion, and partial Monte Carlo techniques, respectively.