Artur Sepp Blog on Quantitative Investment Strategies

Let me present our recent research paper with Parviz Rakhmonov on the stochastic volatility model for Factor Heath-Jarrow-Morton (HJM) interest rate framework (available on SSRN: Stochastic Volatility for Factor Heath-Jarrow-Morton Framework).

Factor Heath-Jarrow-Morton (HJM) model

Under the risk-neutral measure, the interest rate curve can be conveniently modeled using the forward curve f_t(tau) where tau is rate tenor. It is well known that the bond prices can be reconstructed from f_t(tau). When we develop a model for the evolution of forward curve f_t(tau), the HJM framework imposes conditions on the drift of the forward curve f_t(tau) so that the future forward curve reprices (in expectation) the term structure seen today. In practice, this requires using multi-dimensional models and handling path-dependency. See a brief intro to HJM and references in wikipedia.

When we apply a finite dimensional basis for modelling of the forward rate under statistical measure P, the core difficulty is to derive the corresponding factor dynamics under the risk-neutral measure Q. The paper by Lyashenko and Goncharov provides a straightforward way for augmentation of P-dynamics so that the Q-dynamics are arbitrage-free and consistent with the initial term structure of forward rates by construction.

Nelson-Siegel Term Structure model

Nelson-Siegel Term Structure model provides a convenient way to model the forward curve f_t(tau) under the statistical P-measure using just 3 factors for the level, slope, and convexity of the terms structure of forward rates. This model is widely used by central banks due to its intuitiveness and due to its good consistency with time series of rates data.

In our paper, we develop a generic Factor HJM model extended this model with the stochastic volatility using our previous paper for modelling stochastic volatility of one-factor interest rate model. As a base case, we apply the dynamics of Nelson-Siegel factors under P-measure with stochastic log-normal volatility of this factors.

We find that this approach is well aligned with the popularity of Nelson-Siegel model and extends this model for realistic P-modeling of factors with stochastic volatility. It is well-established that stochastic volatility models can model empirical features such as volatility clustering, auto-correlations, and heavy-tails, while log-normality of rates volatility for one-factor models is well documented. We apply our developed log-normal stochastic volatility with quadratic drift as a driver for volatility of Nelson-Siegel factors.

For valuation purposes, we derive the model dynamics under risk-neutral Q-measure. The advantage of our framework is that is fully analytic, and it allows for consistent valuation and risk management of interest rate derivatives including swaps, swaptions, futures rates and options on futures rates.

Simulations of Nelson-Siegel Term Structure model with Log-normal Stochastic Volatility

In this post, I will illustrate some possible outputs from our model using Monte Carlo simulations. First, I apply the inference of  Nelson-Siegel factors using Diebold-Li approach. Then I use the term structure of US rates observed at the end of February 2024 and I apply model parameters calibrated to swaptions data. In Figure 1, I show the term structure of US Treasury yields and fitted Nelson-Siegel curve. The model fit is very good. For pricing purposes under risk-neutral measure Q, we introduce a small deterministic curve so that the given yield curve is fitted exactly.

Figure 1. Initial US Treasury yields and fitted Nelson-Siegel curve

Next I simulate the factors of Nelson-Siegel model under P-measure as shown in Figure 2 for the simulation horizon of one year using the initial Nelson-Siegel curve in Figure 1. For brevity, in Figure 2, I show only 10 paths. Factors 1, 2, 3 are the level, slope, and convexity drivers of the forward curve. Realisations of factor X1 model possible evolution for overall level of rates: we observe a range of outcomes from 1.5% to 5.0% in 1 year. Paths of factor X2 model the (negative) slope: all paths indicate mean reversion back to positive slope indicating upward looking forward curves. Paths of factor X3 show the evolution of the convexity of the forward curve.

Figure 2. Simulated paths of Nelson-Siegel factors for 1y horizon starting from initial values (X1, X2, X3) = (4.36%, 1.3%, -1.0%) with mean-reversion lambda=0.55. See Eq (1) along with Eq (23) in the paper.

Along with the factors I simulate the log-normal stochastic volatility of these factors using the calibrated model. I apply the extension of Karasinki-Sepp stochastic volatility model augmented with the quadratic drift as developed in our paper with Parviz. I show the paths of volatility in Figure 3. The starting value of the stochastic volatility is 100%. For each factors, we apply deterministic volatility scale which is a part of model calibration (see sections 2.1 and 7.5 in the paper). Interestingly, the correlation between different factors and the volatility driver has different signs: the volatility is positively correlated with level factor X1, while the volatility is negatively correlated with  slope factor X2 and convexity factor X3. We see that we can obtain a rich set of realization for both the forward curve and the volatility of factors. We can compare paths of the volatility sigma_t with the move index for implied rates volatility.

Figure 3. Simulated paths of the Log-normal stochastic volatility of Nelson-Siegel factors. See Eq (13) in the paper.

The realisations of Nelson-Siegel factors and their volatilities in 1y allows us to construct the forward curve f_t(tau) under measure Q as seen in 1 year. We observe different shapes of forward curve in 1y compared to the today curve as function of tenor, as shown in Figure 4. Overall, the level of the yield curve is expected to decline following the initial curve shown in Figure 1, yet we have scenarios with higher curve (path 2), inverted U-shape curve (path 5), upward sloping curve (path 1), and flattish curve with different rate levels. Thus, Nelson-Siegel model can generate a rich set of scenarios of the yield curve evolution which can be applied either for valuation of interest rate derivatives or for risk and stress management of fixed-income portfolios.

Figure 4. Realisations of the forward rates in 1 year using simulated paths of Nelson-Siegel factors. See Eq (22) in the paper for factor loadings.

Simulations of the yield curve allow us to construct realisations of interest rate derivatives as swap curves, as shown in Figure 5 for swap rate starting in 1y as function of tenor, and rates futures, which require a convexity adjustment (see Section 3.3 in the paper).

Figure 5. Realisations of swap rate starting in 1Y as function of tenor computed using simulated forward rates. See Eq (29) in the paper.

Furthermore, we can value call and put options on swap futures rates. In Figure 6, I show model implied volatilities for 1y5y swaptions computed using simulated paths. Option strikes are set as fix moneyness in basis points (bps) relative to 1Y5Y swap rate in each paths. We see that the model can generate a variety of implied volatility curves from convex curves (as today), curves with strong positive skeweness (as seen in middle of year 2022), to curves with negative skeweness (as seen during 2010s).

Figure 6. Model implied volatilities of swaptions on 1Y5Y swap rate seen in 1 year as functions of moneyness in bps relative to 1Y5Y swap rate in each path. See Eq (88) in the paper for valuation of swaptions.

Summary

Given the popularity of Nelson-Siegel term structure model, our model can provide a valuable toolkit for building scenarios for the shape of both the yield curve and the implied volatilities and for risk-management of fixed-income derivatives. I emphasize that the model is arbitrage-free and consistent with the initial forward curve by construction and the model can value different interest rate derivatives (swaptions and options on rate futures) consistently.

Enjoy the reading of our paper in full and feel free to provide comments.