A yield curve, also known as a term structure, is the representation of the function which, at a given date, associates the corresponding interest rate level with each maturity. In practice, there is not just one but several yield curves on the markets. We can, however, distinguish two families:

1) Market curves: these are constructed from quotations observed on the financial markets, they are:

2) Implicit curves: these are constructed indirectly from observed market data, such as bond and swap prices. They are mainly zero-coupon yield curves and forwards (instantaneous) rates.

Yield curves can take the form of three shapes:

A shift in the yield curve will occur for a number of reasons, connected not just with the market’s view on interest rates but also factors such as liquidity and supply and demand. The main types of shifts in the yield curve are one of the following:

In practice, we use stochastic models for the shift of the yield curve for two key reasons:

1) Assessment and coverage of interest-rate products delivering random future flows (caplets) are essential. The option seller must be able to provide a price for the product they sell, but more importantly, they must cover the option they sell because they are exposed to unlimited loss. These models are primarily used in trading contexts within market rooms and in risk departments.

2) A portfolio manager makes predictions on the interest rate curve when implementing a strategy. They need to anticipate the scenario of the yield shift. To do this, they use a tool that enables them to consider all possible scenarios of interest rate curve shifts. Since they are not certain that their scenario will come true, they also need to measure the risk they take if that scenario does not occur.

The choice of the perfect model is based on some properties as follows:

1) A rate model shall take into account the empirical properties of the previously identified rate curve.

2) A rate model must be well constructed in the sense that the input of the model is observable on the market or easily estimable, and also frequently re-adjustable.

3) A rate model must be compatible with the market prices of vanilla products (caplets…).

4) Simple enough to allow quick calculations.

5) A model that can be easily manipulated by the user (Traders, Risk Controller).

6) A model that does not forget the risk factor.

7) A model that satisfies the theoretical assumption of the absence of opportunity for arbitration.

8) A model that offers an effective coverage method that allows the seller to cover his product throughout the life of the product.

Various models have been proposed to capture the stochastic dynamics of interest rates.

1) Vasicek Model (1977): This was one of the pioneering models for modeling interest rates, specifically the short rate. It assumes that the short rate follows a Gaussian distribution and is mean-reverting. (See [1] )

2) Cox-Ingersoll-Ross (CIR) Model (1985): Building on the Vasicek model, CIR introduced a non-negative short rate by modeling the rate as a square root process. This model is widely used in fixed income and interest rate modeling due to its ability to capture the volatility of interest rates. (See [2] )

3) Ho-Lee Model (1986): This model assumes that the short rate is driven by a stochastic process, typically calibrated to match observed term structure data. (See [3] )

4) Hull-White Model (1990): This model extends the Vasicek model by adding a time-varying volatility term. It allows for more flexibility in capturing interest rate dynamics and is commonly used in interest rate derivatives pricing; it is equivalent to the G1++ model. (See [4] [5] )

5) Black-Derman-Toy (BDT) Model (1990): Used primarily in the context of interest rate derivatives, this model is a tree-based approach that incorporates mean reversion and stochastic volatility. (See [6] )

6) Black-Karasinski Model (1991): Another tree-based model that extends the CIR model by introducing a stochastic volatility component. These models have different assumptions about interest rate behavior and are used in various financial applications, such as pricing derivatives, risk management, and understanding yield curve dynamics. (See [7] )

Each of these models has its assumptions and characteristics, making them suitable for different applications and market conditions.

A prominent candidate model is the 2-Additive-Factor Gaussian Model (G2++ model)—alternatively known as the 2-Factor Hull-White model) as in [4] [8] [9] . It possesses functional qualities required for various practical purposes such as Asset Liability Management and Trading of interest rate derivatives. The G2++ model is not only considered as a Vasicek two-factor model, but its main strength lies in its ability to precisely fit the market zero-coupon at any given initial value. Such a perfect fit is a primary requirement for practitioners, especially from the front office perspective. Before introducing the G2++ model (an extension of the one-factor model), let us first introduce the one-factor G1++ model as follows.

In G2++, the short-rate process is represented by the sum of two correlated Ornstein-Uhlenbeck processes (denoted as $x_{t;1}$ and $x_{t;2}$, referred to as stochastic factors that are normally distributed to simplify the computation of bond prices and other derivatives.) along with a deterministic shift $\phi \left(t\right)$ added to precisely match the initial zero-coupon curve established using the most liquid market data. The instantaneous short-rate process under the risk-neutral measure $\mathbb{Q}$ is given by:

$r_t=x_{t;1}+x_{t;2}+\phi \left(t\right)$(13)

where $t\mapsto \phi \left(t\right)$ is a (deterministic) function which allows the model to fit the current observed interest rates.

$x_{t;1}$ and $x_{t;2}$ may be viewed as state variables $\left\{x_{t;i};t\ge 0\right\}$ whose the dynamics are assumed to be given by

$\text{d}{x}_{t;1}=-{\kappa }_1{x}_{t;1}\text{d}t+{\sigma }_{t;1}\text{d}{W}_{t;1}{x}_{0;1}=0$(14)

and

$\text{d}{x}_{t;2}=-{\kappa }_2{x}_{t;2}\text{d}t+{\sigma }_{t;2}\text{d}{W}_{t;2}{x}_{0;2}=0$(15)

In (14) and (15), all of these dynamics are given under the risk-adjusted risk-neutral measure $\mathbb{Q}$. Here

The correlation of these two Brownian motions is capturing the possible correlation between different movements in the interest rate.

Integrating Equations (14) and (15) having $s\le t$, then the Equation (13) is given

$\begin{array}{l}{r}_t=x_{s;1}{\text{e}}^{-{\kappa }_1\left(t-s\right)}+x_{s;2}{\text{e}}^{-{\kappa }_2\left(t-s\right)}\\ +{\displaystyle {\int }_s^t\left\{{\sigma }_{u;1}{\text{e}}^{-{\kappa }_1\left(t-u\right)}\text{d}{W}_1\left(u\right)+{\sigma }_{u;2}{\text{e}}^{-{\kappa }_2\left(t-u\right)}\text{d}{W}_2\left(u\right)\right\}}+\phi \left(t\right)\end{array}$(16)

and supposing that $r_t$ have some statistical restrictions (by considering the probability measure $\mathbb{Q}$ that is conditional on the filtration $F_s$), it is possible to define the expected value and the variance of the interest rate $r_t$ by:

$E\left[r_t|F_s\right]=x_{s;1}{\text{e}}^{-{\kappa }_1\left(t-s\right)}+x_{s;2}{\text{e}}^{-{\kappa }_2\left(t-s\right)}+\phi \left(t\right)$

$Var\left[r_t|F_s\right]={\displaystyle {\int }_s^t\left({\sigma }_{u;1}^2{\text{e}}^{-2{\kappa }_1\left(t-u\right)}+{\sigma }_{u;2}^2{\text{e}}^{-2{\kappa }_2\left(t-u\right)}+{\sigma }_{u;1}{\sigma }_{u;2}{\text{e}}^{-{\kappa }_1\left(t-u\right)}{\text{e}}^{-{\kappa }_2\left(t-u\right)}\right)\text{d}u}$(17)

Having defined $r_t$ as in Equation (16), dependent on $x_{t;1}$ and $x_{t;2}$, denoted by $P\left(t,T\right)$ the price at time t of a zero-coupon bond maturing at $T>0$ with unit face value, that is:

$P\left(t,T\right)=E\left[{\text{e}}^{-{\displaystyle {\int }_t^{T}r\left(s\right)\text{d}s}}\right]$(18)

and

$P^{mkt}\left(0,T\right)={\text{e}}^{-{\displaystyle {\int }_0^T{f}^{mkt}\left(0,t\right)\text{d}t}}$(19)

where $f^{mkt}\left(0,t\right)$ is the market instantaneous forward. In other words, is the rate of return for an infinitesimal amount of time $\text{d}t$ measured as at some date (t) for a particular start-value date.

After some statistical work and conditions, and supposing that ${\sigma }_{t;1}={\sigma }_1$, ${\sigma }_{t;2}={\sigma }_2$ positive constants and denoting

$\Upsilon \equiv \left({\kappa }_1,{\kappa }_2,{\sigma }_1,{\sigma }_2,\rho \right).$(20)

So the model perfectly fits the market term structure of the discount factor if and only if for ( $T>0$) and we have:

(21)

then the zero coupon price at time t of maturity T is given by:

$P\left(t,T\right)=A\left(t,T\right){\text{e}}^{-B_1\left(t,T\right)x_{t;1}-B_2\left(t,T\right)x_{t;2}}$(22)

where, $A\left(t,T\right)=\frac{P^{mkt}\left(0,T\right)}{P^{mkt}\left(0,t\right)}{\text{e}}^{\frac{1}{2}\left[V\left(t,T;\Upsilon \right)-V\left(0,T;\Upsilon \right)+V\left(0,t;\Upsilon \right)\right]}$

$B_1\left(t,T\right)=\frac{1-{\text{e}}^{-{\kappa }_1\left(T-t\right)}}{{\kappa }_1},B_2\left(t,T\right)=\frac{1-{\text{e}}^{-{\kappa }_2\left(T-t\right)}}{{\kappa }_2}$(23)

$V\left(t,T;\Upsilon \right)=\underset{i,j=1}{\overset{2}{{\displaystyle \sum }}}{\rho }_{ij}\frac{{\sigma }_i{\sigma }_j}{{\kappa }_i{\kappa }_j}\left(T-t-\frac{1-{\text{e}}^{-{\kappa }_i\left(T-t\right)}}{{\kappa }_i}-\frac{1-{\text{e}}^{-{\kappa }_j\left(T-t\right)}}{{\kappa }_j}+\frac{1-{\text{e}}^{-\left({\kappa }_i+{\kappa }_j\right)\left(T-t\right)}}{{\kappa }_i+{\kappa }_j}\right)$ (24)

Finally

$P\left(t,T\right)=\frac{P^{mkt}\left(0,T\right)}{P^{mkt}\left(0,t\right)}{\text{e}}^{\left\{-B_1\left(t,T\right)x_{t;1}-B_2\left(t,T\right)x_{t;2}+\frac{1}{2}\left[V\left(t,T;\Upsilon \right)-V\left(0,T;\Upsilon \right)+V\left(0,t;\Upsilon \right)\right]\right\}}$(25)

The zero-coupon market prices $P^{mkt}\left(0,t\right)$ and $P^{mkt}\left(0,T\right)$ are assumed to be known.

Putting t = 0 in (25), and taking into account that $P^{mkt}\left(0,0\right)=1$ and $V\left(0,0;\Upsilon \right)=0$ then one obtains

$P\left(0,T\right)=P^{mkt}\left(0,T\right){\text{e}}^{\left\{-B_1\left(0,T\right)x_{0;1}-B_2\left(0,T\right)x_{0;2}\right\}}.$

It means that

$P\left(0,T\right)=P^{mkt}\left(0,T\right)\text{under}\text{the}\text{initial}\text{choice}{x}_{0;1}=0\text{and}{x}_{0;2}=0.$(26)

One appealing aspect of using the G2++ model is its ability to accurately fit the currently observed term structure of market zero-coupon prices at the initial time 0.

At any (future) time t, with 0 < t, resulting from (25), the time-t yield-to-maturity for a maturity T, is readily given by

$y\left(t,T\right)\equiv \frac{B_1\left(t,T\right)x_{t;1}+B_2\left(t,T\right)x_{t;2}-c\left(t,T;\Upsilon \right)}{T-t}$(27)

where

$c\left(t,T;\Upsilon \right)\equiv \text{ln}\left[\frac{P^{mkt}\left(0,T\right)}{P^{mkt}\left(0,t\right)}\right]+\frac{1}{2}\left(V\left(t,T;\Upsilon \right)-V\left(0,T;\Upsilon \right)+V\left(0,t;\Upsilon \right)\right)$(28)

Therefore, Equations (25) and (27) depend on the values of the state variables $x_{t;1}$ and $x_{t;2}$, which are essentially unobservable but are theoretically assumed to be described by Equations (14) and (15).

The G2++ model is only practically meaningful when the yield-to-maturity defined in Equation (27) satisfies $0\le y\left(t,T\right)$, which is equivalent to saying $P\left(t,T\right)\ge 1$. This implies that the time-t values $x_{t;1}$ and $x_{t;2}$ of the state variables should satisfy

$c\left(t,T;\Upsilon \right)\le B_1\left(t,T\right)x_{t;1}+B_2\left(t,T\right)x_{t;2}\text{for}\text{all}0<t<T.$(29)

At t = 0 with the initial values states in (26), then the constraint (29) is reduced to $c\left(0,T;\Upsilon \right)\le 0$. This is always true since $c\left(0,T;\Upsilon \right)=\text{ln}\left[P^{mkt}\left(0,T\right)\right]$ and $V\left(0,0;\Upsilon \right)=0$.

When considering t > 0 as a future time, then the state variable values may be considered as given by the random variables $x_{t;1}$ and $x_{t;2}$ therefore, it is logical to ask about the probability that yield is negative. And consequently it makes sense to ask about the probability that $y\left(t,T\right)\le 0$. Academic literature [11] . has granted very few attention to the computation of the probability that the instantaneous short rate $r_t$ is negative. Generally, people admitted that this probability is nonnegative but it stays to be little, such that this inconvenience is essentially neglected face to the various benefits connected with the use of the model G2 itself.

In this subsection, our contribution is to provide an accurate analysis of the future price of the zero-coupon bond, without resorting to the dynamics of the bond itself, which is typically obtained by applying the two-dimensional Ito’s formula as

$\text{d}P=\left(\frac{\partial P}{\partial t}+\frac{1}{2}\frac{{\partial }^{2}P}{\partial x_1^2}+\frac{1}{2}\frac{{\partial }^{2}P}{\partial x_2^2}\right)\text{d}t+\frac{\partial P}{\partial x_1}\text{d}{x}_1+\frac{\partial P}{\partial x_2}\text{d}{x}_2.$(30)

In the following sections, we assume that

$0<t<T$,

where t represents a future time horizon with respect to the initial time 0.

From time 0, the zero-coupon price at the horizon t is unknown and may be considered as given by a random variable, such that if the price model (25) remains consistent then one should have

$P\left(t,T\right)=P\left(t,T;P^{mkt}\left(0,t\right),P^{mkt}\left(0,T\right);x_{t;1},x_{t;2};\Upsilon \right).$(31)

For various practical purposes (such as valuing positions, measuring risks, managing positions, etc.), it is very beneficial to have a measure of the change in the zero-coupon price,

$P\left(t,T\right)-P\left(0,T\right).$

For example, t may represent one or more days, such as $t=10$ days or $t=90$. In any case, there is no reason to consider t as an infinitesimal number since, from a practical perspective, it has a given and well-defined size. From a theoretical point of view, it is tempting to utilize the dynamics followed by the zero-coupon bond price as in Equation (30) and then discretize the corresponding equation to obtain an approximate value of $P\left(t,T\right)-P\left(0,T\right)$. However, this approach may have limited practical utility since t should be small in size due to the discretization constraint.

Proposition 1. Under the G2++ model, let us consider the uncertainty risk/opportunity factors $x_{\cdot ;1}$ and $x_{\cdot ;2}$ which follow the stochastic differential Equations (14) and (15). Then the future time-t values of these state variables, conditionally on their current values $x_{0;1}$ and $x_{0;2}$, are given by

$x_{t;1}(\cdot )=ℰ\left(t;{\kappa }_1\right)x_{0;1}+{\sigma }_1{ℱ}^{\frac{1}{2}}\left(t;{\kappa }_1,{\kappa }_1\right){\epsilon }_1(\cdot )$(32)

and

$x_{t;2}(\cdot )=ℰ\left(t;{\kappa }_2\right)x_{0;2}+{\sigma }_2{ℱ}^{\frac{1}{2}}\left(t;{\kappa }_2,{\kappa }_2\right)\left\{\omega {\epsilon }_1(\cdot )+\sqrt{1-{\omega }^2}{\epsilon }_2(\cdot )\right\}$(33)

where ${\epsilon }_1(\cdot )$ and ${\epsilon }_2(\cdot )$ are two independent standard normal random variables,

$\omega \equiv \omega \left(t;\rho ,{\kappa }_1,{\kappa }_2\right)=\rho \frac{ℱ\left(t;{\kappa }_1,{\kappa }_2\right)}{{ℱ}^{\frac{1}{2}}\left(t;{\kappa }_1,{\kappa }_1\right){ℱ}^{\frac{1}{2}}\left(t;{\kappa }_2,{\kappa }_2\right)}.$(34)

$ℰ\left(u;k\right)=\text{exp}\left[-ku\right]$(35)

and the quantity $ℱ\left(u;k_i,k_j\right)$ is defined by

$ℱ\left(u;{\kappa }_i,{\kappa }_j\right)\equiv \frac{1}{{\kappa }_i+{\kappa }_j}\left\{1-\text{exp}\left[-\left({\kappa }_i+{\kappa }_j\right)u\right]\right\}.$(36)

Though it is not readily useful in practical simulations, when making reference to our previous model (14) and (15), it may be seen that ${\epsilon }_1(\cdot )$ and ${\epsilon }_2(\cdot )$ are defined such that

$\begin{array}{c}{\epsilon }_1(\cdot )\equiv {\epsilon }_1\left(\cdot ;t,{\kappa }_1\right)\\ =\text{exp}\left[-{\kappa }_{1}t\right]{ℱ}^{\frac{1}{2}}\left(t;{\kappa }_1,{\kappa }_1\right){\displaystyle {\int }_0^t\text{exp}\left[{\kappa }_{1}s\right]\text{d}{W}_{s;1}(\cdot )}\end{array}$(37)

and

$\begin{array}{c}{\epsilon }_2(\cdot )\equiv {\epsilon }_2\left(\cdot ;t,{\kappa }_1,{\kappa }_2,\rho \right)\\ =\frac{1}{\sqrt{1-{\omega }^2}}\{-\omega \text{exp}\left[-{\kappa }_{1}t\right]{ℱ}^{\frac{1}{2}}\left(t;{\kappa }_1,{\kappa }_1\right){\displaystyle {\int }_0^t\text{exp}\left[{\kappa }_{1}u\right]\text{d}{W}_{u;1}(\cdot )}\\ +\text{exp}\left[-{\kappa }_{2}t\right]{ℱ}^{\frac{1}{2}}\left(t;{\kappa }_2,{\kappa }_2\right){\displaystyle {\int }_0^t\text{exp}\left[{\kappa }_{2}u\right]\text{d}{W}_{u;2}(\cdot )}\}\end{array}$(38)

It may be noted that $x_{t;2}(\cdot )$, given in (33), is well defined whenever $\left|\omega \right|<1$, that is

$\left|\omega \left(t,\rho ,{\kappa }_1,{\kappa }_2\right)\right|=\left|\rho \right|\frac{ℱ\left(t;{\kappa }_1,{\kappa }_2\right)}{{ℱ}^{\frac{1}{2}}\left(t;{\kappa }_1,{\kappa }_1\right){ℱ}^{\frac{1}{2}}\left(t;{\kappa }_2,{\kappa }_2\right)}<1.$(39)

Simulations of the state variables $x_{t;1}(\cdot )$ and $x_{t;2}(\cdot )$, at the future time t can be performed using identities (32) and (33). It is worth noting, for example that people often only use Euler discretization of the Stochastic Differential Equations (SDE) governing the state variables. However, for simulation purposes, such discretization (which naturally leads to some errors) is not really necessary since we have completely solved the SDE as presented in identities (32) and (33). Therefore, it is sufficient to generate independent realizations ${\epsilon }_1$ and ${\epsilon }_2$ of the standard Gaussian distribution and then apply these identities. With the corresponding realizations of the state variables, a realization of the zero-coupon bond price $P\left(t,T\right)$ at the future time horizon t can be obtained by using the dependence relations (25), (32) and (33).

Actually with the state variables $x_{t;1}(\cdot )$ and $x_{t;2}(\cdot )$ values, as given in (32) and (33), then the following quantitative expression for the zero coupon price $P\left(t,T\right)(\cdot )$ at the future time-horizon t and for the remaining maturity $T-t$ is quantitatively described by the following result.

Theorem 2. Let us consider the G2++ model as defined in (13), (14) and (15). Let us consider a future horizon time t such that $\omega \equiv \omega \left(t,\rho ,{\kappa }_1,{\kappa }_2\right)$ satisfies the condition (39). Then there are ${\epsilon }_1(\cdot )\equiv {\epsilon }_1\left(\cdot ;t;{\kappa }_1\right)$ and ${\epsilon }_2(\cdot )\equiv {\epsilon }_2\left(\cdot ;t;{\kappa }_1,{\kappa }_2,\rho \right)$ realizations of two independent standard Gaussian random variables such that the future time-t value of any zero-coupon with the maturity T, with $t<T$, conditionally on the values $x_{0;1}$ and $x_{0;2}$ of the state variables, is given by

$P\left(t,T\right)(\cdot )=\text{Θexp}\left[-\left({\lambda }_1{\epsilon }_1(\cdot )+{\lambda }_2{\epsilon }_2(\cdot )\right)\right].$(40)

The terms Θ, ${\lambda }_1$ and ${\lambda }_2$ in (40) are defined as follows

$\begin{array}{c}\text{Θ}\equiv \text{Θ}\left(t,T;P^{mkt}\left(0,t\right),P^{mkt}\left(0,T\right);x_{0,1},x_{0,2};\Upsilon \right)\\ =\text{exp}\left[-\left\{B_1\left(t,T\right)ℰ\left(t;{\kappa }_1\right)x_{0;1}+B_2\left(t,T\right)ℰ\left(t;{\kappa }_2\right)x_{0;2}-c\left(t,T;\Upsilon \right)\right\}\right]\end{array}$(41)

${\lambda }_1\equiv {\lambda }_1\left(t,T;\Upsilon \right)=\left\{{\sigma }_1{B}_1\left(t,T\right){ℱ}^{\frac{1}{2}}\left(t;{\kappa }_1,{\kappa }_1\right)+\omega {\sigma }_2{B}_2\left(t,T\right){ℱ}^{\frac{1}{2}}\left(t;{\kappa }_2,{\kappa }_2\right)\right\}$(42)

${\lambda }_2\equiv {\lambda }_2\left(t,T;{\kappa }_2,{\sigma }_2,\rho \right)=\left\{\sqrt{1-{\omega }^2}{\sigma }_2{B}_2\left(t,T\right){ℱ}^{\frac{1}{2}}\left(t;{\kappa }_2,{\kappa }_2\right)\right\}$(43)

and where $\omega \equiv \omega \left(t;\rho ,{\kappa }_1,{\kappa }_2\right)$ is defined as in (34).

At any (future) time t, with $0<t$, resulting from (25), the time-t yield-to- maturity for a maturity T, is readily given by

$\begin{array}{c}y\left(t,T\right)\equiv \frac{1}{T-t}\{B_1\left(t,T\right)ℰ\left(t;{\kappa }_1\right)x_{0;1}+B_2\left(t,T\right)ℰ\left(t;{\kappa }_2\right)x_{0;2}\\ -c\left(t,T;\Upsilon \right)+{\lambda }_1{\epsilon }_1(\cdot )+{\lambda }_2{\epsilon }_2(\cdot )\}\end{array}$(44)

where

Technically, the future zero-coupon price $P\left(t,T\right)$, with $0<t<T$, generated by the model as written in (40) makes only a practical sense whenever

$0\le {\lambda }_1{\epsilon }_1(\cdot )+{\lambda }_2{\epsilon }_2(\cdot )-\text{ln}\left(\text{Θ}\right)$(45)

or equivalently

$\begin{array}{l}\frac{1}{T-t}c\left(t,T;\Upsilon \right)-\left\{B_1\left(t,T\right)ℰ\left(t;{\kappa }_1\right)x_{0;1}+B_2\left(t,T\right)ℰ\left(t;{\kappa }_2\right)x_{0;2}\right\}\\ \le \left\{{\sigma }_1{B}_1\left(t,T\right){ℱ}^{\frac{1}{2}}\left(t;{\kappa }_1,{\kappa }_1\right)+\omega {\sigma }_2{B}_2\left(t,T\right){ℱ}^{\frac{1}{2}}\left(t;{\kappa }_2,{\kappa }_2\right)\right\}{\epsilon }_1(\cdot )\\ +\left\{\sqrt{1-{\omega }^2}{\sigma }_2{B}_2\left(t,T\right){ℱ}^{\frac{1}{2}}\left(t;{\kappa }_2,{\kappa }_2\right)\right\}{\epsilon }_2(\cdot ).\end{array}$(46)

This corresponds to making restrictions on the values of ${\epsilon }_1(\cdot )$ and ${\epsilon }_2(\cdot )$. That is, arbitrary shocks are not always suitable to run meaningful simulation. We should be aware that a correct use of the model G2++ implies to introduce the restrictions (29) for the time-0 (which are always satisfied with the assumption $x_{0;1}=x_{0;2}=0$) and the new restrictions (46) for the admissible shocks simulations.

Recently, we are faced with market regime changes in which negative interest rates for sovereign bonds and negative spreads are regularly observed. The G2++ model is not designed to take into account such pathological situations. Since there is no prominent model that can properly deal with negative interest rates, a suitable solution would be to use displaced filtered historical simulation. For more information about this aspect of negative yields, refer to [9] .

As mentioned in [11] , the two models 2 HW and G2++ are actually equivalent. Below in Section 4.2, we discuss the relationship between G2++ and some other interest rate models.

1) Vasicek Model: The G2++ model extends the Vasicek model by introducing two factors instead of one. In contrast to the Vasicek model, which characterizes the short rate using a single mean-reverting process, the G2++ model includes an extra factor to account for more intricate term structure dynamics.

2) Hull-White Model: The Hull-White model with one factor is a specific instance of the G2++ model, which focuses on considering only one factor. By incorporating a second factor, the G2++ model expands the Hull-White framework, allowing for enhanced adaptability in modeling the term structure and capturing the changes in interest rates.

3) Heath-Jarrow-Morton (HJM) Framework: The G2++ model is a derivation of the HJM framework, where the HJM framework directly models the complete forward rate curve, while the G2++ model concentrates on the short rate and employs a two-factor method to explain its behavior. It serves as a practical application of the HJM framework, incorporating certain assumptions regarding the underlying factors.

4) Affine Term Structure Models: The G2++ model is categorized as an affine term structure model (ATSM), which means that the bond prices and yields can be represented as exponential-affine functions of the state variables. This characteristic simplifies the calculation of bond prices and other derivatives, leading to the widespread use of the G2++ model in practical applications.

Various processes can be used to calibrate the G2++ model, but the basic idea remains the same: to ensure that observed market prices (such as zero coupon prices) closely match the corresponding prices generated by the model as in [12] - [14] .

Here is a comprehensive guide on enhancing transparency in this process, covering the key aspects: methodology, data, algorithm details, software tools, code, and validation.

1) Objective Function: The objective function is defined as the minimization of the sum of squared differences between the observed zero coupon prices or yields and the model-predicted ones.

$\text{Objective}\text{Function}={\displaystyle \sum {\left(P_{ob{s}_i}-P_{mode{l}_i}\right)}^2}$

where $P_{ob{s}_i}$ are market bond prices and $P_{mode{l}_i}$ are model-predicted bond prices.

2) Optimization Algorithm: We can use algorithms like Levenberg-Marquardt, Nelder-Mead simplex, or gradient descent techniques to effectively reduce the objective function. Establish convergence criteria, such as a threshold for the alteration in the objective function or parameter values between iterations.

3) Initial Parameters: The optimization process can be greatly impacted by the initial guesses. Using historical data or findings from past studies can guide the selection of initial parameters.

4) Software Tools: Specify the software and libraries used for calibration, such as MATLAB, R, or Python (e.g., SciPy for optimization).

5) Simulation of Yield Curves: Describe the procedure for simulating yield curves, including the discretization methods used (e.g., Euler-Maruyama). Analyze how changes in model parameters affect the simulated yield curves and include visualizations.

6) Assessment of Goodness-of-Fit: Define and calculate metrics such as SSE, RMSE, MAE, and MAPE.

Provide visualizations and residual plots to identify patterns or biases.

7) Validation and Testing: Validate the calibrated model against historical data to ensure accuracy. Test the model on new data not used in calibration to assess its predictive power.

To achieve accurate parameter estimation, a systematic approach is required for calibrating the G2++ model. By utilizing historical data, market prices, and robust optimization techniques, practitioners can ensure that the model accurately captures current market dynamics. This, in turn, enables reliable insights for interest rate hedging, derivative pricing, and portfolio optimization. It is essential to regularly validate and recalibrate the model to maintain its effectiveness in financial risk management.

The G2++ model plays a crucial role in pricing interest rate derivatives like swaps, caps, floors, and swaptions due to its accurate modeling of the term structure of interest rates. This feature makes it an essential tool for pricing purposes. For instance, when pricing European swaptions, the G2++ model enables us to calculate closed-form solutions for swaption prices, taking advantage of the model’s flexibility and precision.

In the case of more intricate derivatives such as Bermudan swaptions or interest rate caps and floors with embedded options, numerical methods are often employed for pricing. Methods like Monte Carlo simulations and finite difference methods help us address the path dependency and optionality aspects of these derivatives. Additionally, the G2++ model’s capability to capture correlations between different factors proves to be advantageous in accurately pricing multi-currency derivatives and managing associated risks.

For Interest Rate Derivatives Pricing, the G2++ model is employed to determine the prices of interest rate derivatives, including caps, floors, swaptions, and exotic interest rate products. This model effectively captures the fluctuations in the yield curve, encompassing both short-term rate changes and long-term trends. As a result, it is highly useful in pricing intricate derivative structures.

For Risk management, financial institutions utilize the G2++ framework to effectively manage risks, such as evaluating exposure to interest rate fluctuations and implementing hedging strategies. Through the simulation of various interest rate scenarios using the model’s parameters, risk managers can assess the level of risk within their portfolios and enhance returns by optimizing risk-adjusted strategies.

For Term Structure Modeling the G2++ framework is utilized in term structure modeling, encompassing the anticipation of forthcoming yield curve patterns and term premia. This is imperative for the valuation of bonds, asset-liability management, and the projection of interest rate fluctuations in diverse economic scenarios.

For Asset Allocation and Portfolio Management, portfolio managers rely on the G2++ model to make well-informed asset allocation choices by considering interest rate projections. By integrating yield curve dynamics into asset pricing models, they can effectively enhance portfolio returns while effectively managing risk exposures.

For the Market Risk Analysis, G2++ model is utilized by traders and risk analysts to examine market risk factors associated with interest rates. This encompasses evaluating the influence of yield curve shifts, changes in volatility, and correlation dynamics on investment portfolios and trading strategies.

As a conclusion G2++ yield curve model is of utmost importance in financial markets as it offers a sophisticated framework for interest rate modeling, pricing, risk management, and product innovation. Its capacity to comprehend intricate yield curve dynamics renders it an invaluable tool for financial institutions, investors, and analysts.