Risk factor evolution for counterparty credit risk under a hidden Markov model

Anagnostou, Ioannis; Kandhai, Drona Article Risk factor evolution for counterparty credit risk under a hidden Markov model Risks Provided in Cooperation with: MDPI – Multidisciplinary Digital Publishing Institute, Basel Suggested Citation: Anagnostou, Ioannis; Kandhai, Drona (2019) : Risk factor evolution for counterparty credit risk under a hidden Markov model, Risks, ISSN 2227-9091, MDPI, Basel, Vol. 7, Iss. 2, pp. 1-22, https://doi.org/10.3390/risks7020066 This Version is available at: https://hdl.handle.net/10419/257904 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ risks Article Risk Factor Evolution for Counterparty Credit Risk under a Hidden Markov Model Ioannis Anagnostou 1,2,* and Drona Kandhai 1,2 1Computational Science Lab, University of Amsterdam, Science Park 904, 1098XH Amsterdam, The Netherlands; [email protected] 2Quantitative Analytics, ING Bank, Foppingadreef 7, 1102BD Amsterdam, The Netherlands *Correspondence: [email protected]; Tel.: +31-20-525-6789 Received: 31 March 2019; Accepted: 5 June 2019; Published: 12 June 2019   Abstract: One of the key components of counterparty credit risk (CCR) measurement is generating scenarios for the evolution of the underlying risk factors, such as interest and exchange rates, equity and commodity prices, and credit spreads. Geometric Brownian Motion (GBM) is a widely used method for modeling the evolution of exchange rates. An important limitation of GBM is that, due to the assumption of constant drift and volatility, stylized facts of financial time-series, such as volatility clustering and heavy-tailedness in the returns distribution, cannot be captured. We propose a model where volatility and drift are able to switch between regimes; more specifically, they are governed by an unobservable Markov chain. Hence, we model exchange rates with a hidden Markov model (HMM) and generate scenarios for counterparty exposure using this approach. A numerical study is carried out and backtesting results for a number of exchange rates are presented. The impact of using a regime-switching model on counterparty exposure is found to be profound for derivatives with non-linear payoffs. Keywords: Counterparty Credit Risk; Hidden Markov Model; Risk Factor Evolution; Backtesting; FX rate; Geometric Brownian Motion 1. Introduction One of the main factors that amplified the financial crisis of 2007–2008 was the failure to capture major risks associated with over-the-counter (OTC) derivative-related exposures (Basel Committee on Banking Supervision 2010a). Counterparty exposure, at any future time, is the amount that would be lost in the event that a counterparty to a derivative transaction would default, assuming zero recovery at that time. Banks are required to hold regulatory capital against their current and future exposures to all counterparties in OTC derivative transactions. A key component of the counterparty exposure framework is modeling the evolution of underlying risk factors, such as interest and exchange rates, equity and commodity prices, and credit spreads. Risk Factor Evolution (RFE) models are, arguably, the most important part of counterparty exposure modeling, since small changes in the underlying risk factors may have a profound impact on the exposure and, as a result, on the regulatory and economic capital buffers. It is, therefore, crucial for financial institutions to put significant effort in the design and calibration of RFE models and, in addition, have a sound framework in place in order to assess the forecasting capability of the model. Although the Basel Committee on Banking Supervision has stressed the importance of the ongoing validation of internal models method (IMM) for counterparty exposure (Basel Committee on Banking Supervision 2010b), there are no strict guidelines on the specifics of this validation process. As a result, there is some degree of ambiguity regarding the regulatory requirements that financial institutions are expected to meet. In an attempt to reduce this ambiguity, Anfuso et al. (2014) introduced a complete Risks 2019,7, 66; doi:10.3390/risks7020066 www.mdpi.com/journal/risks Risks 2019,7, 66 2 of 22 framework for counterparty credit risk (CCR) model backtesting which is compliant with Basel III and the new Capital Requirements Directives (CRD IV). A detailed backtesting framework for CCR models was also introduced by Ruiz (2014), who expanded the corresponding framework for Value-at-Risk (VaR) models by the Basel Committee (Basel Committee on Banking Supervision 1996). The most ubiquitous model for the evolution of exchange rates is Geometric Brownian Motion (GBM). Under GBM, the exchange rate dynamics are assumed to follow a continuous-time stochastic process, in which the returns are log-normally distributed. Although simplicity and tractability render GBM a particularly popular modeling choice, it is generally accepted that it cannot adequately describe the empirical facts exhibited by real exchange rate returns (Boothe and Glassman 1987). More specifically, exchange rate returns can be leptokurtic, exhibiting tails that exceed those of the normal distribution. As a result, a scenario-generation framework based on GBM may assign unrealistically low probabilities to extreme scenarios, leading to the under-estimation of counterparty exposure and, consequently, regulatory and economic capital buffers. The main reason for the inability of GBM to produce return distributions with realistically heavy tails is the assumption of constant drift and volatility parameters. In this paper, we present a way to address this limitation without entirely departing from the convenient GBM framework. We propose a model where the GBM parameters are allowed to switch between different states, governed by an unobservable Markov process. Thus, we model exchange rates with a hidden Markov model (HMM) and generate scenarios for counterparty exposure using this approach. A HMM is a mathematical model in which the system being modeled is assumed to follow a Markov chain whose states are hidden from the observer. HMMs have a broad range of applications, in speech recognition (Juang and Rabiner 1991), computational biology (Krogh et al. 1994), gesture recognition (Wilson and Bobick 1999), and in other areas of artificial intelligence and pattern recognition (Ghahramani 2001). HMMs have gained significant popularity in the mathematical and computational finance fields. The application of HMMs in financial and economic time-series was pioneered by Hamilton in Hamilton (1988;1989). Since then, a significant amount of literature has been published, focusing on the ability of HMMs to reproduce stylized facts of asset returns (Bulla et al. 2011;Nystrup et al. 2015;Rydén et al. 1998), asset allocation (Ang and Bekaert 2004;Guidolin and Timmermann 2007; Nystrup et al. 2015), and option pricing (Bollen 1998;Guo 2001;Naik 1993). Our paper expands the counterparty exposure literature by introducing a hidden Markov model for the evolution of exchange rates. We provide a detailed description of HMMs and their estimation process. In our numerical experiments, we use GBM and HMM to generate scenarios for the Euro against two major and two emerging currencies. We perform a thorough backtesting exercise, based on the framework proposed by Ruiz (2014), and find similar performances for GBM and a two-state HMM. Finally, we use the generated scenarios to calculate credit exposure for foreign exhange (FX) options, and find significant differences between the two models, which are even more pronounced for deep out-of-the-money instruments. The remainder of the paper is organized as follows. Section 2provides the fundamentals of HMMs, along with the algorithms for determining their parameters from data. Section 3gives background information on modeling the evolution of exchange rates. Section 4outlines the framework for performance evaluation of RFE models. A numerical study is presented in Section 5. Finally, in Section 6, we draw conclusions and discuss future research directions. 2. An Introduction to Hidden Markov Models The hidden Markov model (HMM) is a statistical model in which a sequence of observations is generated by a sequence of unobserved states. The hidden state transitions are assumed to follow a first-order Markov chain. The theory of hidden Markov models (HMMs) originates from the work of Baum et al. in the late 1960s (Baum and Petrie (1966), Baum and Eagon (1967)). In the rest of this section, we introduce the theory of hidden Markov models (HMMs), following Rabiner (1990). Risks 2019,7, 66 3 of 22 2.1. Formal Definition of a HMM In order to formally define a hidden Markov model (HMM), the following elements are required: 1. N , the number of hidden states. Even though the states are not directly observed, in many practical applications they have some physical interpretation. For instance, in financial time-series, hidden states may correspond to different phases of the business cycle, such as prosperity and depression. We denote the states by X={X1,X2, . . . , XN}, and the state at time tby qt. 2. M , the number of distinct observation symbols per state. These symbols represent the physical output of the system being modeled. The individual symbols are denoted by V={vl , v2 , . . . , vM} . 3. The transition probability distribution between hidden states, A={aij}, where aij =Pqt+i=Xj|qt=Xi, 1 ≤i,j≤N. (1) 4. The observation symbol probability distribution in state j,B={bj(k)}, where bj(k) = Pvkat t|qt=Xj, 1 ≤j≤N,1 ≤k≤M. (2) 5. The initial distribution of the hidden states, π={πi}, where πi=P[q1=Xi], 1 ≤i≤N. (3) The parameter set of the model is denoted by λ= (A , B , π) . A graphical representation of a hidden Markov model with two states and three discrete observations is given by Figure 1. X1X2 a12 v1 b1(1) v2v3 Figure 1. A hidden Markov model (HMM) with two states and three discrete observations, where aij is the probability of transition from state Xi to state Xj and bj(k) is the emission probability for symbol vk in state Xj. In the case where there are an infinite amount of symbols for each hidden state, vk is omitted and the observation probability bj(k), conditional on the hidden state Xj, can be replaced by bj(Ot) = P(Ot|qt=Xj). If the observation symbol probability distributions are Gaussian, then bj(Ot) = φ(Ot|uj , σj) , where φ(·) is the Gaussian probability density function, and uj and σj are the mean and standard deviation of the corresponding state Xj , respectively. In that case, the parameter set of the model is λ= (A,u,σ,π), where uand σare vectors of means and standard deviations, respectively. 2.2. The Three Basic Problems for HMMs The idea that HMMs should be characterized by three fundamental problems originates from the seminal paper of Rabiner (1990). These three problems are the following: Problem 1 (Likelihood) . Given the observation sequence O=O1O2. . .OT and a model λ= (A , B , π) , how do we compute the conditional probability P(O|λ)in an efficient manner? Risks 2019,7, 66 4 of 22 Problem 2 (Decoding) . Given the observation sequence O=O1O2. . .OT and a model λ , how do we determine the state sequence Q =q1q2. . . qTwhich optimally explains the observations? Problem 3 (Learning).How do we select model parameters λ= (A,B,π)that maximize P(O|λ)? 2.3. Solutions to the Three Basic Problems 2.3.1. Likelihood Our objective is to calculate the likelihood of a particular observation sequence, O=O1O2···OT , given the model λ . The most intuitive way of doing this is by summing the joint probability of O and Qfor all possible state sequences Qof length T: P(O|λ) = ∑ all Q P(O|Q,λ)·P(Q|λ). (4) The probability of a particular observation sequence O , given a state sequence Q=q1q2···qT , is P(O|Q,λ) = T ∏ t=1 P(Ot|qt,λ) =bq1(O1)·bq2(O2)···bqT(OT), (5) as we have assumed that the observations are independent. The probability of a state sequence Q can be written as P(Q|λ) = πq1aq1q2aq2q3···aqT−1qT. (6) The joint probability of Oand Qis the product of the above two terms; that is, P(O,Q|λ) = P(O|Q,λ)·P(Q|λ). (7) Although the calculation of P(O|λ) using the above definition is rather straightforward, the associated computational cost is huge. Thankfully, a dynamic programming approach, called the Forward Algorithm, can be used instead. Consider the forward variable αi(t), defined as αt(i) = P(O1O2···Ot,qt=Xi|λ). (8) We can solve for αt(i)inductively using Algorithm 1. Risks 2019,7, 66 5 of 22 Algorithm 1 The Forward Algorithm. 1. Initialization: α1(i) = πibi(O1), 1 ≤i≤N. (9) 2. Induction: αt+1(j) = "N ∑ i=1 αt(i)aij#bj(Ot+1), 1 ≤t≤T−1 1≤j≤N. (10) 3. Termination: P(O|λ) = N ∑ i=1 αT(i). (11) Correspondingly, we can define a backward variable βt(i)as βt(i) = P(Ot+1Ot+2···OT|qt=Xi,λ). (12) Again, we can solve for βt(i)inductively using Algorithm 2. Algorithm 2 The Backward Algorithm. 1. Initialization: βT(i) = 1, 1 ≤i≤N. (13) 2. Induction: βt(i) = N ∑ j=1 aij bj(Ot+1)βt+1(j),t=T−1, T−2, . . .,1 1≤i≤N. (14) 2.3.2. Decoding In order to identify the best sequence Q={q1q2···qT} for the given observation sequence O={O1O2···OT}, we need to define the quantity δt(i) = max q1,q2,...,qt−1P(q1q2···qt=i,O1O2···Ot|λ). (15) By induction, we have δt+1(j) = max iδt(i)aij·bj(Ot+1). (16) To actually retrieve the state sequence, it is necessary to keep track of the argument which maximizes Equation (16) , for each t and j . We do so via the array ψt(j) . The complete procedure for finding the best state sequence is presented in Algorithm 3. Risks 2019,7, 66 6 of 22 Algorithm 3 Viterbi algorithm. 1. Initialization: δ1(i) = πibi(O1), 1 ≤i≤N(17) ψ1(i) = 0. (18) 2. Recursion: δt(j) = max 1≤i≤Nδt−1(i)aijbj(Ot), 2 ≤t≤T 1≤j≤N(19) ψt(j) = argmax 1≤i≤Nδt−1(i)aij2≤t≤T 1≤j≤N. (20) 3. Termination: P∗=max 1≤i≤N[δT(i)] q∗ T=argmin 1≤i≤N [δT(i)]. (21) 4. Sequence back-tracking: q∗ t=ψt+1(q∗ t+1),t=T−1, T−2,···,1. (22) 2.3.3. Learning The model which maximizes the probability of an observation sequence O , given a model λ= (A,B,π) , cannot be determined analytically. However, a local maximum can be found using an iterative algorithm, such as the Baum-Welch method or the expectation-maximization (EM) method (Dempster et al. 1977). In order to describe the iterative procedure of obtaining the HMM parameters, we need to define ξt(i , j) , the probability of being at the state Xi at time t , and the state Xj at time t+ 1, given the model and observation sequence; that is, ξt(i,j) = P(qt=Xi,qt+1=Xj|O,λ). (23) Using the earlier defined forward and backward variables, ξt(i,j)can be rewritten as ξt(i,j) = αt(i)aijbj(Ot+1)βt+1(j) P(O|λ). (24) We define γt(i) = N ∑ j=i ξt(i,j)(25) as the probability of being in state Xiat time t. It is clear that T−1 ∑ t=i γt(i) = expected number of transitions from Xi, and (26) T−1 ∑ t=i ξt(i,j) = expected number of transitions from Xito Xj. (27) Risks 2019,7, 66 7 of 22 Using these formulas, the parameters of a HMM can be estimated, in an iterative manner, as follows: ˆ πi=γ1(i) = expected number of times in state Xiat time t=1; (28) ˆ aij = T−1 ∑ t=i ξt(i,j) T−1 ∑ t=i γt(i) =expected number of transitions from Xito Xj expected number of transitions from Xi ; (29) ˆ bj(k) = T ∑ t=1 1{Ot=vk}γt(j) T ∑ t=1 γt(j) =expected number of times in state jand observing symbol vk expected number of times in state j. (30) If λ= (A , B , π) is the current model and ˆ λ= ( ˆ A , ˆ B , ˆ π) is the re-estimated one, then it has been shown, by Baum and Eagon (1967); Baum and Petrie (1966), that P(O|ˆ λ)≥P(O|λ). In case the observation probabilities are Gaussian, the following formulas are used to update the model parameters uand σ: ˆ uj= T ∑ t=1 γt(j)Ot T ∑ t=1 γt(j) , (31) ˆ σj=v u u u u u u u t T ∑ t=1 γt(j)(Ot−uj)2 T ∑ t=1 γt(j) . (32) 3. Modelling the Evolution of Exchange Rates As discussed in the introduction, the first step in calculating the future distribution of counterparty exposure is the generation of scenarios using the models that represent the evolution of the underlying market factors. These factors typically include interest and exchange rates, equity and commodity prices, and credit spreads. This article is concerned with the modeling of exchange rates. 3.1. Geometric Brownian Motion In mathematical finance, the Geometric Brownian Motion (GBM) model is the stochastic process which is usually assumed for the evolution of stock prices (Hull 2009). Due to its simplicity and tractability, GBM is also a widely used model for the evolution of exchange rates. A stochastic process, St , is said to follow a GBM if it satisfies the following stochastic differential equation: dSt=µStdt +σStdWt, (33) where Wt is a Wiener process, and µ and σ are constants representing the drift and volatility, respectively. Risks 2019,7, 66 8 of 22 The analytical solution of Equation (33) is given by: St=S0expµ−σ2 2t+σWt. (34) With this expression in hand, and knowing that Wt∼N( 0, t) , one can generate scenarios simply by generating standard normal random numbers. 3.2. A Hidden Markov Model for Drift and Volatility One of the main shortcomings ofthe GBM model is that, due to the assumption of constant drift and volatility, some important characteristics of financial time-series, such as volatility clustering and heavy-tailedness in the return distribution, cannot be captured. To address these limitations, we consider a model with an additional stochastic process. The observations of the exchange rates are assumed to be generated by a discretised GBM, in which both the drift and volatility parameters are able to switch, according to the state of an unobservable process which satisfies the Markov property. In other words, the conditional probability distribution of future states depends solely upon the current state, not on the sequence of states that preceded it. The observations also satisfy a Markov property with respect to the states (i.e., given the current state, they are independent of the history). Thus, we consider a hidden Markov model with Gaussian emissions λ= (A , u , σ , π) , as was presented in Section 2.1. We denote the hidden states by X={X1 , X2 , . . . , XN} , and the state at time t as qt . The unobservable Markov process governs the distribution of the log-return process Y={Y2, . . . ,YT}, where Yt=log St St−1,t=2, . . ., T. The dynamics of Yare then as follows: Yt=u(qt) + σ(qt)Zt, (35) where u(qt) = µ(qt)−σ2(qt) 2 and Zt∼N( 0,1 ) are independent standard normal random numbers. The transition probabilities of the hidden process, as well as the drift and volatility of the GBM, can be estimated from a series of observations, using the algorithms presented in Section 2. The number of hidden states has to be specified in advance. In many practical applications, the number of hidden states can be determined based on intuition. For example, stock markets are often characterized as “bull” or “bear”, based on whether they are appreciating or depreciating in value. A bull market occurs when returns are positive and volatility is low. On the other hand, a bear market occurs when returns are negative and volatility is high. It would, therefore, be in line with intuition to assume that stock market observations are driven by a two-state process. The number of states can also be determined empirically; for example, using the Akaike information criterion (AIC) or the Bayesian information criterion (BIC). Once the model parameters have been estimated, scenarios can be generated by generating the hidden Markov chain and sampling the log-returns from the corresponding distributions. 4. RFE Model Performance Evaluation 4.1. Backtesting In this sub-section, we give a brief overview of a framework for the backtesting of RFE models. For a more detailed description, the reader is referred to Ruiz (2014). Backtesting is the process of comparing the distributions given by the RFE models with the realized history of the corresponding risk factors. In accordance with regulatory requirements, RFE models have to be backtested at Risks 2019,7, 66 15 of 22 Table 2. Backtesting results for GBP/EUR with calibration window Tc= 3 years, frequency of re-calibration δc=3 months, and backtesting window Tb=10 years. Time Horizon GBM HMM2 AD CVM KS AD CVM KS 1W 0.9861 0.9816 0.9820 0.8698 0.9280 0.6391 2W 0.9936 0.9919 0.9964 0.9934 0.9920 0.9952 1M 0.9085 0.8965 0.9594 0.9299 0.9140 0.9726 3M 0.8702 0.8273 0.8716 0.8573 0.8014 0.8535 Figure 8shows the 5th and 95th percentiles of the forecast distributions between 2011 and end of 2016. Similarly to the results for USD/EUR, HMM gave slightly more conservative forecasts and the realized time-series fell within the 90% probability region under both models, at the end of the 7 year period. However, in 2016, the realized time-series fell outside the 95th percentile of the GBM distribution, while it was still within this bound for the HMM. 2011 2012 2013 2014 2015 2016 Year 0.2 0.4 0.6 0.8 1.0 1.2 1.4 GBP/EUR Realized GBM 95%ile GBM 5%ile HMM2 95%ile HMM2 5%ile Figure 8. Percentiles of long-term distribution cones for GBP/EUR under GBM and HMM with two states. 5.3.3. RUB/EUR Table 3presents the results of the backtesting exercise for RUB/EUR. It can be seen that both GBM and HMM did not perform very well when the forecasting horizon was 1 week, with HMM having yellow scores under every metric. The results were similar for the 2 week forecasting horizon. In the longer time horizons, however, both models performed better. HMM outperformed the one-state model GBM, achieving green scores in the 1-month horizon. The scores were green for both models when the forecasting horizon was 3 months. Table 3. Backtesting results for RUB/EUR with calibration window Tc= 3 years, frequency of re-calibration δc=3 months, and backtesting window Tb=10 years. Time Horizon GBM HMM2 AD CVM KS AD CVM KS 1W 0.9991 0.9988 0.9988 0.9997 0.9996 0.9996 2W 0.9992 0.9989 0.9992 0.9996 0.9996 0.9988 1M 0.9830 0.9809 0.9446 0.9485 0.9457 0.8898 3M 0.5526 0.1406 0.0651 0.4399 0.3394 0.1624 Risks 2019,7, 66 16 of 22 Figure 9shows the percentiles of the long-term distribution cones for RUB/EUR. It is clear that the difference between GBM and HMM was more pronounced, with the HMM yielding significantly more conservative forecasts. The realized time-series was close to the 95th percentile of the GBM distribution until mid-2014, exceeding it on a number of occasions in 2011 and in 2013. Despite a sharp decline in 2015, the realized time-series remained above the 5th percentile for both models throughout the 7 year period. 2011 2012 2013 2014 2015 2016 Year 0.005 0.010 0.015 0.020 0.025 0.030 Rate Realized GBM 95%ile GBM 5%ile HMM2 95%ile HMM2 5%ile Figure 9. Percentiles of long-term distribution cones for RUB/EUR under GBM and HMM with two states. 5.3.4. MXN/EUR Table 4summarizes the results of the backtesting exercise for MXN/EUR, in terms of scores as well as color bands. Both HMM and GBM had yellow scores for the shorter time horizons (1 and 2 weeks), under all metrics. The models performed better for the longer time horizons (1 and 3 months), achieving green scores. Figure 10 shows the long-term distribution cones. Similar to the the GBP/EUR case, we do not observe a clear difference in performance between GBM and HMM with two states. 2011 2012 2013 2014 2015 2016 Year 0.01 0.02 0.03 0.04 0.05 0.06 0.07 MXN/EUR Realized GBM 95%ile GBM 5%ile HMM2 95%ile HMM2 5%ile Figure 10. Percentiles of long-term distribution cones for MXN/EUR under GBM and HMM with two states. Risks 2019,7, 66 17 of 22 Table 4. Backtesting results for MXN/EUR, with a 3-year calibration window, quarterly re-calibration, and a 10-year backtesting window. Time Horizon GBM HMM2 AD CVM KS AD CVM KS 1W 0.9967 0.9955 0.995 0.9967 0.9956 0.9938 2W 0.9895 0.9864 0.9677 0.9841 0.9768 0.9742 1M 0.5185 0.5963 0.7136 0.5501 0.6071 0.6225 3M 0.7124 0.6643 0.4422 0.7045 0.7373 0.6563 5.4. Impact on Credit Exposure: A Case Study for FX Options 5.4.1. Exposure at Default (EAD) Prior to presenting the case study on FX options, we provide a brief introduction to credit exposure calculation. For a more detailed description, the reader is referred to Zhu and Pykhtin (2007) and Gregory (2012). When a financial institution is permitted to use the IMM to calculate credit exposure, the following steps need to be taken: 1. Scenario Generation. Market scenarios are simulated for a fixed set of exposure dates {tk}N k=1 in the future, using the RFE models. 2. Instrument Valuation. Instrument valuation is performed for each exposure date and for each simulated scenario. The outcome of this process is a set of realizations of credit exposure at each exposure date in the future. One can then estimate the expected exposure EEk as the average exposure at future date tk , where the average is taken across all simulated scenarios of the relevant risk factors. The Expected Positive Exposure (EPE) is defined as the weighted average of the EE over the first year EPE = min(1 year, maturity) ∑ k=1 EEk×∆tk, (43) where the weights ∆tk=tk−tk−1 are the proportion that an individual expected exposure represents over the entire one-year time horizon. In order to account for potential non-conservative aging effects, a modification is necessary. First, an Effective EE profile is obtained from the EE profile by adding the non-decreasing constraint for maturities below one year. Effective EE can be calculated, recursively, as follows: Effective EEk=max {Effective EEk−1−EEk}, (44) where the current date is denoted as t0and EE0equals the current exposure. Effective EPE can, then, be calculated from the Effective EE profile, in the same way that EPE is calculated from the EE profile: Effective EPE = min(1 year, maturity) ∑ k=1 Effective EEk×∆tk. (45) Finally, the Exposure at Default (EAD) is the product of a multiplier αand the Effective EPE EAD =α×Effective EPE. (46) The multiplier α , introduced by Picoult (2002), is a correction coefficient that accounts for wrong-way risk. Under the IMM, α is fixed at a rather conservative level of 1.4. However, banks using Risks 2019,7, 66 18 of 22 the IMM have an option to use their own estimate of α , with the prior approval of the supervisor and a floor of 1.2. 5.4.2. Results In order to study the impact of using a two-state HMM, instead of a GBM, on regulatory and economic capital, we consider the case of FX call options on the RUB/EUR rate. The rationale behind this choice was that the Russian currency suffered a crisis in 2014, which will be included in our calibration data set. Our starting date was 2 Januray 2016. We estimated the parameters of a GBM and a two-state HMM, using three years of data (between January 2013 and December 2015). Following the methodology presented in Section 5.4.1, we generated market scenarios for the following set of future exposure dates: {tk}9 k=1={1 week, 2 weeks, 3 weeks, 4 weeks, 2 months, 3 months, 6 months, 9 months, 1 year}.(47) For each generated scenario and each exposure date, option valuation was performed using the Garman–Kohlhagen model (Garman and Kohlhagen (1983)). The value of a call option at time tis given by the analytical formula Ct=Ste−rf(T−t)N(x+σ√T−t)−Ke−rd(T−t)N(x), (48) where x≡ln(St/K)+rd−rf−σ2/2(T−t) σ√T−t, Stit the spot price of the deliverable currency at time t(domestic units per foreign unit), Kis the strike price of the option (domestic units per foreign unit), T−tis the time to maturity, rdis the domestic risk-free interest rate, rfis the foreign risk-free interest rate, σis the volatility of the spot currency price, and N(·)is the cumulative normal distribution function. Note that, in the formula, both spot and strike price are quoted in units of domestic currency per unit of foreign currency. As a result, the option price will be in the same units, as well. In order to obtain the market value of a position in such an option, it is necessary to multiply by a notional amount Λin the foreign currency. In our example, the foreign and domestic currencies are RUB and EUR, respectively. In order to achieve a candid comparison of the two RFE models for the exchange rate, we do not consider interest rate and volatility as risk factors for FX options. Instead, we make the simplistic assumptions of rd=rf= 0 and constant volatility σ= 0.15 (equal to the supervisory volatility for foreign exchange options in the standardised approach, see Basel Committee on Banking Supervision (2014)). The notional amount Λ is set to RUB 100,000,000. The spot RUB/EUR exchange rate on 2 January 2016 was S0=0.01263. The credit exposure values for out-of-the-money (OTM) call options on the RUB/EUR exchange rate, for a range of strike prices, are illustrated in Figure 11a. The impact of using a two-state HMM, instead of a GBM, is shown in Figure 11b. These results are summarized in Table 5. It is clear that exposure values under HMM exceeded the exposure values under GBM markedly for deep-out-the-money options. This difference would have a direct impact on how these positions would be capitalized against counterparty default, with a difference that could exceed 400% for the strike price K= 0.023. It is also important to note that, given the exchange rate movements over recent years, it is not unrealistic for the moneyness of such options to change dramatically, leading to large Risks 2019,7, 66 19 of 22 unexpected losses. For in-the-money call options, the two models produced identical exposure values. Thus, these results are omitted from this paper. 0.014 0.015 0.016 0.017 0.018 0.019 0.02 0.021 0.022 0.023 Strike price 102 103 104 Exposure Exposure by strike price and model - OTM call options GBM HMM2 (a) 0.014 0.015 0.016 0.017 0.018 0.019 0.02 0.021 0.022 0.023 Strike price 0 100 200 300 400 Percentage points Impact of 2-state HMM on exposure by strike price (b) Figure 11. Credit exposure values for out-of-the-money (OTM) call options on the RUB/EUR exchange rate (a) and the impact of using a two-state HMM, instead of a GBM (b). Risks 2019,7, 66 20 of 22 Table 5. Credit exposure values for out-of-the-money (OTM) options on the RUB/EUR exchange rate. Strike K Credit Exposure Impact (%) GBM HMM2 0.014 29,507.54 29,013.22 −1.68 0.015 13,684.44 12,838.95 −6.18 0.016 5981.10 5939.89 −0.69 0.017 2598.64 3199.17 23.11 0.018 1207.04 1740.47 44.19 0.019 580.34 973.64 67.77 0.020 285.70 595.99 108.61 0.021 143.08 401.04 180.29 0.022 70.20 269.96 284.60 0.023 31.87 165.56 419.47 6. Conclusions In this paper, we presented a hidden Markov model for the evolution of exchange rates with regards to counterparty exposure. In the proposed model, the observations of the exchange rates were assumed to be generated by a discretized GBM, in which both the drift and volatility parameters are able to switch, according to the state of a hidden Markov process. The main motivation of using such a model is the fact that GBM can assign unrealistically low probabilities to extreme scenarios, leading to the under-estimation of counterparty exposure and the corresponding capital buffers. The proposed model is able to produce distributions with heavier tails and capture extreme movements in exchange rates without entirely departing from the convenient GBM framework. We generated exchange rate scenarios for four currency pairs: USD/EUR, GBP/EUR, RUB/EUR, and MXN/EUR. A risk factor evolution model backtesting exercise was performed, in line with Basel III requirements, and the the percentiles of the long-term distribution cones were obtained. The performances of the one-state and two-state models (GBM and the two-state HMM, respectively) were found to be very similar, with the two-state model HMM being slightly more conservative. However, when the generated scenarios were used to calculate exposure profiles for options on the RUB/EUR exchange rate, we found significant differences between the results of the two models. These differences were even more pronounced for deep out-of-the-money options. Our study highlights some of the limitations of backtesting as a tool for comparing the performance of RFE models. Backtesting can be a useful way to objectively assess model performance. However, it can only be performed over short time horizons; with our available data, we could perform a statistically sound test of modeling assumptions for a time horizon of maximum length three months. It is, therefore, important to put effort into the interpretation of backtesting results, before they are translated into conclusions about model performance. Our results show how two models with similar performances in a backtesting exercise can result in very different exposure values and, consequently, in very different regulatory and economic capital buffers. This can lead to regulatory arbitrage and potentially weaken financial stability and, further, turn into a systemic risk. The research presented in this paper can be extended in a number of ways, such as considering the evolution of risk factors other than exchange rates. Another topic worthy of investigation is the enhancement of the backtesting framework presented by Ruiz (2014), by considering statistical tests similar to the ones presented by Berkowitz (2001) and Amisano and Giacomini (2007). Finally, an interesting research direction is the development of an agent-based simulation model with heterogeneous modeling approaches, with regards to the RFE models. This model could potentially give valuable insights into the impact of heterogeneous models in financial stability. Author Contributions: Both authors conceived and planned the research. Ioannis Anagnostou performed the numerical experiments. Both authors discussed the results and contributed to the final version of the manuscript. Risks 2019,7, 66 21 of 22 Funding: This project has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement no. 675044 (http://bigdatafinance.eu/), Training for Big Data in Financial Research and Risk Management. Acknowledgments: The authors are grateful to Jori Hoencamp, Steven van Haren, Marcel Boersma, and the reviewers for the useful comments. Conflicts of Interest: The opinions expressed in this article are solely those of the authors and do not represent in any way those of their current and past employers. The authors declare that there are no conflicts of interest regarding the publication of this paper. References Amisano, Gianni, and Raffaella Giacomini. 2007. Comparing density forecasts via weighted likelihood ratio tests. Journal of Business & Economic Statistics 25: 177–90. Anfuso, Fabrizio, Dimitrios Karyampas, and Andreas Nawroth. 2014. Credit exposure models backtesting for Basel III. 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