## Abstract

In this article, we continue the research of our recent interest rate tree model, called the Zero Black-Derman-Toy (ZBDT) model, which includes the possibility of a jump at each step to a practically zero interest rate. This approach allows a better match with the risk of financial slowdown caused by catastrophic events. We present how to valuate a wide range of financial derivatives using such a model. The classical Black-Derman-Toy (BDT) model and a novel ZBDT model are described, and analogies in their calibration methodology are established. Finally, two cases of applications of the novel ZBDT model are introduced. The first of them is the hypothetical case of an S-shaped term structure and decreasing volatility of yields. The second case is an application in the structure of US sovereign bonds in the 2020 economic slowdown caused by the coronavirus pandemic. The objective of this study is to understand the differences presented by the valuation in both models for exotic derivatives.

**Key Findings**

▪ A general methodology to price a wide range of financial derivatives in the ZBDT model is proposed.

▪ A case study related to the coronavirus recession is considered.

▪ ZBDT model can better match the actions undertaken by the Fed in the presence of the economic downturn than the BDT model.

The increasing participation of agents operating in financial markets has driven the use of complex derivatives. These assets allow investors to carry out financial risk management strategies and establish hedging strategies in case of possible adverse asset price variations. Catastrophic events in financial markets are becoming more and more frequent in different situations and should, therefore, be considered in the valuation of assets.

Bonds are one of the most popular and recognized financial instruments. According to a 2019 report of the Securities Industry and Financial Markets Association, in 2018 the global bond market reached a level of $102 trillion (see SIFMA 2019). Pricing derivatives for interest rate markets is a complicated task that requires a deep understanding of the local and global economy. In recent years, the international financial derivatives market for interest rates has entered the stage of large-scale commercialization. This level is reached due to its great expansion in developed countries, mostly by the over-the-counter (OTC) market. Among all varieties of option-style derivatives, the most recognized are vanilla options (i.e., European and American) and their barrier option equivalents. The European option can be used only at the time of expiration *S* and its American analogue at any time between 0 and *S*. The American option is the most listed in the United States and is one of the most popular derivatives in the rest of the world (see Krzyżanowski and Magdziarz 2020 and references therein).

There exist a large number of exotic derivatives characterized by different payoff functions. These are generally traded on the OTC market and include various variants. In this article, we focus on the simplest exotic options, namely the barrier options: financial derivatives in which the payoff depends on whether the price of the underlying asset reaches a certain threshold over a period of time. The main advantage of these options is that they diminish the effect of possible manipulations of the market that can occur close to the option’s expiration date. They are attractive to some market participants because they are less expensive than standard vanilla options. In particular, barrier options on bonds have become increasingly important and are widely used as hedging instruments for risk management strategies. These complex derivatives are usually difficult to manage, so they cause new problems in pricing and hedging on these contracts. As an example, consider November 1994, when LM International (a US hedge fund) purchased $500 million of “knock-in” put options from Merrill Lynch, and other dealers, for Steinhardt Management, having Venezuelan bonds as assets. After buying the derivatives, the hedge funds tried to boost the prices through long-term call options and the underlying bonds. The transaction ended in a dispute between its counterparts, whether the barrier was activated, and the investigation by the US Securities & Exchange Commission into alleged price manipulation (Frunza 2015).

A global pandemic related to Coronavirus COVID-19 disease started in 2020. It has been an extraordinary phenomenon that generated a deep economic crisis at all levels. Mainly, the financial markets have suffered strong turbulences that were expressed by generalized losses: significant losses occurred at the beginning of March. Monetary authorities immediately responded and, in the face of the most turbulent time, the US Federal Reserve (Fed) cut interest rates to a range of 0% to 0.25%. This expansive monetary policy in the United States may lead to a lower cost of money. A similar decline in U.S. interest rates was observed in the 2008–2009 crisis.

At the time of the COVID-19 pandemic and the increasing probability of global diseases appearing (see, e.g. Bloomfield et al. 2020; Neiderud 2015), new strategies for financial engineering must be developed. Models widely accepted in the world of finance (such as Black-Derman-Toy and Black-Scholes) need to be modified for scenarios such as epidemics, financial crises, or natural disasters. Motivated by such phenomena, an appropriate modification of the classical Black-Derman-Toy (BDT) model on interest rates was proposed (Krzyżanowski et al. 2021). This modification, the Zero Black-Derman-Toy interest rate model (ZBDT), departs from the BDT binary tree model (Black et al. 1990; Bjerksund and Stensland 1996) in that at each time step it includes the possibility of a downward jump with a small probability to a practically zero interest rate value in a zero-interest-rate zone. Additionally, we assume that once the ZBDT process reaches the zero-interest rate zone, it remains there with high probability. In practical terms, the initial BDT binary tree model is modified to a mixed binary-ternary tree model to find consistent interest rates with the market term structure.

The aim of this work is to compare the performance of the classic BDT model and the novel ZBDT model for exotic derivatives applied to an expansive monetary policy of the United States in March 2020 due to the Coronavirus recession. This section is an introduction to the financial markets and economic context in 2020. In the next section, we establish the basis of the BDT model and the way to estimate its interest rates. Moreover, we introduce the economic importance of adding to the interest rate tree a new branch with a low probability into rates following the Fed’s monetary policy in the ZBDT model. The way to model calibration follows the same ideas and algorithms as in the BDT model. In addition, the section provides the class of financial derivatives and the methodology for its valuation. In the following section, we establish a theoretical (S-shaped) term structure and a decreasing volatility structure to compare both models through the option prices and implied volatility. In the section after that, we apply the model with real data from the US term structure on a date that seems relevant to compare both models (before the very high volatility that happened in the financial markets). Our concluding remarks are given in the final section.

## INTEREST RATE MODELS AND FINANCIAL DERIVATIVES VALUATION

In this section, we briefly introduce two models to compare: the Black-Derman-Toy (BDT) with the Zero Black-Derman-Toy (ZBDT). Moreover, we present the general methodology for valuing the options from Exhibit 3 and Exhibit 4 in both models. More detailed descriptions of the BDT and ZBDT models can be found in Black et al. (1990) and Krzyżanowski et al. (2021), respectively. These articles also establish the financial market data to be used and the way to calibrate the corresponding interest rate tree through the yield and its volatility.

### The Black-Derman-Toy Model

The BDT model is one of the most celebrated equilibrium models for interest rates (Black et al. 1990). The model consists in a binary tree with a fixed probability of transition. One assumes that the future interest rates evolve randomly in a binomial tree with two scenarios at each node with a probability equals to 1/2, labeled, respectively, by *d* (for “down”) and *u* (for “up”). It is important to note that it is satisfied that an *u* followed by a *d* take us to the same value as a *d* followed by an *u*. Therefore, after *T* periods, we have *T* + 1 possible states for the stochastic process modelling the interest rate. Besides, the model assumes that the evolution of the interest rate is independent of the past in each period and that the volatility only depends on time (not on the value of the interest rate). These hypotheses give the model the characteristics of being simple and practical for its implementation. More precisely, the model uses the current term structure and computes the yield volatility using a series of consecutive historical term structures. Despite its simplicity and practicality, the BDT model has some limitations (Boyle et al. 2001; Piriol 2015*a*, *b*).

With the aim of simplifying the presentation, we consider that one period is equivalent to one year. Moreover, in the whole article we focus on the zero-coupon bonds (zc-bonds). The corresponding modification to shorter periods of use for bonds with coupons is similar to the way presented by Black et al. (1990). Exhibit 1 presents the BDT model applied with maturity in *T* = 3 years. In Exhibit 1(a), the interest rate tree is considered, where its values *r*_{i,j} for *i* = 0, 1, 2 and *j* = 1, …, *i* + 1 need to be calibrated using yields and volatility data. Through their estimation, it is possible to calibrate the price of the bond by using the interest rate discount technique. In Exhibit 1(b), we present the tree corresponding to the prices of a zc-bond for the Exhibit 1(a). We denote by *B*_{i,j} the zc-bond price corresponding to the period *i* and state *j*. For simplicity, we assume *B*_{T,j} = 100, for *j* = 1, …, *T* + 1 that corresponds to the bond’s face value (FV).

### The Zero Black-Derman-Toy Model

The modification of the classical BDT interest rate tree model consists of adding to the dynamics the possibility of a downward jump to a practically zero interest rate with a small probability at each time step. More precisely, in the ZBDT model, the nodes of the form (*i*, 1) add a third possible downward jump with a small probability *p*. If this downward jump is realized, then the process enters the so-called zero interest rate policy zone (ZIRP zone), meaning that the interest rate becomes a small value *x*_{0} (close to the target of the monetary policy). The other two transition probabilities (up and down) for nodes of the form (*i*, 1) are equal to . When the process is in the ZIRP zone, it remains there at each time step with a high probability (1 − *q*) and exits with a probability *q*. For more information, see Krzyżanowski et al. (2021).

The ZBDT model is a proposal to extend the BDT model with the ideas raised in Eberlein et al. (2018), Lewis (2009), Martin (2018), and Tian and Zhang (2018). The ZBDT model inherits several properties of the original BDT model, such as the positivity of interest rates. Both models allow us to compute the financial derivatives on bonds in a fast and simple way. Moreover, the ZBDT calibration uses the same data and the same hypothesis of independence between the periods as in the case of the BDT model. Similarly, as in the BDT model, it is also assumed that volatility depends only on time.

In Exhibit 2, the interest rate tree (a) and the zc-bond price tree (b) are presented for the ZBDT model. Similar to the BDT model, we denote by *B*_{i,j} the price of the zc bond corresponding to the period *i* and the state *j*. For simplicity, we assume *B*_{T,j} = 100, for *j* = 0, …, *T* + 1 that corresponds to the bond’s face value (FV). In this exhibit, we differentiate three zones. In each zone, different dynamics are conserved and this is manifested by different transition probabilities. The probabilities *p* and *q* have no influence on the first zone (white nodes with gray transition probabilities), therefore, the dynamics of this zone remains the same as in the case of the BDT model (i.e., the probability of transition is *u* = *d* = 1/2). This zone comprises the range from the third bottom row to the top row (labeled (*i*, *j*) with *j* = 2, …, *T* and *i* = 1, …, *T* − 1). In the second zone (red nodes and transition probabilities, labeled (*i*, 1) with *i* = 0, …, *T* − 1, the transition probabilities corresponding to going up and down have the same value and the model includes a small probability *p* of falling into the ZIRP zone. The last zone is the ZIRP zone (blue nodes and transition probabilities, the nodes are marked as *x*_{0}). As mentioned previously, its transition probabilities are *q* for a jump out of the ZIRP zone and 1 − *q* for remaining there.

It is important to note that *p*, *q*, *x*_{0} have a clear interpretation as a probability of the crisis (in the basic period of time, which in this article is 1 year), a conditional probability of economic recovery from the financial crisis (in the basic period of time), and an assumed value of the interest rate in the ZIRP zone, respectively. Estimation of model parameters is not simple because it requires a precise definition of the crisis and the “recovering from the crisis,” so the estimation of these terms may differ depending on the situation under consideration. For example, the parameter *x*_{0} can be estimated by the average interest rates in the ZIRP zone. Furthermore, the values of *p* and *q* can be estimated using the techniques presented, e.g., in Benigno et al. (2020), Gregoriou (2009), Engle and Ruan (2019), Soto and Garcia (2004), and Mecagni et al. (2007). In more detail, it is possible to construct a regression-type model allowing the estimation of the parameter *p*. However, the list of possible factors that could have an impact on a financial crisis is long, so the choice of independent variables is not unique. It is also possible to apply another estimation method using frequency statistics. That is, it is possible to count the number of recessions that have been suffered worldwide and normalize them over the total number of years. The other possible approach to estimate *p* is based on extreme value theory (EVT), which is often used to estimate the probability of events that are more extreme than any that have already been observed (DeHaan and Ferreira 2006; Longin 2016). For the estimation of *q*, it is possible to build medium-term forecasting models that take into account the weaknesses of the global financial system. Again, it is also possible to apply frequency statistics and count the average number of years needed to obtain growth values without an economic recession.

### Selected Financial Options

In this subsection, we recall the payoff functions for financial derivatives considered in the work. Path-dependent options are assets whose payoff depends nontrivially on the price history of an asset. There are two varieties of path-dependent options. In this article, we mainly focus on a soft path-dependent option that bases its value on a single price event that occurs during the life of the option. They play an important role in OTC markets.

In Exhibits 3 and 4, we establish that the payoff function *f*(*Z _{t}*) is the gain of the option holder at the time

*t*for a given underlying instrument

*Z*. In the rest of the article,

*T*is the maturity of the bond and

*S*is the expiration time of the option,

*K*is strike, , in

*t*∈ [0,

*S*] and

*H*

^{+},

*H*

^{−}are the upper and lower barriers, respectively.

### The Valuation of Financial Derivatives

In this subsection, we consider some aspects of the valuation of derivatives introduced in this section. As an extension of the BDT model, the ZBDT model is a way to valuate the different financial derivatives on bonds. Hence, the calibration of the interest rate tree and the corresponding (future) states of its underlying instrument (i.e., bond) is based on the following.

First, we focus on pricing the European-style options (Exhibit 3) on bonds in both models. At the beginning, we take the payoff function of the derivative *f* on the nodes located at the expiration time of the option *S*, whose underlying asset is a bond with maturity in time *T*. Then, using the probability transitions on each branch, we proceed with the backward induction, similarly as was done in Cox et al. (1979). At the end, we obtain *V*_{0,1} which is the price of the derivative (at time *t* = 0) is. Note that the function *f* could be any payoff function of Exhibit 3.

Finally, let us consider the valuation of the American (vanilla) options on bonds in both models. At the beginning, we proceed as in the case of European options, taking the payoff function on the nodes at time *S* to obtain the values *V*_{S,j} = *f*(*B*_{S,j}), where *f* is the payoff function (see Exhibit 2), and *j* = 1, …, *i* + 1 or *j* = 0, …, *i* + 1 for BDT and ZBDT, respectively. Using the probability transitions and the corresponding interest rates, we obtain the values , where *i* = *S* − 1, *j* = 1, …, *i* + 1, or *j* = 0, …, *i* + 1 for the BDT and ZBDT, respectively. Then we have to verify whether immediate exercise of the option is not the most profitable strategy. Therefore, for *i* = *S* − 1, *j* = 1, …, *i* + 1 (for BDT) or *j* = 0, …, *i* + 1 (for ZBDT) we take , where *f _{V}* is the payoff function of the corresponding American (vanilla) option. We proceed in an iterative manner for

*i*=

*S*− 2, …, 0. Finally, we get

*V*

_{0,1}, which is the price of the derivative (at time

*t*= 0).

Note that in both option styles (European and American), we are valuing vanilla and barrier options. However, in the case of the valuation of American barrier options, it is additionally necessary to check if the node of the underlying instrument did not invalidate the contract by exceeding or not exceeding the predetermined barrier(s). The valuation of options in European and American style for both models is summarized in Exhibit 5. The option valuation algorithms are presented in Algorithm 1 and Algorithm 2. In them, *f* represents the payoff function of the option and *f _{V}* represents the payoff function of the corresponding American (vanilla) option. Moreover:

It is important to note that the price *B*_{i,j} can invalidate the contract by exceeding the barrier or interval of the barriers in the case of the knock-out options, or not exceeding the barrier or interval of barriers in the case of the knock-in options.

### Comparison of BDT Model and ZBDT Model

In this section, the classical BDT model and the novel ZBDT model are calibrated to compare the performance in valuing the vanilla and barrier options. In the work, both models are compared by option prices and implied volatility.

To compare both models, an “S-shaped” term structure (i.e., a curve with parts decreasing and increasing) is used. The idea of this curve shape is that it has parts of the three main types of yield curve shapes: normal (upward sloping curve), inverted (downward sloping curve), and flat. For the volatility in each yield value, we consider a decreasing function because it is usually observed in the sovereign debt. We analyze the case for the long term, so for this reason we select one-year nodes (frequency annually) of the 10-year term structure (*T* = 10). The values of yield (%) and volatility (%) can be found in Equation (1). For the ZBDT model, we assume the parameters *x*_{0} = 0.25%, *p* = 0.02, *q* = 0.01. The input data and the set of parameters we call the “example.” In Exhibits 6 and 7, we show the interest rate trees for BDT and ZBDT, respectively. Exhibits 8 and 9 contain the corresponding trees for the prices of the zc-bonds.

From the trees of the prices of zc-bonds in both models (Exhibit 8 and Exhibit 9) it is possible to valuate the financial derivatives introduced previously. We use selected derivatives written at *t* = 0 with an exercise time of five years (*S* = 5). In Exhibits 10 and 12, we present the option prices in dependence on the strike price *K*. Here and in the rest of the work, the red color corresponds to the BDT model and the blue color corresponds to the ZBDT model. In Exhibit 10, we show the prices of (a) European put, (b) American put, and (c) European call option. The nonzero probability of falling into the ZIRP zone causes the prices of the zc-bonds to increase. As a consequence, the prices of the vanilla options in ZBDT are higher than in the BDT model. It follows our objectives because the risk related to the vanilla options put is higher for the ZBDT than for the BDT model.

In the case of the European (vanilla) options valuation and in modeling the risk related to these contracts, it is important to analyze the implied volatility. The implied volatility at time *t* of an European vanilla option written on a zc-bond that matures at time *T*, with strike price *K* and expiry time *S* (*t* < *S* < *T*), is computed via Black’s (1976) formula. Note that, regardless of the model under consideration, we assume that the implied volatility is equal to 0 if the corresponding option is worthless. For more details on the estimation of implied volatility, see McDonald et al. (2006).

In Exhibit 11, we show the implied volatility for the prices of the European vanilla call option for the Example. Panel (a) concerns the dependence between implied volatility σ and strike price *K*. In both models, the implied volatility by similar curves is expressed. Panel (b) concerns the dependence between implied volatility σ and the expiration time of option *S* for *K* = 90. The implied volatility in the BDT model for *S* = 2 and *S* = 3 is assumed to be equal to 0. When analyzing these results, it is concluded that the ZBDT model may be useful. The first reason is the inclusion of the existence of a possible jump up in the price of the zc-bond. Because of this property, the price of those derivatives must be higher for the ZBDT model than for the BDT model. The second reason is the existence of a (feasible) case in which the classical BDT model valuates the derivative with a price equal to 0, since the corresponding price in the ZBDT model is positive. Both results confirm that the risk manifested by the implied volatility is higher for the ZBDT model than for the BDT model.

The prices of the barrier options are collected in Exhibit 12. The left panels are calculated for European put options: (a) knock-double-out, (b) knock-double-in, (e) knock-up-out, and (f) knock-up-in. The right panels (c, d, g, h) represent the prices of American put options that have the same order as their European equivalents. The barriers considered in the Example are *H*^{−} = 70, *H*^{+} = 90. We can observe that, depending on the barrier contract, different relations between the option prices of BDT and the ZBDT model can be obtained. The curve can have different shapes depending on the model and the type of option. Therefore, the price corresponding to ZBDT does not have to be higher than in the BDT model. Because the risk corresponding to the barrier options is lower than for its vanilla analogs (for the issuer of the option), the same relation between the prices of the instruments is conserved.

## APPLICATION TO THE US TERM STRUCTURE BEFORE THE CORONAVIRUS RECESSION

The aim of this section is to analyze the behavior of the model in a real situation, treating the recent events as a case study. The objective of the analysis is to show the advantages of the use of the ZBDT model in aspects of the financial crisis, mainly in the precrisis moments.

In 2019, the international financial markets showed good returns. In particular, short-term interest rates in the United States sovereign market were greater than 2% for a long period. The expectations of the financial agents that there would be an economic crisis in 2020 were very low. These expectations can be observed in the evolution of some stock market indices. Among them is the VIX Index, which measures volatility in Chicago market options (CBOE or Chicago Board Options Exchange) in the S&P 500 Index. In that year, the index was found to have values below 20, which can be interpreted as confidence in the US economy.

However, the COVID-19 pandemic caused instability in all financial markets. During the evolution of the pandemic, there were key moments in the behavior of the main variables in the market. The international markets responded strongly when the first cases of infection appeared in Europe. At the end of February 2020, it was detected that several financial agents disposed of their positions massively (with greater risk) and reinvested their proceeds into “safe” assets. Then followed March, during which a sharp global economic downturn was observed and the stock markets suffered heavy losses. As a result of the situation, on March 15th, the Federal Reserve System embarked on a large-scale bond buying operation and introduced additional measures to support the economy. The objective of such proceedings was the reactivation of consumption and market liquidity.

In Exhibit 13, we present the term structure of the United States in three different stages of the COVID-19 pandemic in 2020. At the beginning of January, there were some records of confirmed infections of COVID-19 (only in China), but there were no records of deaths from COVID-19. The international financial markets were not affected during this period. Second, February 14 was the last day of the period, when Europe had no record of coronavirus deaths. So far, the continent has only had individual cases of confirmed infection. The third of the days considered is March 27—around two weeks after the World Health Organization (WHO) announced the COVID-19 outbreak as a pandemic and when the Fed had been introducing drastic measures to diminish effects of the COVID-19 financial crisis. This day represents a period in which new cases of COVID-19 appeared on a massive scale in different developed countries, including the United States (Statista 2020a, b; Worldometer 2020). In Exhibit 13, we can observe the abrupt drop in the yield rates on the sovereign assets of the United States after the monetary policy of the Fed was imposed. It is worth mentioning that short-term yield rates decreased in a short period of time. For example, the one-year US Treasury yield decreased from approximately 1.5% to 0.2% in approximately three weeks.

We analyze the day before the high volatility in the financial markets appeared. In this application (we call it “Real Case”), we use the first five years of the term structure published in the US Department of the Treasury for February 14th (see the data in US Department of Treasury 2020). These published yields (in %) and the annual volatility of the estimated rates (in %) can be found in Equation 2. For the ZBDT model, we consider the parameter *p* = 0.1 because it corresponds to the higher risk of recession (which is related to the beginning of the COVID-19 pandemic) compared to the value *p* = 0.02 assumed for the more “conservative” example in this section. We also assume that if a crisis occurs, the possible financial recovery (due to health factors) is subject to a lot of uncertainty. Therefore, we impose that the probability of exiting the crisis is ten times less than the probability of entering the crisis, i.e., *q* = 0.01. Moreover, we consider *x*_{0} = 0.25% because it is the upper limit of the ZIRP zone. Exhibits 14 and 15 contain the interest rate and the zc-bond price trees for the BDT and ZBDT models. With these results, it is possible to value the selected financial derivatives. Exhibit 16 presents the option prices in dependence on strike for the BDT model (red) and the ZBDT model (blue) for the selected European call options: (a) knock-up-out, (b) knock-double-out, (c) knock-down-in, (d) knock-down-out, (e) knock-up-in, (f) knock-double-in option. Barriers are *H*^{−} = 93 and *H*^{+} = 98.5. Similarly, as in the case of Exhibit 12, we conclude that the relations between the curves of the option prices corresponding to the BDT and ZBDT models are different depending on the option contract.

From our perspective, the ZBDT model is a good tool for valuing financial derivatives in interest rates before the COVID-19 pandemic. This statement derives from the fact that classical models do not have a probability of such an abrupt decrease over a short period of time.

## CONCLUSIONS

In this article, we propose a general methodology for valuing a wide range of financial derivatives, as defined in the Zero Black-Derman-Toy model proposed in Krzyżanowski et al. (2021). The model is inspired by Lewis’s ZIRP models in continuous time (see Lewis 2009), and also in Duffie and Singleton’s (1999) default framework of bond pricing models. Its novelty is to add a new branch at each period to the classical BDT tree model that includes a small probability of falling into a recession or a catastrophic event, necessitating a rapid interest rate reduction to near-zero rates. The methodology to estimate interest rates *r _{ij}*, price bonds, and price their derivatives in the ZBDT model follows the same ideas as in the BDT model.

In the context of the hypothetical case study, we considered a decreasing–increasing term structure and decreasing volatility. We valued different vanilla and barrier options in both models. Moreover, we compared the implied volatility determined for European call options using the Black’s formula. As a result, significant differences are observed between both models. We can conclude that a novel ZBDT model can better match market conditions than the classical model.

To apply the model to a real US-term structure, we studied the financial crisis generated by the coronavirus. It is widely known that financial markets suffered great turbulence due to the health and economic crisis caused by the pandemic. On March 14, the Federal Reserve System made the decision to cut the interest rate to values close to 0% and promised to buy $700 billion in Treasury-backed securities and mortgages. These measures were meant to prevent market disruptions from aggravating what is likely to be a severe slowdown in the pandemic. This situation leads the financial industry to improve the valuation of financial derivatives to consider catastrophic events in a more realistic way. To analyze both models (classical BDT and novel ZBDT) in the current situation, they were calibrated the day before the first death caused by the coronavirus in Europe, which immediately affected the international Stock Exchanges. The BDT model and the ZBDT model differ in their interest rate/bond/derivative trees. Both models can be applied for economic crises or catastrophic events, but in our opinion, the ZBDT model better matches the actions taken by the Fed in the presence of the economic downturn.

## ACKNOWLEDGMENTS

The research of G.K. was partially supported by NCN Sonata Bis 9 grant nr 2019/34/E/ST1/00360.

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