On a Transform Method for the Efficient Computation of Conditional VaR (and VaR) with Application to Loss Models with Jumps and Stochastic Volatility

Measuring risks is certainly one of the core competence of any financial institution, even from a regulatory perspective. An efficient and reliable computation of risk measures is consequently a primary concern of any modern risk management activity, dramatically highlighted during the recent financial crisis. Value-at-Risk and Conditional (or Average) Value-at-Risk (henceforth VaR and CVaR) are without doubt among the best known monetary risk measures. Since its introduction, VaR rapidly has become a benchmark in the financial industry both for regulatory purposes and in the practice of risk management, mainly due to its simplicity. On the other side, it is sharply criticized for the lack of sub-additivity and the inability to quantify the severity of an exposure to rare events. For these reasons, alternative risk measures are considered as the CVaR, which turns out to be one of the well known example of coherent risk measure (see [1],[2]).

In this paper we present a method for their computation based on a Fourier transform technique. The Fourier representation of distribution functions and expected values of random variables is well-known in the financial and actuarial literature: Lèvy and Gil-Pelaez inversion formulas and more recently the method introduced by Carr and Madan in their seminal paper [5] for contingent claim valuation through Fast Fourier transform, nowadays are standard computational techniques to efficiently solve statistical and pricing problems. Here we consider a parametric framework, thus assuming a probabilistic description for the quantity we are interested in, namely profit/loss random variables. Aside the standard definitions for VaR and CVaR, in this paper we exploit an alternative characterizations of these risk measures in which transform representation can play an important tool for their efficient calculation. In particular, their property of being the solution of an elementary optimization problem of convex type in one dimension permits to evaluate both measures by solving numerically a unique simple univariate minimization problem, in which the objective function can be efficiently computed by means of Fourier representation. A quick and dirty solution is then available through (fractional) Fast Fourier Transform algorithms.

Analytical calculation of VaR by using inversion formulas has been considered in Duffie and Pan [9] while the use of generalized Fourier transform and the FFT algorithm is more recent, see e.g. Le Courtois and Walter [17] who applied such a technique in a Variance Gamma model, Kim et al. [16], Scherer et al. [24] where the class of tempered stable and infinitely divisible distribution were considered or Bormetti et al. [4] for an application to a stochastic volatility model. The techniques considered here can be applied to all models having an analytic (and computable) characteristic function. This is the case of many financial dynamic models of returns emerged in the literature of the last decades for the analysis and the management of portfolio risks, such as Lèvy finite/infinite activity and stochastic volatility models; an application to an univariate loss model in such a framework will be presented in the numerical section. But they can also be applied to that stochastic models which find their application in actuarial science, such as the compound Poisson loss distribution, used to model the aggregate claims of an insurance-risk business (see [10]).

The paper is organized as follows: VaR and CVaR are briefly introduced in Section 2, where their main characterizations are outlined. Transform technique is reviewed in Section 3 together with the use of fast and fractional Fourier transform algorithms and finally a set of numerical experiment is reported in Section 4 to show the effectiveness of the proposed procedures. In particular, the Fourier based techniques are applied to a univariate loss model of portfolio returns characterized by dynamical risk factors with jumps and stochastic volatility.

Let $(\Omega,F,{\mathbb{P}})$ be a probability space, $L$ a real-valued random variable and $F_{L}(x)={\mathbb{P}}(L\leq x)$ its distribution function. In the following we suppose that $\mathbb{E}[|L|]<\infty$. To measure the risk of a financial position characterized by an uncertain future value over a given time horizon, the quantiles of its distribution function are commonly used. Given a confidence level $\alpha\in(0,1)$, the set of $\alpha$-quantiles of the random variable $L$ is the interval $[q^{-}_{\alpha}(L),q^{+}_{\alpha}(L)]$ where

$$q^{-}_{\alpha}(L)=\inf\{q\in\mathbb{R}|P(L\leq q)\geq\alpha\},\ \ q^{+}_{\alpha}(L)=\inf\{q\in\mathbb{R}|P(L\leq q)>\alpha\}.$$

In this paper the random variable $L$ describes the loss of a financial position. Given $L$, VaR is defined as the lower $\alpha$-quantile, $q^{-}_{\alpha}(L)$:

$$\mathrm{VaR}_{\alpha}(L):=\inf\{q\in\mathbb{R}|P(L\leq q)\geq\alpha\}.$$

In financial terms, VaR is “the maximum possible loss which is not exceeded with probability $\alpha$”, or “the smallest amount of capital which, if added to the current position, keeps the probability of a non-negative outcome below the level $1-\alpha$”. For a random variable having continuous and strictly increasing distributions function, $q^{-}_{\alpha}(L)=q^{+}_{\alpha}(L)\equiv q_{\alpha}(L)=F^{-1}_{L}(\alpha)$, the ordinary inverse of $F$, i.e. VaR solves the equation

$${\mathbb{P}}(L\leq\mathrm{VaR}_{\alpha}(L))=\alpha\ \ \ \ \ (\mbox{or equivalently}\ \ {\mathbb{P}}(L\geq\mathrm{VaR}_{\alpha}(L))=1-\alpha).$$

Although widely used, it is well known that VaR is not a coherent risk measure (see [1], [2]), in particular for being not sub-additive. To overcome the weakness of VaR, several alternative risk measures have been proposed in literature, among which the Conditional, or Average, Value-at-Risk, which does satisfy the axioms of coherence (see e.g. [1],[2]). Several equivalent definitions have been proposed in the literature: given the confidence level $\alpha\in(0,1)$, the basic idea is to average all the possible losses exceeding $\mathrm{VaR}_{\alpha}(L)$, that is

$$\mathrm{CVaR}_{\alpha}(L):=\frac{1}{1-\alpha}\int_{\alpha}^{1}\mathrm{VaR}_{u}(L)\ du.$$

Alternatively, it may be convenient to define the CVaR as the expectation of $L$ under the (scaled) distribution function (see [23])

$$F_{L,\alpha}(x)=\left\{\begin{array}[]{cc}0&\mbox{for}\ \ x<q_{\alpha}^{-}(L)\\ (F_{L}(x)-\alpha)/(1-\alpha)&\mbox{for}\ \ x\geq q_{\alpha}^{-}(L).\\ \end{array}\right.$$

When $F$ is continuous and strictly increasing then $\mathrm{VaR}_{\alpha}(L)=F^{-1}_{L}(\alpha)$ and

$$\int_{\alpha}^{1}\mathrm{VaR}_{u}(L)\ du=\int_{F^{-1}_{L}(\alpha)}^{+\infty}x\ dF_{L}(x)=\mathbb{E}[X\mathbb{I}_{X\geq F^{-1}_{L}(\alpha)}];$$

therefore $\mathrm{CVaR}_{\alpha}(L)=\mathbb{E}[L|L\geq\mathrm{VaR}_{\alpha}(L)]$¹¹1For general distribution functions this is not true: see [23] and [14] for a detailed discussion about alternative definitions.. Furthermore, since $xI_{x\geq q}=(x-q)^{+}+qI_{x\geq q}$ we get

$$\mathrm{CVaR}_{\alpha}(L)=\mathrm{VaR}_{\alpha}(L)+\frac{1}{1-\alpha}\mathbb{E}[(X-\mathrm{VaR}_{\alpha}(L))^{+}].$$

(1)

More generally, it is possible to prove that (1) is still valid for any $\alpha$-quantile of $L$: that is

$$\mathrm{CVaR}_{\alpha}(L)=q+\frac{1}{1-\alpha}\mathbb{E}[(L-q)^{+}]$$

for any $q\in[q^{-}_{\alpha}(L),q^{+}_{\alpha}(L)]$, which evaluated for $q=\mathrm{VaR}_{\alpha}(L)$, clearly gives (1). The quantity $SL_{L}(q)\equiv\mathbb{E}[(L-q)^{+}]$ is known as the stop-loss transform of $L$. The following approach can therefore be considered for computing VaR and CVaR:

Algorithm 1. - Two Steps 1. compute $\mathrm{VaR}_{\alpha}(L)$, i.e. find $q^{*}$ such that ${\mathbb{P}}(L\leq q^{*})=\alpha$ (possibly not a unique solution); 2. compute $\mathbb{E}[(L-q^{*})^{+}]$ to get $\mathrm{CVaR}$ through formula (1).

An alternative characterization of VaR and CVaR for an arbitrary loss $L$, due to Rockafellar and Uryasev [22], [23], is obtained as follows. For a given value $\alpha\in(0,1)$ let us introduce the real function

$$G_{L,\alpha}(x)=x+\frac{1}{1-\alpha}\mathbb{E}[(L-x)^{+}],\ \ x\in\mathbb{R}$$

and let $\Gamma=\mathrm{argmin}_{x}G_{L,\alpha}(x)$ be the set of $x$ for which the minimum value of $G_{L,\alpha}$ is attained, with $\Gamma^{-}$ and $\Gamma^{+}$ respectively equal to the lower and the upper endpoint of $\Gamma$. The proof of the following Theorem can be found in [23] (but see also [11], where an alternative proof based on the Fenchel-Legendre transform is proposed (Lemma 4.6)):

This theorem suggests to compute VaR and CVaR by solving an unique optimization problem:

Algorithm 2. - NL-Min 1. Solve the non-linear minimization problem $\min_{x}G_{L,\alpha}(x)$.

In both the algorithms, the expectations required to compute VaR and CVaR must be evaluated for different values of a parameter $q$, namely ${\mathbb{P}}(L\leq q)=\mathbb{E}[\mathbb{I}_{\{L\leq q\}}]$ and $\mathbb{E}[(L-q)^{+}]$. This can be efficiently done by means of the Fourier transform technique. Before recalling the basic properties of this class of computational methods, we now briefly introduce a simple loss model which will be used to test the computational procedures in the final Section.

Fourier transform methods are efficient techniques emerged in recent years in the financial practice as one of the main methodology for the evaluation of derivatives. In fact, the no-arbitrage price of an European style contingent claim can be represented as the (conditional) expectation of the derivative payoff under a proper risk-neutral measure (see e.g. [3]). These methods essentially consist on the representation of such an expectation as the convolution of two Fourier transforms. Since the value of most derivatives depends on a trigger parameter, two main variants have been developed depending on which variable of the payoff is transformed into the Fourier space. In our setting, due to the functional (exponential) form of the transformed functions we consider, the formulas we get by applying the two approaches are essentially the same: we can pass from one to the other by simply changing the integration contour (see [21]). In the following we choose to work with the generalized Fourier transform (GFT) w.r.t. the trigger parameter $v$. In essence, given a function $H(y,v)$, the quantity that we want to compute is $h(v)=\mathbb{E}[H(Y,v)]$, where the expectation is taken over a given probability measure ${\mathbb{P}}$. Let us consider the GFT with respect to $v$: formally we have for $z=u+\mathrm{i}\nu\in\mathcal{C}\subset\mathbb{C}$

$$\hat{h}(z)=\int_{\mathbb{R}}\mathrm{e}^{\mathrm{i}zv}h(v)dv=\int_{\mathbb{R}}\mathrm{e}^{\mathrm{i}zy}\left(\int_{\mathbb{R}}H(y,v)\mathbb{P}(dy)\right)dv=\int_{\mathbb{R}}\widehat{H}^{(v)}(z,y)\mathbb{P}(dy)=\mathbb{E}[\widehat{H}^{(v)}(z,Y)]$$

where we have defined

$$\widehat{H}^{(v)}(z,y)=\int_{\mathbb{R}}\mathrm{e}^{\mathrm{i}zv}H(y,v)dv.$$

Notice that $\hat{h}$ corresponds to the classical Fourier transform of the $\nu$-damped expectation, as introduced in [5]: Fourier inversion gives

$$h(v)=\frac{1}{2\pi}\int_{\mathrm{i}\nu-\infty}^{\mathrm{i}\nu+\infty}\mathrm{e}^{-\mathrm{i}zv}\widehat{h}(z)dz=\frac{1}{2\pi}\int_{\mathrm{i}\nu-\infty}^{\mathrm{i}\nu+\infty}\mathrm{e}^{-\mathrm{i}zv}\mathbb{E}[\widehat{H}^{(v)}(z,Y)]\ dz$$

for $\nu$ in some strip of $\mathbb{C}$. The previous equalities must be justified under the appropriate conditions on the function $H$, its transform and the characteristic function of the underlying random variables (see e.g. [18] for a thorough discussion on the subject). For our application we consider the following functions:

$$H_{1}(y,v)=(y-v)^{+},\ \ \ H_{2}(y,v)=\mathbb{I}_{\{y\leq v\}},$$

$$H_{3}(x,k)=(e^{x}-e^{k})^{+},\ \ \ {\bar{H}}_{3}(x,k)=(e^{k}-e^{x})^{+}.$$

The reason for considering an exponential transformation of the basic risk factor is the fact that many financial models are usually introduced in the form $\exp(X)$, as in Example (2.1). Their GFT are readily obtained by means of standard (complex) integration: we summarize such results in the following

Let $\phi_{Y}(z)=\mathbb{E}[\mathrm{e}^{\mathrm{i}zY}],\ \ \ z\in\mathbb{C}$ be the (generalized) characteristic function of the r.v. $Y$. The following result can be proved as in [18].

Under the hypothesis of Theorem (2.1) we finally obtain an integral representation for the function $G_{L,\alpha}$:

	$\displaystyle G_{L,\alpha}(x)$	$\displaystyle=$	$\displaystyle x-\frac{1}{(1-\alpha)2\pi}\int_{\mathrm{i}\nu-\infty}^{\mathrm{i}\nu+\infty}\mathrm{e}^{-\mathrm{i}zx}\frac{\phi_{Y}(z)}{z^{2}}\ dz$		(14)
		$\displaystyle=$	$\displaystyle x-\frac{\mathrm{e}^{\nu x}}{(1-\alpha)\pi}\int_{0}^{+\infty}\mathrm{Re}\left(\mathrm{e}^{-\mathrm{i}ux}\frac{\phi_{Y}(u+\mathrm{i}\nu)}{(u+\mathrm{i}\nu)^{2}}\right)\ du,\ \ \ \nu<0$

or, in the case of exponential models,

$$G^{(e)}_{L,\alpha}(x)=x+\frac{1}{(1-\alpha)2\pi}\int_{\mathrm{i}\nu-\infty}^{\mathrm{i}\nu+\infty}\mathrm{e}^{-\mathrm{i}zx}\frac{\phi_{X}(z-\mathrm{i})}{\mathrm{i}z-z^{2}}\ dz$$

$\displaystyle=x+\frac{\mathrm{e}^{\nu x}}{(1-\alpha)\pi}\int_{0}^{+\infty}\mathrm{Re}\left(\mathrm{e}^{-\mathrm{i}ux}\frac{\phi_{X}(u+\mathrm{i}(\nu-1))}{\nu^{2}-\nu-u^{2}+\mathrm{i}u(1-2\nu)}\right)\ du,\ \ \nu>1,\ \mbox{or}\ \ \nu<0.$

(15)

As it is widely known, the transform method deserves for an efficient evaluation of expectations by means of the FFT algorithm for a proper range of the trigger parameter. Actually, if only one value has to be evaluated for a fixed parameter $v$ or $k$, there is no need to use FFT and a proper quadrature algorithm suffices to compute the required expectations.

The numerical procedures outlined in Section 2 for the computation of VaR and CVaR both require to evaluate the distribution function and/or the stop-loss expectation of the random variable $L$. Algorithm 1 firstly calls for a zero-finding routine that needs to compute (a sequence of) values of $F_{L}(\cdot)$ and then $SL(L)$ must be evaluated. Algorithm 2 must solve iteratively a univariate minimization problem requiring at each step to compute an expectation. Efficient numerical quadratures for computing Fourier integrals suffice for implementing the two algorithms without the need of calling for fast Fourier transforms. Error bounds on the approximation obtained clearly depends on the computational methods chosen for numerical integration, zero-finding and minimization routines. A third algorithm can be outlined, providing a quick-and-dirty solution, based on FFT/FRFT. Since we want to minimize w.r.t. $x$ the functions $G_{L,\alpha}(x)$ or $G^{(e)}_{L,\alpha}(x)$ we can approximate them for a whole range of values $\mathbf{x}$ by applying once FFT/FRFT: from the integral representation (14) and (15), we get

$$G_{L,\alpha}(\mathbf{x})\approx\hat{G}_{\alpha}(\mathbf{x})\equiv\left\{\begin{array}[]{l}\mathbf{x}-\frac{\mathrm{e}^{\nu\mathbf{x}}}{(1-\alpha)\pi}\odot FFT(\mathbf{x},\mathbf{h})\\ \mathbf{x}-\frac{\mathrm{e}^{\nu\mathbf{x}}}{(1-\alpha)\pi}\odot FRFT(\mathbf{x},\mathbf{h},\eta)\end{array}\right.$$

and

$$G^{(e)}_{L,\alpha}(\mathbf{x})\approx\hat{G}^{(e)}_{\alpha}(\mathbf{x})\equiv\left\{\begin{array}[]{l}\mathbf{x}+\frac{\mathrm{e}^{\nu\mathbf{x}}}{(1-\alpha)\pi}\odot FFT(\mathbf{x},\mathbf{h})\\ \mathbf{x}+\frac{\mathrm{e}^{\nu\mathbf{x}}}{(1-\alpha)\pi}\odot FRFT(\mathbf{x},\mathbf{h},\eta)\end{array}\right.$$

where $\mathbf{h}$ is the vector defined in (17), evaluated according to the functions in (14) or (15). Hence, the minimum and the corresponding minimizer of the vector $\hat{G}_{L,\alpha}$ will provide approximated values for CVaR and VaR, respectively.

Algorithm 3. - FFT/FRFT 1. Compute the vector $\hat{G}_{L,\alpha}$ through FFT/FRFT algorithm for a proper grid $\mathbf{x}$; 2. Find the minimum and the corresponding minimizer of $\hat{G}_{L,\alpha}$