Estimating value-at-risk using quantile regression and implied volatilities

Petter E. de Lange, Morten Risstad and Sjur Westgaard

Abstract

In this paper we propose a semi-parametric, parsimonious value-at-risk forecasting model based on quantile regression and readily available market prices of option contracts from the over-the-counter foreign exchange interbank market. Explanatory variables are implied volatilities with plausible economic interpretation. The forward-looking nature of the model, induced by the application of implied moments as risk factors, ensures that new information is rapidly reflected in value-at-risk estimates. The proposed model outperforms traditional benchmark models when evaluated in-sample and out-of-sample on EUR/USD data. The model is relatively easy to estimate, which facilitates practical application. Our quantile regression implied moments model is subjected to extensive risk model validation by means of backtesting, using both coverage tests and loss functions. Thus, his paper is relevant for both risk modeling and risk model validation in the context of foreign exchange risk.

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Introduction

1 Introduction

Value-at-risk (VaR) is an estimate of the loss that will be exceeded with a small probability during a fixed holding period. Besides a prominent role in regulatory frameworks, VaR continues to be important for financial institutions as a measure of market risk.

Effective VaR estimation relies on successfully predicting the conditional distribution of the return series, more specifically the tails of the distribution. We propose a particularly parsimonious VaR forecasting model based on quantile regression (QR) (Koenker and Bassett 1978) and readily available market prices of option contracts from the over-the-counter (OTC) foreign exchange (FX) interbank market. The OTC FX interbank market is organized in a manner that makes it particularly useful for the empirical investigation we report in this paper. In the interbank FX options market, market makers quote a set of new options every day, with standardized maturities and moneyness. This is in contrast to exchange-traded options for which new maturities are introduced once a month or even less frequently. In OTC FX options we therefore have consistent time series to investigate (eg, one-week implied volatilities over time), which leaves less room for error in the estimation of time series properties. In addition, the OTC FX options market has the beneficial property of quoting market prices in terms of implied volatilities directly. Thus, implied volatilities need not be inverted from option premiums. This is possible only since this market, unlike any other market, has standardized the option formula used for quoting implied volatilities.¹¹ 1 See Reiswich and Wystup (2010) for a guide to FX options quoting conventions. These institutionalized features of OTC FX options interbank trading make this market especially convenient for empirical analysis.

As is common in the literature, we validate our proposed risk model by both loss functions and coverage tests. When evaluated in-sample and out-of-sample on EUR/USD data, the model outperforms traditional nonparametric and parametric methods frequently used for VaR forecasting, including RiskMetrics (RM), filtered historical simulation (FHS), extreme value theory (EVT), Glosten–Jagannathan–Runkle generalized autoregressive conditional heteroscedasticity (GJR-GARCH) and conditional autoregressive value-at-risk (CAViaR).

This paper contributes to the literature by being (to the best of our knowledge) the first to use information in the volatility surface – more specifically at-the-money (ATM) volatility and risk reversals (RRs) as proxies for higher-order moments – combined with QR, to provide accurate VaR estimates. We develop a new model that

1. (1)

   uses readily available market prices as explanatory variables;

2. (2)

   uses market-based forward-looking information about the future distribution of FX returns;

3. (3)

   demonstrates strong performance when subjected to appropriate risk model validation techniques (both in-sample and out-of-sample for both long and short positions);

4. (4)

   avoids interpolation schemes or complex optimization routines, which reduces model risk; and

5. (5)

   can be easily implemented.

Further, the paper contributes to the strand of academic empirical research based on OTC FX options data, which is relatively scarce.

2 Literature review

2.1 Existing methods for estimating VaR

VaR is a quantile of the estimated distribution of returns. Hence, the different modeling approaches differ in the way they construct the conditional density function. Nieto and Ruiz (2016) classify VaR models into three main categories. Nonparametric models assume that historical returns represent the true data generating process. Fully parametric models assume a particular error distribution. Semi-parametric models impose a parametric structure on the dynamics of VaR through their relationship with lagged information, but they require no assumptions about the conditional distribution of financial returns.

Historical simulation (HS) defines the VaR estimate as the $q$-quantile of historical raw returns (Butler and Schachter 1997).

GARCH models are flexible enough to capture important features of returns data and flexible enough to accommodate specific aspects of individual assets. In the normal GARCH(1,1) model from Engle (1982) and Bollerslev (1986), daily returns are normally distributed, and the variance of day $t$ depends on the squared return, the conditional variance estimate of the previous day and the long-term variance. The GJR-GARCH model from Glosten et al (1993) allows for an asymmetric response of conditional volatility to negative and positive returns through a leverage parameter. GARCH models can accommodate a wide range of assumptions with regards to the distribution of residuals, but they require the maximum likelihood estimation of parameters, as well as complex nonlinear optimization techniques.

FHS was introduced by Barone-Adesi et al (2008), and it seeks to combine the best of the model-based approaches with the best of the model-free approaches. The distribution of residuals is derived from the empirical distribution of standardized returns. The empirical distribution of innovations can be combined with conditional volatility estimates. Thus, FHS can generate large losses in the forecast period even without having observed a large loss in the recorded past returns.

The RM methodology (JP Morgan–Reuters 1996) assumes conditionally normal returns and handles time-varying volatility using fixed parameters.

EVT states that the extreme tail of a wide range of distributions can be approximated by the generalized Pareto distribution (GPD). McNeil and Frey (2000) suggest fitting a GARCH model to the time series or returns and then applying EVT to the standardized residuals.

The CAViaR methodology from Engle and Manganelli (2004) models VaR as an autoregressive process. CAViaR explicitly models the dynamics of conditional quantiles, without assumptions about the return distribution. A generic CAViaR model allows for use of lagged estimates of VaR or other variables as explanatory variables. The models are estimated using QR but require the use of complex numerical optimization algorithms.

The two methods most widely used by practitioners to forecast VaR are HS and FHS (Pérignon and Smith 2010). Though these two methods are popular, they fail to capture the stylized properties of asset returns in the energy market (Westgaard et al 2019). Alexander and Sheedy (2008) and Giot and Laurent (2003) find that the Student $t$ GARCH and GJR-GARCH models perform well. The CAViaR methodology shows mixed results in empirical applications (Engle and Manganelli 2004; Kuester et al 2006; Haugom et al 2016). Buczyński and Chlebus (2019) find that parametric models perform better than nonparametric models across volatility states.

2.2 Implied distributions, QR and VaR

Implied volatilities can be interpreted as risk-neutral expectations of future realized volatility. The OTC FX market quotes option prices in terms of Black–Scholes–Merton (BSM) volatilities. The standard BSM option-pricing model assumes Gaussian distributed asset-price returns, lognormally distributed asset prices and constant cross-sectional volatilities. This is in contrast to empirical distributions of asset price returns, which typically have high peaks and fat tails. This empirical fact is factored into market prices and reflected in the volatility smile, in the form of different implied volatilities for different strikes at a given expiry date. The existence of the volatility smile, along with the term structure of implied volatility, is recognized in academia and in practice, and it is discussed in a large number of papers (see, for example, Hull and White 1987; Haug et al 2010; Ornelas and Mauad 2019).

A number of studies (Breeden and Litzenberger 1978; Jackwerth 2000; Galati et al 2005; Ross 2015) show that implied probability distributions can be inferred from option prices. Chang et al (2013) provide a survey of different forecasting objectives using options data, including option-implied individual moments. Barone-Adesi et al (2019) and Huggenberger et al (2018) show how to use options data to compute forward-looking VaR and conditional VaR measures.

Several studies find that methods based on option-implied information generally outperform historical-based estimates of real-world return densities (Shackleton et al 2010; Christoffersen et al 2013; Crisóstomo and Couso 2018). Chong (2004), however, compares daily VaR estimates derived from univariate and multivariate time series models with those implied by currency option prices, reporting that time-series-induced VaR models perform better than those based on implied volatility.

The informational content of option combinations, such as RRs, has been studied in the context of arbitrage-free option pricing and hedging (Bossens et al 2010; Vaidyanathan 2012). Christoffersen and Mazzotta (2005) is one of the very few papers that use data on OTC FX option combinations, such as RRs and strangles, to obtain density forecasts. Their findings show that implied volatility is not a particularly useful input in density and interval forecasts, since it fails to capture the tail behavior of the distribution. They nevertheless make parametric assumptions to estimate the tails of the distribution, which is fundamentally different to the approach in this paper.

A few papers have applied QR in the context of forecasting volatility or VaR of FX rates. Taylor (1999) finds that a QR approach delivers a better fit to multiperiod data than variations of the GARCH(1,1) model when forecasting volatility. Huang et al (2011) use QR to forecast FX rate volatility. Jeon and Taylor (2013) include implied volatility as an additional regressor in CAViaR models and find increased precision in FX VaR estimates.

Taylor (2000), Chen et al (2012), Pradeepkumar and Ravi (2017) and Christou and Grabchak (2019) investigate alternatives to the linear QR model and find that more advanced estimation methods, such as nonlinear machine learning algorithms, might improve VaR estimates.

3 The proposed model and its motivation

Christoffersen et al (2013) provide a description of situations when option-implied forecasts are likely to be most useful for risk quantification, such as when the option market is highly liquid. This resembles the interbank FX option market, which is among the largest derivatives markets in the world.

We base our quantile regression implied moments (QR-IM) model on the hypothesis that the volatility surface for OTC FX options contains information that can be used to improve the accuracy of VaR estimates compared with existing methods. To extract this information, we analyze data for ATM options and RRs. Additionally, we experiment with other option combinations, such as butterfly spreads. However, these variables prove to be statistically insignificant when added to the QR model alongside ATM and RRs.

ATM options are struck at the FX forward rate. ATM options have an initial delta of 50%. The ATM implied volatility is the risk-neutral expectation of spot rate volatility over the remaining life of the option.

RRs involve the simultaneous sale of a put option and purchase of a call option. The two options are struck at the same delta. At the outset a 25% delta RR will have a combined delta of 50% and very little sensitivity to gamma and vega, because of the offsetting effect of the long and short position. RRs are usually quoted as the difference in implied volatility of the call options and that of the put options, on the base currency with the same delta. The RR reflects the difference in the demand for out-of-the money options at high strikes compared with low strikes. Thus, it can be interpreted as a market-based measure of skewness: the most likely direction of the spot movement over the expiry period.

The QR method, originally introduced by Koenker and Bassett (1978), is particularly well suited for the purpose of this paper, for several reasons. First, the institutional features of the OTC FX options market, with high-quality time series data resulting from deep liquidity and directly quoted implied volatilities, lends itself to application within a regression framework without introducing interpolation schemes. This reduces model risk, compared with many of the existing models described in Section 2. Second, QR does not require complex optimization routines, is relatively simple to estimate and can readily be applied by practitioners. Third, QR allows for a parsimonious specification, which reduces the risk of overfitting. Fourth, the flexible nature of QR does not require specific parametric assumptions on the explanatory variables themselves or on the residuals. Finally, regression quantile estimates are known to be robust to outliers, a desirable feature in general and for applications to financial data in particular.

In the general case, the simple linear QR model is given by

$$Y_q=\alpha +\beta X+{\epsilon }_q,$$

(3.1)

where the distribution of ${\epsilon }_q$ is left unspecified. The expression for the conditional $q$-quantile, , is defined as any solution to the minimization problem

$$\underset{\alpha ,\beta }{\min }\sum _{t=1}^T(q-I_{Y_t\le \alpha +\beta X_t})(Y_t-(\alpha +\beta X_t)),$$

(3.2)

where

(3.3)

In the QR-IM model, the conditional quantile function can be expressed as

$${\widehat{\mathrm{VaR}}}_{q,t+1}={\widehat{\alpha }}_q+{\widehat{\beta }}_q^{\mathrm{ATM}}{\mathrm{ATM}}_t+{\widehat{\beta }}_q^{\mathrm{RR}}{\mathrm{RR}}_t+{\epsilon }_{q,t}.$$

(3.4)

A unique vector of regression parameters $[\widehat{\alpha },{\widehat{\beta }}^{\mathrm{ATM}},{\widehat{\beta }}^{\mathrm{RR}}]$ can be obtained for each quantile of interest, and the whole return distribution can be found, given observed values for the ATM volatility and the RRs.

To the best of our knowledge, our paper is the first to approximate higher-order moments of the return distribution by explicitly utilizing observable market prices from OTC FX options (as expressed by ATM and RR) to forecast VaR in a QR framework. Our paper distinguishes itself from the existing literature in several respects. First, we use observable forward-looking market volatilities as explanatory variables in the QR model. This circumvents sampling errors indisputably present in historical time return series data. Second, we avoid any interpolation of market data, complex estimating techniques and arbitrary modeling assumptions, which reduces model risk. Third, we investigate the empirical performance and practical relevance of the model via backtesting.

4 Data

To investigate the empirical properties of the QR-IM model, defined in (3.4), we consider EUR/USD data. Daily exchange rates and implied volatility quotes covering the period from January 2009 to September 2020 are sourced from Bloomberg.

4.1 Spot rate returns

Figure 1: EUR/USD exchange rates and returns. (a) EUR/USD spot exchange rates. (b) Daily log returns. Time period: January 2009 to September 2020. Source: Bloomberg.

Table 1: Descriptive statistics for daily EUR/USD log returns. [Time period: January 2009 to September 2020. Source: Bloomberg.]

$n$	2930
Mean	$-$0.0001
Standard deviation	$-$0.0053
Skewness	$-$0.0330
Kurtosis	$-$4.7508
Min	$-$0.0229
Max	$-$0.0295

We calculate daily returns as $r_t=\ln (S_t/S_{t-1})$, where $S_t$ is the spot exchange rate at time $t$. Table 1 displays descriptive statistics for daily log returns from January 2009 to September 2020, and Figure 1 plots the corresponding time series. The return series exhibits the stylized facts that have been widely documented in the financial economics literature, with unconditional means close to 0, clustering of volatilities and fat-tailed return distributions.

4.2 Implied volatility market quotes

Figure 2: Option-implied moments. Time series and empirical distributions for implied moments of one-week-to-expiry options. (a), (b) ATM volatilities. (c), (d) 25-delta RR. Time period: January 2009 to September 2020. Source: Bloomberg.

Figure 2 displays time series for ATM volatilities and 25-delta RR for European EUR/USD options with one week to expiry, and it reveals the stochastic nature of the volatility surface. ATM levels have spiked around important economic events, such as the Brexit referendum in June 2016 and the Covid-19 outbreak in March 2020. Part (c) shows that the sign of the RR has changed over time and has taken both positive and negative values, which is in itself an interesting observation. If the RR reflects the relative probability of depreciation and appreciation of a currency, the time-varying sign and magnitude of the RR can be interpreted as an indication of the time-varying probability of tail events. Parts (b) and (d) display empirical distributions, which are skewed and leptokurtic for both variables. ATM volatility is naturally bounded below by 0, but it spikes during periods of market turmoil and thus causes a heavy right tail. The RR displays a highly nonnormal, heavy-tailed empirical distribution.

5 Empirical investigation

In this section we estimate the proposed QR-IM model, provide economic interpretation and discuss the statistical significance of the coefficients. To assess the empirical performance and practical relevance we perform both in-sample and out-of-sample evaluations of the preferred QR specification, and we compare it with a set of benchmark VaR models, using loss functions and coverage tests. We use a fixed estimation window from July 1, 2009 to December 31, 2017. The out-of-sample period covers January 1, 2018 to September 28, 2020.

5.1 Estimating the QR VaR model

Table 2: Quantile regression coefficients for the QR-IM model. [The table shows estimated QR coefficients. The model is estimated on daily Bloomberg data over the in-sample period from July 2009 to December 2017. Regression coefficients are scaled by 100 for readability. ${}^{*}$ and ${}^{**}$ denote statistical significance at the 1% and 5% confidence levels, respectively.]

	Quantiles (%)
EUR/USD	1.0	2.5	5.0	10.0	90.0	95.0	97.5	99.0
Constant	$-$0.61${}^{*}$	$-$0.21	$-$0.07	$-$0.04	$-$0.03	$-$0.09	$-$0.05	$-$0.12
ATM	$-$0.06${}^{*}$	$-$0.09${}^{*}$	$-$0.08${}^{*}$	$-$0.07${}^{*}$	$-$0.08${}^{*}$	$-$0.09${}^{*}$	$-$0.12${}^{*}$	$-$0.15${}^{*}$
25-delta RR	$-$0.21${}^{*}$	$-$0.16${}^{*}$	$-$0.10${}^{*}$${}^{*}$	$-$0.03	$-$0.16${}^{*}$	$-$0.16${}^{*}$	$-$0.30${}^{*}$	$-$0.38${}^{*}$

Figure 3: Quantile regression coefficients for the QR-IM model. (a) Constant. (b) ATM volatility. (c) 25-delta RR. The model is estimated on daily Bloomberg data over the in-sample period from July 2009 to December 2017. Coefficients are scaled by 100 for readability.

Table 2 displays regression coefficients for different quantiles of the EUR/USD return distributions. Standard errors are calculated using the bootstrap method proposed by Feng et al (2011). Figure 3 provides a corresponding graphical representation of the estimated coefficients.

As for the sign, magnitude, shape and statistical significance of regression coefficients, the results are consistent for both the ATM variable and the RR variable. The estimated ATM coefficients support our interpretation of ATM as a level indicator of overall market risk: there is a nonlinear, increasing relationship between quantiles and ATM regression coefficients. The absolute value of the regression coefficients is higher for the extreme quantiles than the quantiles closer to the median. The loadings on ATM are negative in the left tail of the distribution and positive in the right tail. These estimation results are reasonable, given that the model attempts to project the shape of the tails of the conditional return distribution. Further, the ATM regression coefficients are fairly symmetrical in the two tails of the return distribution. Still, the absolute value of ATM coefficients is slightly higher in the right tail, which indicates a nonlinear relationship between implied volatility and returns.

Figure 3(c) reveals that the RR coefficients display a U-shaped pattern. The coefficients are positive for all quantiles. They are generally strongly significant, except for the 10% quantile. The estimated RR coefficients have two interpretations. First, the high statistical significance implies that including the RR as a risk factor in the QR model, in addition to ATM volatility, will improve model accuracy. Second, the U-shaped pattern of coefficients implies that the RR is relatively more important for estimating the extreme tails of the return distribution. In summary, the estimation results support our economic interpretation of the RR as a market-based measure of implied skewness. Further, estimation results conform well to the well-known stylized fact of nonnormal returns. ATM volatility, by definition, implicitly assumes normally distributed returns. Including measures of higher-order moments should intuitively improve model fit to the tails of the distribution. The consistent estimation results strongly indicate that the RR has predictive power when estimating conditional tail risk, both for negative and positive returns, and justifies including the RR as a risk factor in our QR VaR-model.

5.2 Estimating the benchmark VaR models

As a basis for comparing the out-of-sample performance of our QR model, we estimate a set of benchmark VaR models.²² 2 Parameter estimates for the remaining benchmark models are available from the authors upon request. These models, which are described in Section 2.1, include FHS, EVT, RM, the symmetric absolute value specification of CAViaR and the Student $t$ GJR-GARCH (GJR-GARCH-$t$).

Inspired by Jeon and Taylor (2013), we filter conditional volatility in the FHS and EVT VaR models by both GARCH(1,1) estimates and ATM volatility (implied volatility (IV)). These benchmark models are referred to as FHS-GARCH, FHS-IV, EVT-GARCH and EVT-IV, respectively.

In the normal GARCH(1,1) model from Engle (1982) and Bollerslev (1986), daily returns are normally distributed. The conditional variance estimate of day $t$ depends on the squared return, $r_{t-1}^2$, the conditional variance estimate of the previous day, ${\sigma }_{t-1}^2$, and the long-term variance, ${\sigma }_{\mathrm{L}}^2$:

(5.1)

Parameters $\alpha$ and $\beta$ are estimated using maximum likelihood, and they determine the relative weighting of the preceding variance estimate and squared returns. The GJR-GARCH model from Glosten et al (1993) allows for the asymmetric response of conditional volatility to negative and positive returns through the leverage parameter $\gamma$:

(5.2)

The VaR estimate from GARCH models is defined as

$${\mathrm{VaR}}_t=Q_q^D{\sigma }_t,$$

(5.3)

where $Q_q^D$ is the number of standard deviations corresponding to the $q$-quantile of distribution $D$, and ${\sigma }_t$ is the conditional volatility estimate. For GJR-GARCH VaR estimates, we assume a symmetric Student $t$ error distribution.

To obtain VaR estimates from the FHS model, we start by calculating standardized returns $z_t\sim \mathrm{iid}D(0,1)$ over the in-sample period:

$$z_t=\frac{r_t}{{\sigma }_t},$$

(5.4)

where ${\sigma }_t$ is the filtered conditional volatility estimate. The VaR estimate from FHS is

$${\mathrm{VaR}}_t=Q_q^{\mathrm{FHS}}{\sigma }_t,$$

(5.5)

where $Q_q^{\mathrm{FHS}}$ is a scalar corresponding to the $q$-quantile of empirical standardized returns.

In the RM model, the conditional volatility estimate is given by

${\sigma }_t^2=\lambda {\sigma }_{t-1}^2+(1-\lambda )r_{t-1}^2.$

(5.6)

We follow the common approach of setting $\lambda =0.94$. The VaR estimate is thus

${\mathrm{VaR}}_t=Q_q^N{\sigma }_t,$

(5.7)

where $Q_q^N$ is the number of standard deviations associated with the $q$-quantile of the standard normal $N(0,1)$ distribution.

In the EVT-GARCH VaR model we combine EVT and time-varying conditional volatility by following the procedure outlined in Christoffersen (2011). The daily VaR estimate is defined as

$${\mathrm{VaR}}_t={\sigma }_t{F}_{1-q}^{-1}={\sigma }_{t}u{\left[\frac{q}{T_u/T}\right]}^{-\eta },$$

(5.8)

where ${\sigma }_t$ is the filtered conditional volatility estimate of standardized returns (5.4), $F_{1-q}^{-1}$ is the inverse cumulative distribution, $u$ is the threshold value separating the extreme tail from the rest of the distribution and $\eta$ is the tail index parameter controlling the shape of the tail. To balance bias and variance, we follow Christoffersen (2011) and set $T_u=50$. Excess kurtosis in our return data ensures a strictly positive $\eta$, which enables us to apply the closed-form Hill estimator to approximate the GPD:

$$\eta =\frac{1}{T_u}\sum _{i=1}^{T_u}\ln \left(\frac{r_i}{u}\right).$$

(5.9)

A generic CAViaR model (Engle and Manganelli 2004) has the following expression:

$$q_t(\beta )=f(r_{t-1},{𝚽}_{t-1}+\mathrm{\cdots }+r_t,{𝚽}_t;𝜷),$$

(5.10)

where $𝜷$ is a vector of the parameters, $r_t$ is the return and $𝚽$ is a vector of explanatory variables. Engle and Manganelli (2004) consider four different CAViaR specifications. In this paper, we apply the symmetric absolute value specification, in which the estimated VaR responds symmetrically to the absolute value of realized returns:

$${\mathrm{VaR}}_{q,t+1}={\beta }_0+{\beta }_1{\mathrm{VaR}}_{q,t}+{\beta }_2|r_t|.$$

(5.11)

5.3 In-sample evaluation

To compare the in-sample fit of our proposed QR-IM model with the benchmark models, we implement the model confidence set (MCS) procedure from Hansen et al (2011).

5.3.1 Model confidence set

The MCS procedure consists of a sequence of tests that construct the set of “superior” models, where the null hypothesis of equal predictive ability (EPA) is not rejected at a certain confidence level. This procedure is a top-down approach in which a null hypothesis of the EPA of competing models is tested at each step. If the hypothesis is rejected, the model with the worst relative loss is excluded, and the testing procedure is repeated until the null of EPA cannot be rejected at a given confidence $1-\alpha$ (we set $\alpha =0.05$).³³ 3 In an ideal situation, the MCS procedure continues to reject the null of equal predictive performance until one model is superior to all other models in the set. In most cases, however, the MCS will at one point not be able to distinguish between the last remaining $k$ models in the set in terms of a statistically significant differential in the loss function. In this case, the MCS procedure stops and reports the $k$ models as the superior set of models. The MCS algorithm eliminates the model with the worst test statistic $T_{\max }$ in each iteration. Therefore, the order of model evaluation should not impact the ranks. Let the loss function for the $i$th model be defined as:

$$L(i,t)=L({\widehat{\mathrm{RV}}}_{i,t},{\mathrm{RV}}_t),$$

where ${\mathrm{RV}}_t$ and ${\widehat{\mathrm{RV}}}_{i,t}$ are observed and estimated values of the metric of interest, respectively.⁴⁴ 4 The metric of interest could, for instance, be a volatility loss function, a mean-squared error or a VaR loss function, as in the Abad–Benito–López loss function (FABL) metric in Section 5.3.2. When two competing models $i$ and $j$ are compared, a loss differential $D(i,j,t)$ is evaluated as follows:

$$D(i,j,t)=L(i,t)-L(j,t).$$

Let $m$ denote the number of competing models and $D(i,\cdot ,t)$ denote the average loss of model $i$ at time $t$ with respect to the remaining models as follows:

$${\widehat{d}}_{(i,\cdot )}=\frac{1}{m-1}\sum _{i\ne j}D(i,j,t).$$

Given that ${\overline{d}}_{(i,\cdot )}$ is an average of the average loss differential of the $i$th model with respect to the other competing models, the statistic is defined as

$$t_{(i,\cdot )}=\frac{{\overline{d}}_{(i,\cdot )}}{\sqrt{\overline{\mathrm{var}}({\overline{d}}_{(i,\cdot )})}}.$$

The denominator is a block bootstrap estimator of the variance of ${\overline{d}}_{(i,\cdot )}$. Finally, the test statistic is defined as

$$T_{\max }=\underset{(i)}{\max }{t}_{(i,\cdot )}.$$

The distribution under the null hypothesis of the test statistics $T_{\max }$ is bootstrapped as for the variance. If the null hypothesis of EPA is rejected, then the model with the highest $t_{(i,\cdot )}$ is eliminated, and the testing procedure is repeated until the null is not rejected.

5.3.2 Choice of VaR loss function

The test statistics for EPA are flexible, which implies that the MCS is equally applicable in-sample and out-of-sample and for arbitrary loss functions. This is particularly relevant when evaluating VaR models by loss functions. As thoroughly discussed by Abad et al (2015), the choice of an appropriate loss function is highly user and context dependent and is thus somewhat arbitrary. A number of papers propose loss functions that penalize the VaR model only when breeches occur (Lopez 1999; Caporin 2008). Others point out that overcapitalization represents an opportunity cost, which implies some kind of model penalty in nonbreech scenarios as well. Abad et al (2015) propose a loss function that captures the opportunity cost of reserved capital:

(5.12)

Abad et al (2015) do not define $\beta$ numerically, but they argue that the parameter should capture “the real cost of capital”. This diverges somewhat from Sarma et al (2003), who in a comparable approach interpret $\beta$ as the cost of collateral, ie, as a short-term money market interest rate. Our view resembles that of Abad et al (2015) in that the required rate of return on equity capital is the relevant cost, since capital reserves to cover market risk are typically viewed as equity capital, from both a regulatory perspective and an economic perspective. For the purpose of this paper, we set $\beta$ to 10%.⁵⁵ 5 Based on experience, equity hurdle rates applied by banks lie within a range of 10–15%. As a robustness check, we have performed similar calculations with $\beta$ equal to 5% and 15%, which does not materially change the main results.

5.3.3 In-sample results

In this section we combine the FABL loss function and the MCS procedure to identify which of the alternative models adequately capture the risk characteristics of the EUR/USD spot exchange rate.

Table 3 shows that the QR-IM model belongs to the superior set of models in five out of the six quantiles. The only exception is the 5% quantile, where the $p$-value of the MCS procedure is close to 0. Further, the QR-IM model is highly ranked among the superior set of models. We interpret this as evidence that the proposed QR-IM models are capable of adequately capturing the true data-generating process.

As for the benchmark models, an interesting observation is that the implied volatility filtered models (FHS-IV and EVT-IV) seem to perform better than their GARCH filtered counterparts. Further, neither the parametric GJR-GARCH model nor the semi-parametric CAViaR model seems to fit the data very well in-sample.

Table 3: Model confidence set: EUR/USD, in-sample

	1%		2.5%		5%		95%		97.5%		99%
Benchmarks	Rank	Loss	Rank	Loss	Rank	Loss	Rank	Loss	Rank	Loss	Rank	Loss
QR-IM	1	3.71	2	3.71			1	3.26	1	3.32	1	3.82
FHS-GARCH							8	3.53	6	3.59
FHS-IV	4	3.80	1	3.18	1	2.98	3	3.40	4	3.52
EVT-GARCH			4	3.42			5	3.49
EVT-IV	3	3.77	3	3.32			2	3.31	2	3.42	3	3.85
CAViaR-SAV							4	3.47	3	3.49
RiskMetrics	2	3.72	5	3.44			7	3.53	5	3.58	2	3.84
GJR-GARCH							6	3.50
(Student $t$)
MCS $p$-value	0.13		0.08		0.01		0.16		0.07		0.87

5.4 Out-of-sample VaR forecasts and backtesting

In this section we backtest benchmark models and the QR-IM model by employing the conditional coverage test from Christoffersen (1998) and the dynamic quantile (DQ) test from Engle and Manganelli (2004). All models are estimated on daily data from July 1, 2009 to December 31, 2017. Model parameters are held fixed over the out-of-sample period, which covers January 1, 2018 to September 28, 2020.

Table 4: VaR backtesting: EUR/USD, out-of-sample. [The table shows results for out-of-sample backtesting of the QR-IM model with benchmarks. The fixed estimation window is July 2009 to December 2017; the out-of-sample period is from January 2018 to September 2020. Part (a) shows hit percentage. Parts (b) and (c) show $p$-values for the Christoffersen test and dynamic quantile test (with four lags).] [The table shows results from the MCS procedure from Hansen et al (2011) using the FABL loss function from Abad et al (2015) applied to the in-sample period from July 2009 to December 2017. The MCS procedure selects the models of EPA among the eight models evaluated. Blank rows indicate that the model has inferior predictive ability and hence has been eliminated from the set of superior models. “Rank” displays the ranking of the model and “Loss” the average value of the VaR loss function at a given confidence level. The number of bootstrapped samples is 5000 and the confidence level parameter $\alpha$ is set to 5%.]

(a) Hit percentage
	Quantiles
	1%	2.5%	5%	95%	97.5%	99%
QR-IM	0.7	2.1	6.3	95.0	97.5	99.3
FHS-GARCH	0.4	1.4	4.1	95.7	98.2	99.3
FHS-IV	1.0	2.0	6.3	94.0	97.2	99.0
EVT-GARCH	0.6	1.4	4.1	96.8	98.2	99.3
EVT-IV	1.4	2.5	4.6	94.7	97.2	98.7
CAViaR-SAV	0.3	1.1	3.8	94.5	96.9	99.0
RiskMetrics	1.4	3.5	5.6	94.4	96.5	98.6
GJR-GARCH (Student $t$)	0.6	1.4	4.2	96.5	98.2	99.3
(b) Christoffersen test ($p$-value)
	Quantiles
	1%	2.5%	5%	95%	97.5%	99%
QR-IM	0.67	0.57	0.26	0.99	0.63	0.67
FHS-GARCH	0.21	0.00	0.38	0.31	0.38	0.67
FHS-IV	0.93	0.48	0.26	0.43	0.49	0.93
EVT-GARCH	0.43	0.00	0.38	0.03	0.38	0.67
EVT-IV	0.52	0.63	0.18	0.92	0.49	0.71
CAViaR-SAV	0.07	0.00	0.20	0.17	0.00	0.93
RiskMetrics	0.52	0.11	0.75	0.47	0.27	0.52
GJR-GARCH (Student $t$)	0.43	0.00	0.50	0.09	0.38	0.67
(c) DQ test ($p$-value)
	Benchmarks
	1%	2.5%	5%	95%	97.5%	99%
QR-IM	0.89	0.95	0.30	0.67	0.30	0.98
FHS-GARCH	0.82	0.00	0.69	0.62	0.71	0.00
FHS-IV	0.96	0.62	0.37	0.74	0.02	0.04
EVT-GARCH	0.90	0.00	0.70	0.23	0.70	0.00
EVT-IV	0.54	0.85	0.57	0.99	0.02	0.12
CAViaR-SAV	0.71	0.00	0.61	0.36	0.00	0.00
RiskMetrics	0.79	0.10	0.65	0.55	0.25	0.07
GJR-GARCH (Student $t$)	0.88	0.00	0.93	0.41	0.71	0.00

The out-of-sample backtests displayed in Table 4 reveal that our proposed QR-IM model, which uses ATM volatility and the 25-delta RR as explanatory variables, delivers convincing results across quantiles. Part (a) indicates that most models do a reasonable job in terms of unconditional coverage, as expressed by the hit percentage. However, superior to all benchmark models, the QR-IM model performs very well under the Christoffersen and DQ tests across quantiles, as evidenced by the high $p$-values in parts (b) and (c).

The GJR-GARCH (Student $t$) model delivers mixed results and is not able to deliver consistent, reliable VaR estimates. The CAViaR-SAV is not dynamically well specified and does not perform well.

Based on the in-sample evaluation of Section 5.3.1, the IV-filtered FHS-IV and EVT-IV models are expected to be the closest competitors to the QR-IM models. However, when evaluated by the DQ test out-of-sample, the distinction between GARCH and IV filtered specifications becomes less clear. None of the EVT- or FHS-based models performs well across quantiles. In particular, EVT-IV and FHS-IV do not provide reliable VaR estimates in the right tail of the return distribution.

Rapach and Strauss (2008) find that the exchange rate volatility series often contain structural breaks, which is supported by the findings in Coudert et al (2011). Holmes (2008) documents nonstationarity of real exchange rates. As pointed out by Engle and Manganelli (2004), the unreliability of VaR forecasts could result from the parameter estimation procedure rather than model misspecification: in general, the accuracy of VaR forecasting is influenced by the number of observations in the tail of the distribution of the returns. When the data is scarce in the tail, the performance of the VaR estimation methods, and even the estimates of the unconditional empirical quantiles, can be unreliable. Biased parameter estimates is one possible explanation for the relative underperformance of the EVT, FHS, GJR-GARCH-$t$ and CAViaR-SAV models.

There are several plausible explanations for the highly satisfactory performance of the QR-IM model. We have already highlighted the utilization of forward-looking information derived from options as a fundamental feature of the QR-IM model to estimate the conditional return distribution. The implied volatility surface quickly factors in new information. The forward-looking nature of the QR model will immediately use this information in VaR estimates. This is in contrast to the CAViAR and GARCH-filtered benchmark models, which are autoregressive in nature. The FHS-IV and EVT-IV models share one of the features of the QR-IM models, by virtue of the implied volatility filtering. However, an important difference lies in the construction of the distribution of residuals. EVT is a parametric approach, while FHS relies entirely on historical return distributions. The QR-IM model, on the other hand, uses the RR as a proxy for higher-order moments – an approach which is neither parametric nor contingent on historical information. The latter is likely to be important under certain conditions.

In this context, it is relevant to discuss the sensitivity of model errors to structural breaks and rapidly changing market conditions. Generally, successful application of nonparametric and parametric models requires that the assumed innovation distribution appropriately represents the true data-generating process. For instance, the inherent model risk of a GJR-GARCH-based VaR model includes parameter estimation uncertainty and possibly incorrect innovation assumptions. Hence, in the presence of structural breaks or extreme market conditions leading to altered expectations of the true return distribution, the model might be misspecified and parameters might be biased, which would lead to unsatisfactory VaR estimates. Frequent reestimation to change the dynamics of the model could potentially reduce this challenging problem, but it is unlikely to capture a structural break completely. We argue that our proposed QR model is less exposed to this type of model risk. First, a general feature of QR is that the method is robust to outliers and does not require specific assumptions about the distribution of variables or residuals. Second, our specification uses both the second moment (approximated by ATM volatility) and higher-order moments (approximated by the RR) as explanatory variables to model the return distribution. These variables can be thought of as Markovian, since they are forward-looking and independent of previous realizations. Thus, any event causing altered expectations for the shape of the conditional return distribution, due to either long-term structural effects or short-term factors, will immediately impact VaR estimates through updated observed variables for the explanatory variables. As such, the need to reestimate model parameters becomes less urgent.⁶⁶ 6 This does not imply that the model would not benefit from a more representative estimation sample. However, the dynamics of the QR model and our choice of risk factors mitigate the effect of biased model parameters. Hence, due to both general features of the QR framework and our particular specification, the proposed model appears to be relatively robust.

6 Conclusions

In this paper we proposed a simple QR model to forecast daily exchange rate VaR in liquid FX markets. We validated the risk model, which is forward-looking and uses directly observable option prices as explanatory variables, through coverage tests and loss functions. In empirical applications to EUR/USD data, the model outperforms nonparametric and parametric models and compares well with the more complicated CAViaR model, both in-sample and out-of sample. In addition to delivering accurate estimates, the model is parsimonious, relies on simple optimization routines and does not require data beyond readily available interbank option prices. As such, the model can be easily implemented by practitioners.

We see several possible directions for future research, which has potential to improve the proposed model further. A first step would be to assess QR-IM model performance in informationally less efficient markets than EUR/USD. Nonlinear models and machine-learning-based estimation methods might be appropriate. To reduce parameter uncertainty, a distributed lag structure (or other combinations of readily available implied volatilities) might reduce noise in parameter estimates. Further, including explanatory variables beyond the volatility surface, such as macroeconomic variables, measures of risk aversion or intraday tick data, might prove helpful. Technically this is straightforward and can be done by adding relevant variables to the matrix of predictors. However, care should be taken to avoid multicollinearity and to preserve a parsimonious specification; regularization or variable selection procedures are advisable. Also, it is likely that the model will contribute positively in dynamic averaging modeling strategies. We leave this for further research.

Declaration of interest

The authors report no conflicts of interest. The authors alone are responsible for the content and writing of the paper.

Acknowledgements

The research presented in this paper is funded by SpareBank 1 Markets.

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• Keywords
• Foreign exchange
• Implied volatility
• Over-the-counter (OTC) derivatives
• Value-at-risk (VAR)
• Original research