Consider a generic financial portfolio described by a complete set of cashflows $\text{𝐂𝐅}={(t_i,{\mathrm{CF}}_i)}_{i\in \mathcal{4}}$, where $\mathcal{4}$ is an appropriate index set. Each cashflow ${\mathrm{CF}}_i$ is determined from values of “underliers” ${𝑿}_s$ for $s\le t$, where the underliers ${𝑿}_t$ represent a certain stochastic process. Formally, ${\mathrm{CF}}_i$ is measurable as of time $t_i$ with respect to a certain filtration ${ℱ}_t$, which is generated by ${𝑿}_s$ for $s\le t$. Examples of the financial portfolio could range from a single European option on a single underlier to a large portfolio representing a complete netting set of many financial instruments depending on many underliers.

In addition, we suppose that the value of the outstanding cashflows of the portfolio at time $t$ (in an appropriate sense, such as replication by the bank or a computation by some accepted central calculation agent) is given as a stochastic process $V_t$ or as an exact or approximate function $V(t,{𝑿}_t)$ (or $V(t,{𝑿}_t,{𝑼}_t)$, where ${𝑼}_t$ is an extra Markovian state necessary to compute the value, eg, a barrier breach indicator for barrier options). For approximate computation, we can view it as $V(t,{𝑹}_t)$, where the ${𝑹}_t$ are an appropriate set of features measurable as of ${ℱ}_t$. We also assume that the $V_t$ for different $t$ are discounted to a common time or otherwise made comparable so that they can be added to or subtracted from each other and any ${\mathrm{CF}}_i$ meaningfully.

We can interpret $V_t$ as the value to the bank (“us”): positive ${\mathrm{CF}}_i$ would be interpreted as payments to us, while negative ${\mathrm{CF}}_i$ are interpreted as payments by us to some counterparty.

Assuming some realization of ${𝑿}_t$ under some measure $ℳ$, indicated by an argument $\omega$ (as in ${𝑿}_t(\omega )$), there is one corresponding realization $V_t(\omega )$ and there is one realization $\mathrm{CF}(\omega )$. When comparing different measures, if $V$ is given by a function, it will be the same function but its arguments will follow their dynamics under the measure $ℳ$, while the processes $V_t$ will be different. In our examples here, the ${𝑿}_t$ will be given as realizations of a system of stochastic differential equations (SDEs).

Consider the change of value over the time period $t$ to $t+\delta$ defined by

If the measure $ℳ$ or function $f$ are clear from the context, we omit the corresponding superscripts.

Some simple examples for the function $f(x,t_i,t)$ are $x$, 0, $x^{+}$ or $x^{-}$. The interpretation would be that cashflows from the portfolio during that period are assumed to be not made or only partially made. In the counterparty credit risk applications, we could assume that the period from $t$ to $t+\delta$, the margin period of risk (MPoR), describes the period from the first uncertainty about the default of the counterparty to its resolution by default and liquidation and/or replacement of the portfolio. Often we assume that the bank will make payments until default is almost certain, while the counterparty will avoid payments, for example, by engaging in disputes (see Andersen et al (2017) for a more careful modeling of the MPoR). For market risk purposes, all payments are typically assumed to be made and received.

$\mathrm{\Delta }{V}_t^{\delta ,f,ℳ}(\omega )$ is now a random variable. In one common setting, the pertinent parts of the filtration ${ℱ}_t$ are ${𝑿}_t$ and ${𝑼}_t$. Each possible $\omega$ corresponds to (different) realizations ${𝑿}_s$ and ${𝑼}_s$ for $s\in [t,t+\delta ]$, which in turn lead to realizations of CF and $V_s$ in that interval.

In market risk (for market risk value-at-risk (VaR)) and counterparty credit risk (for initial margin requirements), we are interested in quantiles for that random variable, denoted by $Q_{\alpha }(\mathrm{\Delta }{V}_t^{\delta ,f,ℳ}\mid {ℱ}_t)$, and computed or approximated as functions $q_{\alpha }(t,{𝑿}_t,{𝑼}_t)$ or $q_{\alpha }(t,{𝑹}_t)$. In the first $q_{\alpha }$ expression, we find the function given the complete Markovian state. In the second, we find it as a function of a set of regressors ${𝑹}_t$. Often, we try to use as few regressors as possible to minimize computational requirements, for example, by using only $V_t$ itself or together with a few main risk factors.

In the initial margin requirement case, the percentile $\alpha$ is typically 1% (or 99% if seen from the other side). For market risk VaR, $\alpha$ is often 3%, 1% or 0.3% (or 97%, 99% or 99.7% if seen from the other side).

Often initial margin and market risk VaR are computed with $t$ representing today. However, to understand the impact of future initial margin (FIM) requirements or future VaR (fVaR), we consider $t$ representing times beyond the valuation date to model FIM requirements or fVaR. Then, we compute future expected margin requirements (or VaR) or quantiles of the future expected margin requirements (or VaR). The first is called dynamic initial margin (DIM) for future expected margin requirements, which can be written as

$\mathbb{P}$ would be a pricing measure if the future expected margin requirements are computed for pricing purposes, or it could be a historical measure if computed for risk management purposes.

However, once computed, the initial margin requirements or VaR can be used in follow-up computations. For initial margin requirements, we can compute the expected cost to fund FIM requirements (margin value adjustment). VaR (and now also expected shortfall (ES)) is used in the computation of required regulatory or economic capital, and further, the cost to fund it (capital value adjustment). While we do not cover these applications here, the methods described here can and have been applied to them, and they will be described elsewhere.

In this paper, we assume that the stochastic process for $V_t$ (respectively, the functions for $V$) are given (respectively, approximated) independently before we apply the methods given in the paper. In a follow-up paper, we describe how the $V_t$ can be computed through backward deep backward stochastic differential equation (deep BSDE) methods and how to implement methods to compute the quantiles in that particular case.

The rest of the paper is organized as follows. Section 2 briefly outlines the assumptions and requirements for the methods studied. Section 3 briefly compares the VaR and initial margin computations as studied here with regulatory requirements and other methods that have been implemented in the industry. Section 4 describes the nested Monte Carlo procedure to estimate the FIM, which will be used as a benchmark to compare other methods against. Section 5 describes methods using moment-matching techniques implemented on a cross section of the outer simulation paths, such as Johnson least-square Monte Carlo (JLSMC) and Gaussian least-square Monte Carlo (GLSMC) in order to fit an appropriate distribution for the computation of the initial margin. Section 6 describes the quantile regression method and its implementation. Section 7 describes the Delta-Gamma methods that utilize the analytic sensitivities of the instruments to estimate conditional quantile and FIM. Section 8 analyzes the effect of adding a limited number of inner samples for the estimation of FIM and DIM. While the methods described in the earlier sections (with the exception of nested Monte Carlo) all work on the cross section of outer paths, the estimation of the properties of the distribution could be refined further by the addition of a limited number of inner paths, which is the subject of this section. Section 9 presents a comparison of running times and root mean square errors (RMSEs) across many methods for the call and call combination examples, and discusses their performance. We end with our conclusions and discussion in Section 10.

Moment-matching least square Monte Carlo methods can be outlined as follows.

• •

  Generate $M$ portfolio scenarios (outer simulations) corresponding to the $k$ underlying assets, given the various risk factors and correlations from time 0 to maturity $T$.

  

• •

  Define the appropriate change in portfolio value corresponding to sample path $m$ as

  $$\mathrm{\Delta }{V}_m(t,\delta )=V_m(t+\delta )+{\mathrm{DCF}}_m(t,t+\delta )-V_m(t),$$

  (5.1)

  where $V_m$ is the pathwise portfolio value in scenario $m$, and ${\mathrm{DCF}}_m$ is the value of the fully or partially included net cashflow between the counterparties from time $t$ to $t+\delta$.

• •

  Based on the distribution(s) to be fitted, compute the first four raw sample moments $\mathrm{\Delta }{V}_m$, ${(\mathrm{\Delta }{V}_m)}^2$, ${(\mathrm{\Delta }{V}_m)}^3$ and ${(\mathrm{\Delta }{V}_m)}^4$ (for Johnson) or the first two raw sample moments $\mathrm{\Delta }{V}_m$ and ${(\mathrm{\Delta }{V}_m)}^2$ (for Gaussian) that will be used to estimate the actual raw moments.

• •

  Accordingly, estimate the first four raw moments $r_1$, $r_2$, $r_3$, $r_4$, or first two raw moments $r_1$, $r_2,$ as a function of the underlying states ${𝑿}_t$ or the portfolio value $V_t$ (or other regressors ${𝑹}_t$), by a suitable parametric function such as polynomial regression:

  	$r_i$	$=f(𝜷,𝑿)\approx E[{(\mathrm{\Delta }V)}^i\mid {𝑿}_t=𝑿]$
  or
  	$r_i$	$=f(𝜷,V)\approx E[{(\mathrm{\Delta }V)}^i\mid V_t=V],$

  where $𝜷$ is a vector of polynomial regression coefficients and $i=1,2,3,4$.

• •

  For GLSMC (Caspers et al 2017), the estimated raw moments $r_1$ and $r_2$ can be used to approximate the mean and variance of the distribution $\mathrm{\Delta }{V}_t$ (portfolio value change of interest during MPoR) for an assumed normal distribution of $\mathrm{\Delta }{V}_t$. The normal distribution with the estimated mean and variance is then used to estimate the conditional quantile.

• •

  For JLSMC (McWalter et al 2018), the estimated first four raw moments, $r_1$, $r_2$, $r_3$, $r_4$, can be used to fit an appropriate Johnson distribution via the moment-matching method (JMM) (Hill et al 1976) with (approximately) these first four moments. The Johnson’s distribution so obtained is then used to estimate the conditional quantile.

An example of raw moments from the portfolio value change and the corresponding regressed moments with portfolio value $V_t$ via polynomial regression for the first four moments for the call combination instrument is shown in Figure 4.

A detailed procedure and the settings used for JLSMC is presented in McWalter et al (2018) and is not repeated here for brevity. However, it is noted here that the moment-matching algorithm (Hill et al 1976), originally designed for smaller sets of observations, is quite involved. Therefore, the total number of given sets of moments that must be fitted via the JMM algorithm is equal to the number of outer simulation paths, $N_{\mathrm{O}}=\mathrm{100\hspace{0.17em}000}$, at each time step where VaR is required. This poses a heavy computational burden in the absence of additional redesign and parallelization of the moment-matching algorithm. Hence, in order to reduce the computational load and obtain results with comparable accuracy, only moments at select values of the outer simulation values, either certain values of the underlying risk factors ${𝑿}_t$ or certain quantiles of the portfolio values $V_t$ at time $t$, are used for moment matching. Let $v_j=Q_{j/N}(V_t),j=0,\mathrm{\dots },N$, be the $j/N$th quantile of the samples $V_t$ at time $t$ and let

be the corresponding raw moments obtained via parametric regression at the specific values of $V_t$ or ${𝑿}_t$. The parameters of Johnson’s corresponding distribution obtained by the moment-matching procedure are given by

where $\mathrm{𝚡𝚒}(𝚓),\mathrm{𝚕𝚊𝚖𝚋𝚍𝚊}(𝚓),\mathrm{𝚍𝚎𝚕𝚝𝚊}(𝚓),\mathrm{𝚐𝚊𝚖𝚖𝚊}(𝚓)={\xi }_j,{\lambda }_j,{\delta }_j,{\gamma }_j$ are the parameters of Johnson’s distribution, and $\mathrm{𝚝𝚢𝚙𝚎}(𝚓)$ represents the type or family of the distribution. The probability density function (pdf) of Johnson’s distributions belonging to different families is given by Johnson (1949) and McWalter et al (2018) as follows:

• •

  $S_{\mathrm{L}}$ family,

  (5.2)

• •

  $S_{\mathrm{B}}$ family,

  (5.3)

  and

• •

  $S_{\mathrm{U}}$ family,

  	$p(y)$	$={\displaystyle \frac{\delta }{\sqrt{2\pi }}}{\displaystyle \frac{1}{\sqrt{y^2+1}}}\exp \{-\frac{1}{2}{[\gamma +\delta \log (y+\sqrt{Y^2+1})]}^2\},$
  				(5.4)

for $y=(X-\xi )/\lambda$.

The above methods were compared for a realistic IR swap example, with one party making annual fixed rate payments and the counterparty making quarterly floating rate payments over a 15-year period. Here, party $A$ pays a floating rate with a constant spread of 0.9% every three months over the current floating IR (London Interbank Offered Rate/prime, etc). Party $B$ pays a fixed annual rate of 4.5% every year based on the IR agreed at the start of the contract. The outstanding notional value varies from USD36.2 million in the first year to USD63.2 million at the end of year 15. The MPoR is set to 10 business days or 0.04 years, with a total of 375 intervals until maturity. The IR and risk factors were simulated assuming G1$++$ short rate processes, and a sample path for cashflows, portfolio values and the risk factors is shown for a sample scenario in Figure 5.

The conditional 1% quantiles obtained via GLSMC and JLSMC at time $t=8.38$ years are compared in Figure 6 and are found to be in good agreement.

Finally, the time evolution of DIM for the same instrument over the entire 15-year period as defined in (1.2) is compared for GLSMC and JLSMC. The results are shown in Figure 7, where the first four moments are regressed with Laguerre polynomials of degree $7$. In part (a), cashflows during MPoR are included in the change of value and part (b), MPoR cashflows are not included. It can be seen that not assuming that all cashflows are paid can make the margin requirements much more spiky. The results for the different methods are seen to be in good agreement with each other.

We present another instrument, namely an FX call option, in order to highlight the differences in the results obtained via the GLSMC and JLSMC methods. The Black–Scholes FX call is simulated and priced with a constant domestic risk-free rate $r_{\mathrm{d}}=0.08$ and a constant foreign rate of $r_{\mathrm{f}}=0.02$, along with an FX rate volatility of $\sigma =0.3$. The current spot FX rate is $S_0=100.0$. The option has a maturity of $T=1.0$ year and the strike $K=105.0$. Similar to the other examples, the MPoR is fixed at 0.04 years, with 25 MPoR intervals until maturity.

An algorithm to fit an appropriate Johnson’s distribution given the first four raw or central moments is presented in Hill et al (1976). Raw moments are estimated by independent polynomial regression on the first four powers of the change in portfolio values, as shown in Figure 4.

Johnson’s analysis of the types of distributions in Johnson (1949) also outlines the constraints for various moments, under which a valid distribution can be found. The relationship between skew (${\beta }_1$) and kurtosis (${\beta }_2$) requires that ${\beta }_2\ge {\beta }_1+1$. However, since the moments are regressed independently without imposing any joint constraints, there is no guarantee that relationship between skew and kurtosis will be satisfied everywhere when the regressed form of the moments is used. In fact, it can be seen that in some examples under some regression models, the skew–kurtosis relationship fails to hold at some points, and therefore we are unable to find a distribution that matches the moments, as shown in Figure 8. This is elaborated further in the following subsection.

As a first approximation, the change in portfolio values can be considered to be normally distributed, with mean and variance obtained from (7.6). Here, the discounted asset return $R$ is assumed to be normally distributed with mean zero and variance $\mathrm{\Omega }={\sigma }^2$, ie, $R\sim 𝒩(0,\mathrm{\Omega })$:

Further, with the approximation that $\mathrm{\Delta }V$ follows a normal distribution, the corresponding $\alpha$th quantile of the distribution can be written as

where $z_{\alpha }$ is the $\alpha$th quantile of the standard normal distribution.

However, this approximation, which could significantly deviate from the actual $\mathrm{\Delta }V\mid V$ distribution, could lead to overestimation or underestimation of the margin requirements. This deviation due to the approximation is apparent from the actual sample distribution shown in Figure 2(a), which is skewed and could be heavy-tailed depending on the value of the portfolio and the risk factors (Figure 12). A comparison of DIM obtained via this approximation against other methods for the call combination discussed earlier is shown in Figure 13. Hence, although the normal approximation of the $\mathrm{\Delta }V\mid V$ distribution allows for rapid estimation of initial margin requirements, it may be not be suitable for complex portfolios or portfolios with more complex instruments.

A further refinement of the previous method is to relate the quantiles of an empirical distribution to the quantiles of standard normal distribution via the higher-order moments of the empirical distribution and the so-called Cornish–Fisher expansion. Given the raw moments and the corresponding central moments, ${\mathrm{cm}}_1$, ${\mathrm{cm}}_2$, ${\mathrm{cm}}_3$, ${\mathrm{cm}}_4$, ${\mathrm{cm}}_5$ from raw moments, compute the skew (${\kappa }_3$), kurtosis (${\kappa }_4$) and ${\kappa }_5={\mathrm{cm}}_5^5/{\mathrm{cm}}_2^{5/2}$ from the central moments. Hence, the Delta-Gamma method approximates the quantiles of the portfolio value changes $\mathrm{\Delta }V\mid V$ in terms of its raw moments, and it uses the Cornish–Fisher expansion to compute the $\alpha$th percentile of $\mathrm{\Delta }V\mid V$:

where $z_{\alpha }$ is the $\alpha$th quantile of a standard normal distribution and ${\kappa }_3$, ${\kappa }_4$, ${\kappa }_5$ are the skew, kurtosis and higher-order moments of the distribution, respectively. The Delta-Gamma approach approximates the above quantities from the corresponding portfolio sensitivities and asset returns over the MPoR.

Similar to the Delta-Gamma-normal method, the discounted asset return is assumed to be normally distributed with mean zero and variance $\mathrm{\Omega }$, ie, $R\sim 𝒩(0,\mathrm{\Omega })$. The corresponding raw moments written as powers of $\mathrm{\Delta }V$ in terms of ${\delta }^{\$}(=\delta ),{\gamma }^{\$}(=\gamma ),R$ and $\mathrm{\Omega }$ are

Using the first 10 moments of the normal distribution, the expectations of the above powers of $\mathrm{\Delta }V$ can be written as

The central moments, ${\mathrm{cm}}_1$, ${\mathrm{cm}}_2$, ${\mathrm{cm}}_3$, ${\mathrm{cm}}_4$, ${\mathrm{cm}}_5$ from the above raw moments, the skew (${\kappa }_3$), kurtosis (${\kappa }_4$) and ${\kappa }_5={\mathrm{cm}}_5^5/{\mathrm{cm}}_2^{5/2}$ derived from above terms are then used to determine the corresponding $\alpha$th quantile from (7.8). The comparison of DIM and VaR estimates obtained via this method for the call combination discussed earlier is shown in Figure 13, which is seen to be in much closer agreement with the benchmark nested Monte Carlo method, while requiring far lower computational resources than the same. Hence, the above method is suitable for instruments and portfolios with given analytical or approximate first- and second-order sensitivities.

Based on the GLSMC and JLSMC methods presented in earlier sections, the moment-matching algorithms are applied to moment estimates of $\mathrm{\Delta }V\mid V$ that use an additional limited number of inner samples. The raw moments ${(\mathrm{\Delta }{V}_m)}^i$ for $i=1,2,3,4$ from a single outer path are replaced by $(1/N_{\mathrm{I}}){\sum }_{j=1}^{N_{\mathrm{I}}}{(\mathrm{\Delta }{V}_{jm})}^i$ for $i=1,2,3,4$, where $N_{\mathrm{I}}$ is the number of inner simulations for the outer portfolio scenario $m$. Here, the number of inner simulations is chosen to be rather small ($N_{\mathrm{I}}=50$), in order to limit the needed computational resources. The comparison of raw moments in Figure 14 at a specific time ($t=0.16$ years) across all the outer paths shows that additional inner samples decrease the variance of the raw moments. The time evolution of DIM obtained via GLSMC and JLSMC methods using these raw moments is shown in Figure 15. Although the time evolution obtained via the GLSMC method has not changed much, as it depends only on the first two moments, the JLSMC method seems to exhibit a smoother time evolution as it depends on and is sensitive to higher moments.

The Johnson percentile-matching method fits an Sl-, Su- or Sb-type Johnson distribution (“Sl” denotes logarithmic, “Sb” denotes bounded, “Su” denotes unbounded and “Sn” denotes normal) given a limited number of inner samples, based on a discriminant computed from particular percentiles of the inner samples data. The procedure is outlined in Slifker and Shapiro (1980) and George and Ramachandran (2011). In short, a fixed value $z$ () is chosen and the corresponding percentiles $P_{3z}$, $P_{-3z}$, $P_z$ and $P_{-z}$ of the standard normal distribution corresponding to the four points $\pm 3z$ and $\pm z$ are computed. The values of data points corresponding to the percentiles are then determined from the sample distribution as $Q_{P_i}(X)$, where $i=3z,z,-z,-3z$, as $x_{3z}$, $x_z$, $x_{-z}$, $x_{-3z}$.

The value of the discriminant $d=mn/p^2$ is then used to select the type of Johnson distribution, where $p=x_z-x_{-z}$, $m=x_{3z}-x_z$ and $n=x_{-z}-x_{-3z}$. The Su Johnson distribution is fitted to the data if $d>1.001$, the Sb Johnson distribution is chosen if and the Sl Johnson distribution is chosen if $0.999\le d\le 1.001$. This method provides a closed-form solution to the Johnson distribution parameters $\xi$, $\gamma$, $\delta$ and $\lambda$ in terms of $m$, $n$ and the discriminant $d$, hence requiring far fewer computational resources than the moment-matching algorithm.

The time evolution of DIM obtained via this method with a much reduced $N_{\mathrm{I}}=200$ number of inner simulations is compared with nested Monte Carlo and Delta-Gamma approaches in Figure 13 and is seen to be in good agreement with the nested Monte Carlo approach.

The conditional quantile regression approach can also take advantage of additional inner samples. Similarly to the other methods, samples from limited inner simulations were added to the training samples for quantile regression. A comparison of the time evolution of DIM obtained via quantile regression with a different number of inner samples is shown in Figure 17. Although the inner samples help refine the distribution, the time evolution thus obtained is still comparable in terms of accuracy and scale to that obtained with a single inner sample. This indicates that with a sufficient number of outer samples (here, $N_{\mathrm{O}}=100$K) the quantile regression procedure can be robust with respect to variations in inner distributions across the underlier at any given time or at different times. Comparing the results of a single inner sample versus the others, it seems that adding at least a few inner samples results in a somewhat smoother evolution of the DIM.

As the process of running nested inner simulations for a large number of outer simulations is computationally very expensive, an alternative method is to approximate inner simulations with available nearby outer simulation values around a neighborhood of $X_t$ or $V_t$. Hence, under some circumstances the values from outer simulations can be considered as proxies for samples from the actual distribution of portfolio values or pseudo inner samples.

The distribution $(\mathrm{\Delta }V\mid x_t=X_t)$ is approximated by observations from portfolio scenario $m$, with $\mathrm{\Delta }{V}_m$ given by (5.1) such that $x_t^{(m)}\in {ℬ}_{\epsilon }(X_t)$ or $V_t^{(m)}\in {ℬ}_{{\epsilon }_1}(V_t)$, where the underlying asset or portfolio value is observed to be within a neighborhood ${ℬ}_{\epsilon }$ of $X_t$ or $V_t$. Figure 18 presents the time evolution of initial margin estimates obtained via a limited number of pseudo inner samples considered around the neighborhood of $X_t$, for the JPP method, along with regular inner samples also implemented for the same method. It can be seen that both JPP with inner samples and JPP with pseudo inner samples agree well with DIM obtained via nested Monte Carlo for the call combination and FX call instruments. Additionally, the DIM obtained by estimating quantiles from the raw pseudo samples is also presented, which is seen to be in good agreement with that obtained via nested Monte Carlo as well.

It is illustrative to reflect on the requirements and constraints of each of these methods in light of the above results. The methods with pseudo samples outperform the traditional JLSMC and quantile regression methods in terms of accuracy and speed for these particular financial instruments with one-dimensional risk-factors ($X_t$) and portfolio values ($V_t$). However, with multiple risk factors in higher dimensions the number of pseudo samples that can be borrowed from within a higher-dimensional bin would be much sparser, thus leading to noisy estimates of quantiles. Second, the number of higher-dimensional bins that need to be considered grows rapidly with the dimensions, thus forcing us to resort to $V_t$-based pseudo samples for quantile estimation. Further, even for one-dimensional problems, it is necessary to have a sufficient number of samples within a neighborhood of $V_t$ or $X_t$, in order to accurately capture the properties of the distribution. The number of active samples within any neighborhood of $V_t$ or $X_t$ becomes sparse with a higher volatility and time horizon, again leading to noisy estimates. Hence, the pseudo-sample-based methods are ideally suited for problems with rich outer-simulation data.

Methods with moment regression (JLSMC) give medium accuracy, wherein it is necessary to have a good estimate of moment values, with methods and distributions that can fit higher moments typically performing better (JLSMC performing better than GLSMC).

Any inconsistencies in moment estimation leads to undefined distributions, thus necessitating moment correction. Hence, moment regression methods are particularly sensitive to the type of moment regression and the corresponding moment-matching procedure. We also see that moment regression methods are relatively sensitive to the exact way they are set up, thresholds, etc.

The quantile regression (LGBM quantile regression and deep quantile regression) procedures perform best with a sufficient number of outer samples as they aim to minimize the quantile regression loss function. However, the computational requirements grow with the number of available samples (outer paths) while requiring sophisticated optimization frameworks. It is seen that LGBM quantile regression produces results comparable to pseudo sample methods, as it employs a series of tree-based optimizations and empirical estimation of quantiles in order to minimize the quantile loss function at the cost of a longer running time. On the other hand, the deep quantile regression procedure involves minimizing the quantile loss function via neural networks over several thousand minimization steps, which can become impractical for a large number of DIM steps, unless the training of and optimization over the DNN can be sped up substantially.

The objective of this study is to present the details and procedures of various methods used for estimation of fVaR and initial margin, along with an objective evaluation of their accuracies and performance. The choice of method for a particular problem or financial instrument depends on a variety of constraints and requirements as outlined in Table 1, and there is probably not a single method that can be applied for all financial instruments under all scenarios.