Analysing Quantiles in Models of Forward Term Rates

Abstract

The class of forward-LIBOR market models can, under certain volatility structures, produce unrealistically high long-dated forward rates, particularly for maturities and tenors beyond the liquid market calibration instruments. This paper presents a diagnostic tool for analysing the quantiles of distributions for forward term rates in a displaced lognormal forward-LIBOR model (DLFM). In particular, we provide a quantile approximation that can be used to assess whether the modelled term rates remain within realistic bounds with a high probability. Applying this diagnostic tool (verified using Quasi-Monte Carlo (QMC) simulations), we show that realised forward term rates for long time horizons may be kept within realistic limits by appropriately damping the tail of the DLFM volatility function.

1. Introduction

Early one-factor short rate models were able to keep long-dated interest rates within realistic limits by using a mean reversion feature (see, e.g., Vasiček 1977; Cox et al. 1985; or Hull and White 1990). Mean reverting short rates tend to decrease when high and increase when low. This property is imposed in the risk-neutral instantaneous short rate dynamics by a component of the drift dependent on the level of the short rate. However, if one considers the resulting dynamics of these models for, say, quarterly forward rates with simple compounding, when viewed under the respective forward measures to the end of their accrual periods, mean reversion manifests itself in the fact that the volatility of these rates decay with increasing time to maturity; there is no “mean reverting” component in the drift of these rates because simply compounded forward rates must be martingales (i.e., driftless) under the forward measure to the end of the accrual period in any arbitrage-free model.

Historically, simply compounded forward rates have had the label “forward LIBOR”, and models constructed directly on the basis of such market-quoted rates were called “LIBOR Market Models” (LMM).1 While the London Interbank Offer Rate (LIBOR) and similar interest rate benchmarks in other jurisdictions are widely2 being replaced by benchmarks based on overnight rates, such as the Secured Overnight Funding Rate (SOFR) in the United States, simply compounded term rates are still key market objects. Consequently, the LMM and its extensions remain important tools for the modelling of interest rate risks.3 In keeping with long-standing convention, we shall speak of simply compounded forward rates as “forward LIBORs.”

In such models, a decay of forward rate volatility with increasing time to maturity must be introduced explicitly (as opposed to implicitly via the mean reversion feature in short rate models) in order to prevent unrealistic long-term behaviour of the model dynamics. This is also true of models specified in terms of the volatilities of instantaneous forward rates in the Heath et al. (1992) (HJM) framework; however, in the present paper, we focus on models specified in terms of volatilities of simply compounded forward rates, i.e., LMM-type models.4 When choosing a volatility specification, there is no readily available algorithm to determine whether the choice will lead to realistically bounded realised forward LIBORs, other than using time-consuming Monte-Carlo methods. This is particularly important when extrapolating beyond the reach of current market calibration instruments when, for example, modelling the risk of longer-dated liabilities.5 In this paper, we develop and show the efficacy of a diagnostic tool for analysing the distributional behaviour of realised forward LIBORs arising from volatility specifications. This is useful for detecting poorly performing volatility parametrisations and evaluating the effectiveness of mitigation strategies.

We focus on a displaced diffusion extension of the LMM, namely the displaced lognormal forward-LIBOR model (DLFM), since it is more flexible (discussed in Section 2 below) yet still analytically tractable. In particular, we provide an efficient quantile approximation using the moment approximations derived in Van Appel and McWalter (2018, 2020), which are adapted here to the more generally applicable spot risk-neutral measure.6 Quasi-Monte Carlo (QMC) simulations show that our quantile approximation is an accurate and useful tool in analysing the distributional behaviour of long-dated forward-LIBOR rates. We then show that unrealistically high forward-LIBOR rates may be kept within realistic limits by appropriately damping the long-dated background volatility tail using, for example, a sigmoid function.

The paper is structured as follows. Section 2 describes some requirements of a modern term structure model and provides an overview of the DLFM under the spot risk-neutral measure. Section 3 introduces our quantile approximation with an outline of the Van Appel and McWalter (2020) moment approximation algorithm adapted to the spot measure. The accuracy of our quantile approximation is demonstrated under different scenarios in Section 4 using QMC simulations. Furthermore, we describe a useful volatility damping methodology to keep long-dated forward-LIBOR rates within realistic limits. We conclude with Section 5.

2. The Model

Complex derivatives have highlighted the need for term structure models that can produce changes in the shape, both in the gradient and curvature, of the yield curve. A model may gain this ability through the mechanisms of time-dependent volatilities or multiple driving factors. The latter allows financially convincing instantaneous forward rate correlation structures while the former is just as financially sound and capable of producing the desired terminal de-correlation, even in a one-factor model (see, e.g., Fries 2007, sct. 21.2.1). We do not consider the effectiveness of either mechanism individually since the class of forward-LIBOR market models inherently possesses both.

Our focus is on the DLFM, firstly, for its ability to capture skew effects and remain analytically tractable. Secondly, it can produce CEV-like skews with a parsimonious parametrisation (see, e.g., chp. 3 of Brace 2008 or Svoboda-Greenwood 2009). Furthermore, under a specific parameter specification, the DLFM may describe any discretized Gaussian model within the HJM framework (see, e.g., chp. 24 of Fries 2007; chp. 3 of Brace 2008; or chp. 14 of Andersen and Piterbarg 2010b). This is accomplished by setting the DLFM displacement parameter equal to the reciprocal of the accrual period of each rate, integrating the instantaneous forward rate volatility function over the discrete forward LIBOR tenors, and specifying the appropriate correlation structure. Lastly, the DLFM can produce negative forward LIBORs under a suitable parametrisation, which has become a desirable feature in periods of extremely low interest rates.

An important modelling consideration is the pricing measure. We may choose between the T-forward, terminal or spot measures. The latter is induced by a discretely compounded analog of the continuously compounded bank account numéraire, while the terminal measure corresponds to a special case of the T-forward measure in which the numéraire is the last modelled maturity zero-coupon bond. The terminal and spot measures are suitable when it comes to derivatives that require a numéraire that remains alive over the entire tenor structure. We, however, prefer the spot measure since it can be extended to any time horizon by simple compounding and it is more efficient when it comes to simulation. More specifically, under the terminal measure very large forward LIBOR realizations produce very small zero-coupon bond prices, resulting in normalized payoffs becoming very large (see, e.g., Example 5.1 in Brace 2008).

To fix notation, definitions, and terminology used throughout the paper, let tenor dates be given by $0=:T_0<T_1<\dots <T_M<T_{M+1}$. The time-t price of a zero-coupon bond paying a unit of currency at maturity $T_i$ is denoted $P(t,T_i)$. Define the accrual factors as ${\tau }_i=T_{i+1}-T_i$, in which case the time-t value of the forward LIBOR from $T_i$ to $T_{i+1}$ is given by

$$\begin{array}{c}\hfill F_i\left(t\right):=F\left(t;T_i,T_{i+1}\right)=\frac{1}{{\tau }_i}\left(\frac{P\left(t,T_i\right)}{P\left(t,T_{i+1}\right)}-1\right).\end{array}$$

(1)

We denote by ${\mathbb{Q}}^{\mathcal{B}}$ the spot measure associated with the numéraire asset7

$$\begin{array}{c}\hfill \mathcal{B}\left(t\right):=P\left(t,T_{\vartheta \left(t\right)}\right)\prod _{i=0}^{\vartheta \left(t\right)-1}\left(1+{\tau }_i{F}_i\left(T_i\right)\right),\end{array}$$

(2)

where $\vartheta \left(t\right):=min\{i:T_i\ge t\}$. Then the DLFM dynamics of $F_i\left(t\right)$ under ${\mathbb{Q}}^{\mathcal{B}}$ are

$$\mathrm{d}{F}_i\left(t\right)={\tilde{\mu }}_i\left(t\right)(F_i\left(t\right)+a_i)\mathrm{d}t+{\sigma }_i\left(t\right)(F_i\left(t\right)+a_i)\mathrm{d}{W}_i^{\mathcal{B}}\left(t\right),$$

(3)

for $t\le T_i$, where

$${\tilde{\mu }}_i\left(t\right):={\sigma }_i\left(t\right)\sum _{j=\vartheta \left(t\right)}^i\frac{{\rho }_{ij}{\sigma }_j\left(t\right){\tau }_j(F_j\left(t\right)+a_j)}{1+{\tau }_j{F}_j\left(t\right)},$$

(4)

$a_i$ are the constant displacement factors, ${\sigma }_i\left(t\right)$ is the instantaneous volatility of $F_i\left(t\right)$, and $W_i^{\mathcal{B}}\left(t\right)$ is the ith component of an M-dimensional ${\mathbb{Q}}^{\mathcal{B}}$ Brownian motion with instantaneous correlations given by:

$$\mathrm{d}{〈W_i^{\mathcal{B}},W_j^{\mathcal{B}}〉}_t={\rho }_{ij}\mathrm{d}t.$$

(5)

The correlation matrix formed by the elements ${\rho }_{ij}$ is denoted by $\rho$ and is assumed to be of rank n. If $n=M$, then $\rho$ is full rank.

It is worth remarking that the forward rates in (3) have an unknown distribution but are approximately displaced-lognormally distributed, and as a consequence, $F_i\left(t\right)+a_i$ are approximately lognormally distributed. Under the assumption of lognormality of $F_i\left(t\right)+a_i$, Van Appel and McWalter (2020) derive a computationally tractable algorithm that accurately and efficiently approximates the moments of the forward rate distribution under the T-forward measure. In the next section, we adapt this algorithm to the spot measure and provide a quantile approximation.

3. Quantile Approximation

In this section, we provide an algorithm for a computationally tractable quantile approximation of the entire forward rate curve under the spot measure, ${\mathbb{Q}}^{\mathcal{B}}$. We closely follow the terminology and notation set out in Van Appel and McWalter (2020) by defining a vector of displaced forward rates, $\mathit{H}\left(t\right)$, in terms of vectors $\mathit{F}\left(t\right)$ and $\mathit{a}$ with the elements $F_i\left(t\right)$ and $a_i$, for $1\le i\le M$ and $t\le T_i$, as

$$\begin{array}{c}\hfill \mathit{H}\left(t\right):=\mathit{F}\left(t\right)+\mathit{a}.\end{array}$$

(6)

Let $\overline{\mathit{H}}\left(t\right)$ and ${\mathsf{\Sigma }}_{}^{\mathit{H}}\left(t\right)$ denote the mean and covariance of $\mathit{H}\left(t\right)$. Then, we assume that

$$\begin{array}{cc}\hfill \mathit{Y}\left(t\right)& :=log\left(\mathit{H}\left(t\right)\right)\hfill \end{array}$$

(7)

is normally distributed as

$$\begin{array}{c}\hfill \mathit{Y}\left(t\right)\sim \mathcal{N}\left(\overline{\mathit{Y}}\left(t\right),{\mathsf{\Sigma }}_{}^{\mathit{Y}}\left(t\right)\right),\end{array}$$

(8)

where $\overline{\mathit{Y}}\left(t\right)$ and ${\mathsf{\Sigma }}^{\mathit{Y}}\left(t\right)$ are the mean and covariance, respectively, so that $\mathit{H}\left(t\right)$ is lognormally distributed as

$$\begin{array}{c}\hfill \mathit{H}\left(t\right)\sim log\mathcal{N}\left(\overline{\mathit{Y}}\left(t\right),{\mathsf{\Sigma }}_{}^{\mathit{Y}}\left(t\right)\right).\end{array}$$

(9)

Therefore, the quantile of each realised forward rate, $F_i\left(T_i\right)$, for $1\le i\le M$ is

$$\begin{array}{c}\hfill Q_i^{\mathit{F}}\left(p;{\overline{Y}}_i\left(T_i\right),{\mathsf{\Sigma }}_{ii}^{\mathit{Y}}\left(T_i\right)\right)=exp\left({\overline{Y}}_i\left(T_i\right)+\sqrt{2{\mathsf{\Sigma }}_{ii}^{\mathit{Y}}\left(T_i\right)}{\mathrm{erf}}^{-1}(2p-1)\right)-a_i,\end{array}$$

(10)

where

$$\begin{array}{c}\hfill p:={\mathbb{Q}}^{\mathcal{B}}\left(F_i\left(T_i\right)<Q_i^{\mathit{F}}\left(p;{\overline{Y}}_i\left(T_i\right),{\mathsf{\Sigma }}_{ii}^{\mathit{Y}}\left(T_i\right)\right)\right)\end{array}$$

(11)

with ${\overline{Y}}_i\left(T_i\right)$ and ${\mathsf{\Sigma }}_{ij}^{\mathit{Y}}\left(T_i\right)$ determined recursively through time in increments of $\Delta$. We outline this algorithm in the following steps:

• For $T_{\vartheta \left(t\right)-1}\le t<T_{\vartheta \left(t\right)}$ and $\vartheta \left(t\right)\le \{i,j\}\le M$, calculate

  $$\begin{array}{cc}\hfill {\overline{Y}}_i\left(t\right)& =log\left({\overline{H}}_i\left(t\right)\right)-\frac{1}{2}{\mathsf{\Sigma }}_{ii}^{\mathit{Y}}\left(t\right)\hfill \end{array}$$

  (12)

  and

  $$\begin{array}{cc}\hfill {\mathsf{\Sigma }}_{ij}^{\mathit{Y}}\left(t\right)& =log\left(\frac{{\mathsf{\Sigma }}_{ij}^{\mathit{H}}\left(t\right)}{{\overline{H}}_i\left(t\right){\overline{H}}_j\left(t\right)}+1\right),\hfill \end{array}$$

  (13)

  where $\overline{\mathit{H}}\left(0\right)=\mathit{H}\left(0\right)$, $\overline{\mathit{Y}}\left(0\right)=\mathit{Y}\left(0\right)$ and ${\mathsf{\Sigma }}_{}^{\mathit{H}}\left(0\right)={\mathsf{\Sigma }}_{}^{\mathit{Y}}\left(0\right)=\mathbf{0}$.
• Determine

  $$\begin{array}{cc}\hfill {\overline{Y}}_i(t+\Delta )& =U_i\left(\overline{\mathit{X}}\right)+\frac{1}{2}\mathrm{tr}\left({\nabla }^2{U}_i\left(\overline{\mathit{X}}\right){\mathsf{\Sigma }}_{}^{\mathit{X}}\right)\hfill \end{array}$$

  (14)

  and

  $$\begin{array}{cc}\hfill {\mathsf{\Sigma }}_{ij}^{\mathit{Y}}(t+\Delta )& =\nabla U_i{\left(\overline{\mathit{X}}\right)}^{\top }{\mathsf{\Sigma }}_{}^{\mathit{X}}\nabla U_j\left(\overline{\mathit{X}}\right)+\frac{1}{2}\mathrm{tr}\left({\mathsf{\Sigma }}_{}^{\mathit{X}}{\nabla }^2{U}_i\left(\overline{\mathit{X}}\right){\mathsf{\Sigma }}_{}^{\mathit{X}}{\nabla }^2{U}_j\left(\overline{\mathit{X}}\right)\right)\hfill \end{array}$$

  (15)

  for $\vartheta \left(t\right)\le \{i,j\}\le M$, where $\mathrm{tr}(·)$ is the trace operator using

  $$\begin{array}{c}\hfill \begin{array}{cc}\hfill U_i\left(\mathit{X}\right)& :=[Y_i\left(t\right)+{\mu }_i\left(t\right)\Delta +\frac{1}{2}{\mathcal{L}}^0{\mu }_i\left(t\right){\Delta }^2\hfill \\ \hfill & +\sum _{h=1}^n\left\{{\sigma }_i\left(t\right){\tilde{\rho }}_{ih}\Delta B_h+{\mathcal{L}}^h{\mu }_i\left(t\right)\Delta {\tilde{B}}_h\right\}{]}_{\mathit{X}}\hfill \end{array}\end{array}$$

  (16)

  with

  $${\mu }_i\left(t\right)={\sigma }_i\left(t\right)\sum _{j=\vartheta \left(t\right)}^i\frac{{\rho }_{ij}{\sigma }_j\left(t\right){\tau }_j(F_j\left(t\right)+a_j)}{1+{\tau }_j{F}_j\left(t\right)}-\frac{1}{2}{\sigma }_i^2\left(t\right),$$

  (17)

  and where ${\mathcal{L}}^0{\mu }_i\left(t\right)$ and ${\mathcal{L}}^h{\mu }_i\left(t\right)$ are defined in (A1) and (A2), respectively. The random variable

  $$\begin{array}{c}\hfill \mathit{X}:={[\mathit{Y}\left(t\right),\mathit{B}]}^{\top }\end{array}$$

  (18)

  is composed of two independent multivariate normal random variables, $\mathit{Y}\left(t\right)$ and $\mathit{B}\sim \mathcal{N}\left(\mathbf{0},{\mathsf{\Sigma }}_{}^{\mathit{B}}\right)$, where

  $$\begin{array}{c}\hfill \mathit{B}:=\left[\begin{array}{c}\Delta B_1\\ \Delta {\tilde{B}}_1\\ \Delta B_2\\ \Delta {\tilde{B}}_2\\ ⋮\\ \Delta B_n\\ \Delta {\tilde{B}}_n\end{array}\right]\mathrm{and}{\mathsf{\Sigma }}_{}^{\mathit{B}}:=\left[\begin{array}{ccccccc}\Delta & \frac{1}{2}{\Delta }^2& 0& 0& \cdots & 0& 0\\ \frac{1}{2}{\Delta }^2& \frac{1}{3}{\Delta }^3& 0& 0& \cdots & 0& 0\\ 0& 0& \Delta & \frac{1}{2}{\Delta }^2& \ddots & 0& 0\\ 0& 0& \frac{1}{2}{\Delta }^2& \frac{1}{3}{\Delta }^3& & 0& 0\\ ⋮& ⋮& \ddots & & \ddots & ⋮& ⋮\\ 0& 0& 0& 0& \cdots & \Delta & \frac{1}{2}{\Delta }^2\\ 0& 0& 0& 0& \cdots & \frac{1}{2}{\Delta }^2& \frac{1}{3}{\Delta }^3\end{array}\right].\end{array}$$
  Thus, $\mathit{X}\sim \mathcal{N}\left(\overline{\mathit{X}},{\mathsf{\Sigma }}_{}^{\mathit{X}}\right)$ with

  $$\begin{array}{c}\hfill \overline{\mathit{X}}=\left[\begin{array}{c}\overline{\mathit{Y}}\left(t\right)\\ \mathbf{0}\end{array}\right]\mathrm{and}{\mathsf{\Sigma }}_{}^{\mathit{X}}=\left[\begin{array}{cc}{\mathsf{\Sigma }}_{}^{\mathit{Y}}\left(t\right)& \mathbf{0}\\ \mathbf{0}& {\mathsf{\Sigma }}_{}^{\mathit{B}}\end{array}\right].\end{array}$$
  The elements of the gradient, $\nabla U_i\left(\overline{\mathit{X}}\right)$, Hessian, ${\nabla }^2{U}_i\left(\overline{\mathit{X}}\right)$, and their corresponding derivatives can be found in Appendix A.2. It is worth remarking that since (10) only requires the diagonal elements of ${\mathsf{\Sigma }}^{\mathit{Y}}$ we may significantly reduce the execution time by computing

  $$\begin{array}{c}\hfill {\mathsf{\Sigma }}_{ii}^{\mathit{Y}}(t+\Delta )=\nabla U_i{\left(\overline{\mathit{X}}\right)}^{\top }{\mathsf{\Sigma }}_{}^{\mathit{X}}\nabla U_i\left(\overline{\mathit{X}}\right)+\frac{1}{2}\mathrm{tr}\left({\mathsf{\Sigma }}_{}^{\mathit{X}}{\nabla }^2{U}_i\left(\overline{\mathit{X}}\right){\mathsf{\Sigma }}_{}^{\mathit{X}}{\nabla }^2{U}_i\left(\overline{\mathit{X}}\right)\right)\end{array}$$

  (19)

  and using

  $$\begin{array}{c}\hfill {\mathsf{\Sigma }}_{ij}^{\mathit{Y}}(t+\Delta )\approx \sqrt{{\mathsf{\Sigma }}_{ii}^{\mathit{Y}}(t+\Delta )}{\rho }_{ij}\sqrt{{\mathsf{\Sigma }}_{jj}^{\mathit{Y}}(t+\Delta )}\end{array}$$

  (20)

  to approximate the off-diagonal elements. The resulting trade-off, along with a lower-order approximation, is fully explored in Van Appel and McWalter (2020).
• Compute

  $$\begin{array}{cc}\hfill {\overline{H}}_i(t+\Delta )& =exp\left({\overline{Y}}_i(t+\Delta )+\frac{1}{2}{\mathsf{\Sigma }}_{ii}^{\mathit{Y}}(t+\Delta )\right)\hfill \end{array}$$

  (21)

  and

  $$\begin{array}{cc}\hfill {\mathsf{\Sigma }}_{ij}^{\mathit{H}}(t+\Delta )& ={\overline{H}}_i(t+\Delta ){\overline{H}}_j(t+\Delta )\left(exp\left({\mathsf{\Sigma }}_{ij}^{\mathit{Y}}(t+\Delta )\right)-1\right).\hfill \end{array}$$

  (22)
• If $t+\Delta <T_M$, repeat steps 1 through to 3.
• Once all the rates are evolved to expiry, the quantiles $Q_i^{\mathit{F}}\left(p;{\overline{Y}}_i\left(T_i\right),{\mathsf{\Sigma }}_{ii}^{\mathit{Y}}\left(T_i\right)\right)$ are computed using (10) for each $F_i$ and $T_i$, with $1\le i\le M$.

In the following section, we demonstrate the capability of the quantile approximation in (10) as a diagnostic tool to analyse the long-dated behaviour of forward LIBORs implied by the model.

4. Numerical Results

To verify the capability of the quantile approximation, given in (10), we used QMC simulations. In particular, we use Sobol’ sequences (see Joe and Kuo (2003, 2008)), Brownian-bridging and a predictor-corrector algorithm (see Hunter et al. 2001), in conjunction with $2^{20}-1$ sample paths of $F_i\left(t\right)$, for $1\le i\le M=199$ and ${\tau }_i=0.25$, at times $t\in \{0,\Delta ,2\Delta ,\dots ,T_i\}$ with increments of $\Delta =0.25$. Furthermore, to allow the replication of our results, we used the flexible and well-known instantaneous volatility parametrisation

$${\sigma }_i\left(t\right):=\sigma (t,T_i,a,b,c,d)=\left[a+b\left(T_i-t\right)\right]e^{-c(T_i-t)}+d,$$

(23)

while making the piecewise constant assumption

$${\sigma }_{ij}^2:=\frac{1}{{\tau }_j}{\int }_{T_j}^{T_{j+1}}{\sigma }_i^2\left(t\right)\mathrm{d}t,$$

(24)

for $j=0,1,\cdots ,\alpha -1$.

The next subsection takes advantage of the fact that a special DLFM specification yields a discretized Gaussian model from within the HJM framework. This allows us to demonstrate the forward rate quantile, provided in (10), for well-behaved forward LIBORs under a mean reverting Gaussian short rate process described in the HJM framework. This example further illustrates the forward rate mechanism that keeps long-dated forward rates bounded and realistic.

4.1. DLFM Specification of a Discretized HJM Model

The model in this section is for illustrative purposes. For its simplicity, we considered a one-factor mean-reverting Vasiček model, which has an initial instantaneous forward rate curve that is given by

$$\begin{array}{c}\hfill f(0,T)={\mu }_{\mathrm{V}}+e^{-{\alpha }_{\mathrm{V}}T}\left(r\left(0\right)-{\mu }_{\mathrm{V}}\right)-\frac{{\sigma }_{\mathrm{V}}^2}{2{\alpha }_{\mathrm{V}}^2}{\left(1-e^{-{\alpha }_{\mathrm{V}}T}\right)}^2,\end{array}$$

(25)

where r denotes the instantaneous short rate. The model parameters $\{{\alpha }_{\mathrm{V}},{\mu }_{\mathrm{V}},{\sigma }_{\mathrm{V}}\}$ and $r\left(0\right)$ are positive constants with their classical interpretation; that is, $r\left(0\right)$ is the initial short rate or spot rate, ${\alpha }_{\mathrm{V}}$ is the mean reversion speed, ${\mu }_{\mathrm{V}}$ is the long-run mean of the process and ${\sigma }_{\mathrm{V}}$ is the short rate volatility. Then, in the terms of the HJM framework, the Vasiček model evolves the instantaneous forward rate according to

$$\begin{array}{c}\hfill \mathrm{d}f(t,T)=\left({\sigma }_f(t,T){\int }_t^T{\sigma }_f(t,s)\mathrm{d}s\right)\mathrm{d}t+{\sigma }_f(t,T)\mathrm{d}{B}^{\mathbb{Q}}\left(t\right),\end{array}$$

(26)

where $B^{\mathbb{Q}}$ is a standard Brownian motion under the risk-neutral measure, which is associated with the usual bank account numéraire and

$$\begin{array}{c}\hfill {\sigma }_f(t,T)={\sigma }_{\mathrm{V}}exp\left(-{\alpha }_{\mathrm{V}}(T-t)\right)\end{array}$$

(27)

is the instantaneous forward rate volatility.

The volatility function in (27) provides a key insight for forward rates derived from a mean-reverting short rate process; the time-homogeneous forward rate volatility structure must exponentially decay to zero with increasing time to maturity.

The above HJM model description can be discretized to yield a DLFM using a particular specification as follows (for details on linking the HJM and DLFM, see, e.g., chp. 24 of Fries 2007; chp. 3 of Brace 2008; or chp. 14 of Andersen and Piterbarg 2010b). For $1\le i\le M$, set the displacement parameter equal to the reciprocal of the accrual period,

$$\begin{array}{c}\hfill {\alpha }_i=\frac{1}{{\tau }_i},\end{array}$$

(28)

and construct the discrete tenor volatility by integrating the instantaneous forward rate volatility in (27) over the LIBOR tenor structure to produce

$$\begin{array}{c}\hfill {\sigma }_i\left(t\right):={\int }_{T_i}^{T_{i+1}}{\sigma }_f(t,s)\mathrm{d}s=\sigma \left(t,T_i,\frac{{\sigma }_{\mathrm{V}}}{{\alpha }_{\mathrm{V}}}\left(1-exp(-{\alpha }_{\mathrm{V}}{\tau }_i)\right),0,{\alpha }_{\mathrm{V}},0\right),\end{array}$$

(29)

which is a simple reformulation of (23). Then, determine the initial forward LIBOR curve, $F_i\left(0\right)$, with the relationship in (1) using (25) to compute

$$\begin{array}{c}\hfill P(0,T_i)=exp\left(-{\int }_0^{T_i}f(0,s)\mathrm{d}s\right)=exp\left(V_1(0,T_i)-V_2(0,T_i)r\left(0\right)\right),\end{array}$$

(30)

where

$$\begin{array}{cc}\hfill V_1(0,T_i)& =\left(V_2(0,T_i)-T_i\right)\left[{\mu }_{\mathrm{V}}-\frac{{\sigma }_{\mathrm{V}}^2}{2{\alpha }_{\mathrm{V}}^2}\right]-\frac{{\sigma }_{\mathrm{V}}^2}{4{\alpha }_{\mathrm{V}}}{V}_2{(0,T_i)}^2\hfill \end{array}$$

(31)

and

$$\begin{array}{cc}\hfill V_2(0,T_i)& =\frac{1}{{\alpha }_{\mathrm{V}}}\left(1-exp(-{\alpha }_{\mathrm{V}}{T}_i)\right).\hfill \end{array}$$

(32)

To present this Vasiček example of well behaved realised long-dated forward LIBORs, we used the following model input parameters: ${\alpha }_{\mathrm{V}}=0.15$, ${\mu }_{\mathrm{V}}=0.09$, ${\sigma }_{\mathrm{V}}=0.025$, and $r\left(0\right)=0.07$. Figure 1a shows the Vasiček realised forward LIBOR standard deviation, denoted as $\sqrt{{\mathsf{\Sigma }}_{ii}^{\mathit{F}}\left(T_i\right)}:=\sqrt{{\mathsf{\Sigma }}_{ii}^{\mathit{H}}\left(T_i\right)}$, for $1\le i\le M$ computed using (22) (red dots).8 Here, the rate of increase in the forward rate standard deviation decreases and flattens out for longer maturities. This is a direct result of the time-homogeneous forward rate volatility, which exponentially decays to zero with increasing time to maturity. This can be seen in Figure 2a, which is a visualisation of the integrated instantaneous Vasiček forward rate volatility, (27), given in (29). For $1\le i\le M$, Figure 1b shows the realised Vasiček forward LIBOR quantile, $Q_i^{\mathit{F}}\left(p;{\overline{Y}}_i\left(T_i\right),{\mathsf{\Sigma }}_{ii}^{\mathit{Y}}\left(T_i\right)\right)$, given in (10), for $p=\{0.01,0.5,0.99\}$ along with the mean, ${\overline{F}}_i\left(T_i\right)={\overline{H}}_i\left(T_i\right)-{\alpha }_i$, computed using (21). The $p=\{0.01,0.99\}$ quantiles flatten out with increasing time to maturity, showing that the realised forward LIBORs remain bounded. Again, this is due to the time-homogeneous forward rate volatility, which exponentially decays to zero with increasing time to maturity, which prevents the forward rate process from diffusing when it is far from maturity. This effect can be seen in Figure 2b, which displays $2^6-1$ sample path realisations of $F_i\left(t\right)$ for $t\le T_i$ with a fixed $i=M=199$. Figure 1 shows that our diagnostic tool is accurate when compared to the QMC sample estimates (black circles), which we take to be our benchmark, given the relatively large sample size along with the use of Brownian bridging and a predictor-corrector algorithm.

Given that we have demonstrated the link between well behaved realised forward LIBORs and a decaying volatility function, ${\sigma }_f(t,T)$, under a simple scenario, we next explore the accuracy of our diagnostic tool in a more sophisticated multifactor scenario.

4.2. DLFM with a Background Level of Volatility

We consider an $n=5$ factor DLFM scenario with input parameters taken from Van Appel and McWalter (2020). The initial forward-LIBOR rate curve is specified as:

$$\begin{array}{c}\hfill F_i\left(0\right)=\frac{log\left(T_i+1\right)}{y}+x,\mathrm{for}{F}_0\left(0\right)=0.08\mathrm{and}{F}_M\left(0\right)=0.04\end{array}$$

(33)

and $\rho$ is the correlation matrix, which is the rank-n reduced version of the Schoenmakers and Coffey (2003) formulation, ${\rho }^{\mathit{SC}}$, with elements specified by

$${\rho }_{ij}^{\mathit{SC}}=exp\left[-\frac{|i-j|}{M-1}\left(-log{\rho }_{\infty }+\zeta \frac{M-i-j+1}{M-2}\right)\right],$$

(34)

where, $0\le \zeta \le -log{\rho }_{\infty }$ and $\{i,j\}\le M$ are tenor indices. The rank reduction method used is the approach of zeroing the smallest $M-n$ eigenvalues and rescaling (see, e.g., Brigo and Mercurio 2006, sct. 6.9.2). For the displacement factors, we chose the following CEV-inspired parametrisation (see, e.g., chp. 3 of Brace 2008 or Svoboda-Greenwood 2009):

$$\begin{array}{c}\hfill a_i=\frac{1-\gamma }{\gamma }{F}_i\left(0\right),\end{array}$$

(35)

where $\gamma =0.45$ is the CEV elasticity parameter since it is parsimonious and produces CEV-like skews.

Table 1. Summary of the DLFM volatility and correlation input parameters.

a	b	c	d	${\mathit{\rho }}_{\mathbf{\infty }}$	$\mathit{\zeta }$
0.03	0.08	1.35	0.07	0.40	0.8

provides a summary of the DLFM volatility and correlation input parameters for this scenario.

Here the realised forward LIBOR standard deviation does not flatten as in the Vasiček illustration of Section 4.1, but rather continues to increase with an upward curvature for long-dated maturities (as seen by comparing Figure 3a to Figure 1a). This is as a result of the level-d background volatility present in (23), which is responsible for diffusion of the long-term forward rates even when they are far from maturity. Figure 3b shows that this parameter set could be considered realistic for maturities out to approximately 35 years, with a $p=0.99$ quantile of around 40%. For longer maturities, however, there is a clear upward curvature developing in the $p=0.99$ realised forward LIBOR quantile (as with the realised forward LIBOR standard deviation in Figure 3a), which is an indication that forward LIBORs with longer maturities may encounter extreme outlier realisations. Further support for this is evident in the divergence between the mean and median realised forward LIBOR for the long maturities, which indicates that the large (outlier) sample paths become prominent. Exploding forward rates are well documented in the literature (see, e.g., Andersen and Piterbarg 2010a, sct. 4.5.3), with one solution being to restrict the volatility. This is the subject of the next section.

Figure 3 indicates that the approximations made by our diagnostic tool start to become less accurate for maturities longer than 35 years. This is seen in Figure 3a by comparing the realised forward rate standard deviation approximation, $\sqrt{{\mathsf{\Sigma }}_{ii}^{\mathit{F}}\left(T_i\right)}$, (red dots) to the benchmark QMC sample estimates (black circles). A similar observation can be made in Figure 3b for the quantile approximation, $Q_i^{\mathit{F}}\left(0.99;{\overline{Y}}_i\left(T_i\right),{\mathsf{\Sigma }}_{ii}^{\mathit{Y}}\left(T_i\right)\right)$, (magenta line). This is due to the gradual breakdown of the lognormal forward-LIBOR rate approximation, which can be seen in Figure 4, with the lognormal Q–Q plots for two different maturities. In particular, Figure 4a is a lognormal Q–Q plot of the realised sample QMC forward-LIBOR rate maturing at 25 years ($i=99$); here, the assumption of approximately lognormal forward-LIBOR rates is reasonable. However, the lognormal Q–Q plot in Figure 4b for the realised sample QMC forward rate maturing at 50 years ($i=M=199$) shows that the lognormal assumption erodes as a result of large outlier forward LIBOR realisations.

While the diagnostic tool in Figure 3 shows a loss of accuracy for long-dated forward LIBORs, this only occurs when the realised rates were already unrealistically high (greater than 50%). Thus, the quantile approximation is fit for the purpose of identifying unrealistically high rates, which only occur when the lognormal assumption breaks down. The next section describes a volatility-damping methodology that keeps long-dated forward LIBORs within realistic limits.

4.3. DLFM with Damped Background Volatility

We now demonstrate an approach that could be used to ensure realistic behaviour in the DLFM. This entails damping the level-d background volatility in (23) after a set cut-off time, $\tilde{T}$, using a sigmoid function with damping intensity, $\theta$, defined as

$${\tilde{\sigma }}_i\left(t\right)={\sigma }_i\left(t\right){\varsigma }_i\left(t\right),$$

(36)

where

$$\begin{array}{c}\hfill {\varsigma }_i\left(t\right):=1+{\mathbb{I}}_{\left\{T_i-t>\tilde{T}\right\}}\left[\frac{\mathrm{erf}\left(2-\theta \left\{T_i-t-\tilde{T}\right\}\right)-\mathrm{erf}\left(2\right)}{2}\right].\end{array}$$

(37)

For illustration purposes, we set $\tilde{T}=8$ years and $\theta =0.11$. Figure 5a compares the un-damped volatility in (23), which includes the level-d background (red line), to the damped volatility in (36) (blue dashed line).

Using the damped instantaneous volatility function in (36), we reproduce the results from the DLFM scenario presented in Section 4.2. Firstly, by comparing Figure 5b and Figure 3b (which are generated on the same scale), it is clear that the damping of the instantaneous forward rate volatility has kept the long-dated realised forward LIBOR quantiles more tightly bound and realistic to the 50-year maturity. The realised forward LIBOR standard deviation in Figure 6a flattens out for longer maturities and does not continue to increase with an upward curvature as in Figure 3a. These results are reminiscent of those in Section 4.1 for the Vasiček illustration, with a similar bounding effect in the realised forward LIBOR quantiles and standard deviation. Finally, Figure 6b is a lognormal Q–Q plot of the sample QMC realised forward rate maturing at 50 years ($i=M=199$) and shows that the assumption of approximately lognormal forward LIBORs is now more plausible when compared to Figure 4b.

In summary, the parametrisation in (23) may provide flexibility while calibrating to the short end of the market with a background level, d, of volatility. However, this may lead to unrealistic outlier rate realisations for longer maturities and tenors, particularly those beyond the reach of current calibration instruments. In such cases, the quantile approximation, given in (10), may be used to identify if a given calibration produces realistic behaviour in the long-dated forward LIBORs.

5. Conclusions

The diagnostic tool presented in this paper allows for a tractable analysis of the distribution of long-dated forward term rates implied by the DLFM under different forward rate volatility parametrisations. This is particularly relevant for long-term asset/liability management (needed, for example, in the pension and life insurance industry), where one needs to model interest rates for maturities and tenors beyond the reach of current market calibration instruments. When poor-performing volatility specifications are detected, we have demonstrated that damping the instantaneous volatility function ameliorates the situation.

Migrating the forward LIBOR moment approximation approach of Van Appel and McWalter (2020) to the spot risk-neutral measure has the advantage of better numerical stability, and it also means that it is immediately clear that any reasonable market price of interest rate risk results in a negligible impact of carrying out the analysis under a risk-neutral rather than the statistical (a.k.a. “real-world”) measure. The change in drift between these two distributions is given by the market price of risk only (rather than being conflated with the volatility of some numéraire asset), and since we are only interested in quantiles of unrealistically high interest rates, this change in drift has a negligible effect on the outcome of the analysis using our tool.

QMC simulations show that our approach yields accurate quantiles when compared to benchmark estimates under realistic volatility specifications. When a volatility specification leads to a violation of the lognormal assumption, our method is less accurate. Even though accuracy is somewhat diminished in these cases, our quantile approximation remains viable for identifying unrealistic LIBOR ranges. In conclusion, our quantile approximation may be used to verify whether or not a DLFM calibration produces realistic behaviour.

Our approach of ensuring the realistic long-term behaviour of forward term rates is particularly useful for scenario generation in the pension and insurance industries, where claims must be priced beyond liquid market instruments. For example, the European Insurance and Occupational Pensions Authority (see EIOPA 2022) mandates that participants price claims beyond the last liquid point (LLP) in market rates. They require that the forward rate term structure is extrapolated to a constant ultimate forward rate (UFR), which is stipulated by regulation at regular intervals. This may well lead to the need to further dampen the modelled volatility of long-term forward rates;9 our tool can assist in implementing this in a realistic fashion. We anticipate conducting further research in this area by applying our approach under both the risk-neutral measure for pricing claims and the real-world measure for risk management.