We examine the predictability of the model-free implied volatility from swaptions on the future realized volatility of the underlying swap rates. The model-free implied volatility demonstrates significant predictability of the future realized volatility of the swap rate along a broad cross-section of tensors. We focus on investigating the predictive power of the model-free implied volatility on the future realized volatility of the underlying swap rates.
Furthermore, the model-free implied volatility from the swaps is an upwardly biased predictor of future realized volatility. The superior predictability of model-free implied volatility is robust conditional on different market states. We briefly describe the methodology used to calculate model-free implied volatility for swap rates.
Panel B reports the summary statistics for the swap rate model-free implied volatility we estimated.
Information content of SRMFIV
Moreover, all the SRMFIV series have lower standard deviations and larger autocorrelations than those for the realized volatility. Thus, SRMFIV contains significant information about future realized volatility of exchange rates over a wide range of tenors. LargeF statistics, ranging from 42.34 to 176.81, show that SRMFIV is a biased forecast of future realized volatility for exchange rate.
The estimates show that the significant information content of the SRMFIV arising from swaptions about the realized volatility of the underlying swap rates is persistent over different sample periods. In summary, we show that, by estimating a univariate predictive regression as in equation (6), we show that the implied volatility without a swap rate model, derived from the swaptions over various strike rates, contains significant information about the future realized volatility of swap rates over a broad spectrum. range of tenors.
Comparison with other predictors
These results show that the model-free implied volatility contains significant information for future realized volatility and includes the information content of lagged realized volatility. Then, with the built-in GARCH(1,1) model, we simulate the model 20 steps ahead, which is about one month ahead and 1000 sample paths. Finally, monthly swap rate change is obtained by aggregating each sample path and the standard deviation across the 1000 simulated points is estimated as a predictor of future realized volatility.
Moreover, the predictive power of conditional volatility estimated from the GARCH model is even weaker than that of lagged realized volatility from Table 4. Panel B shows the estimates when both V GARCH and SRMFIV are included in the predictive regression. Taken together, all these results show that the model-free implied volatility derived from swaptions is superior to the conditional volatility estimated from the GARCH model in predicting future realized volatility of swap rates.
As a final comparison, we examine the predictive power of the model-free implied volatility while controlling for all the other alternative predictors. RVt+1=β0+β1SRM F IVt+β2RVt+β3V GARCHt+t+1, (11) where all the variables are defined in the same way as in the previous sections. All the columns in Table 6 show that the model-free implied volatility has superior predictive power for future realized volatility.
All the alternative predictors: lagged realized volatility and conditional volatility estimated with the GARCH model, have no marginal predictability with model-free implied volatility included in the predictive regression. In summary, in this section we show by various comparisons that the model-free implied volatility estimated from swaptions traded over-the-counter is a superior predictor of future realized volatility of swap rates across different tenors. With the model-free implied volatility included, alternative predictors do not provide much additional predictability.
Out-of-sample performance
To investigate the performance of predictive regression with different predictors, we compare the RMSE of Model 1, Model 2, and Model 3 with the RMSE of the benchmark model. The comparison model could also be considered as a constrained model of each of the three models, where the slope coefficient is constrained to zero. In addition, to demonstrate the superior predictability of model-free implied volatility due to swaps, we compare the RMSEs of Model 1 with those of Models 2 and 3.
The difference between RMSEs from different models (such as Model 1 vs. Model 2 or Model 1 vs. Model 3) is tested based on Diebold and Mariano (2002). All three models produce lower RMSE values than the benchmark model reported in Panel A. Additionally, Model 1, which has model-free implied volatility as a univariate predictor, is associated with the lowest RMSE values across all terms.
The positive and high DM statistics show that both Model 2 where lagged realized volatility is used as the univariate predictor and Model 3 where conditional volatility from a GARCH model is used as the univariate predictor generate higher RMSEs than Model 1. Adding lagged realized volatility or conditional volatility from GARCH model does not weaken the contribution of. Model 5” is the model when both the model-free implied volatility and conditional volatility estimated from the GARCH model are included in the predictive regression from Eq.(10).
Model 6" is a model in which model-free implied volatility, lagged realized volatility, and conditional volatility estimated from the GARCH model are all included in the predictive regression according to equation (11). By including model-free implied volatility in the linear predictive model, neither realized neither the lagged volatility nor the conditional volatility estimated from the GARCH model provide a rather marginal contribution to predicting realized volatility one month ahead in the future.In summary, we show that the superior predictability of model-free implied volatility to realized volatility in the future is not significant only in the sample , but also important when studied outside the sample.
Additional analysis
In each panel, rows 1-2 are for Model 1 when the model-free implied volatility is used as the predictor in the predictive regression of Eq. 6), rows 3-4 are for Model 2 when lagged realized volatility is used as the predictor in the predictive regression of Eq. 7) and rows 5-6 are for Model 3 when GARCH-type conditional volatility is used as the predictor in the predictive regression of Eq. From Panel B, we could see that the difference between R2OOS,REC and R2OOS,EXP is smaller than that of other models, implying the superior performance of SRMFIV when predicting future realized volatility in recessions. Panel D shows that the difference between R2OOS,HIGH and R2OOS,LOW is much smaller than that of other models, implying the superior performance of SRMFIV when forecasting future realized volatility when monetary policy uncertainty is high.
RVt+1,t+h=β0+β1SRM F IVt,t+h+β2RVt−h+1,t+β3V GARCHt,t+h+t+1,t+h, (15) where RVt+1, t+ h is realized volatility from month t+ 1 to t+h, SRM F IVt,t+h is model-free implied volatility estimated at the end of month t, from time swaps to maturity h months, RVt−h+1,tis lagged realized volatility from month−h+ 1 to. First, expanding forecast horizons, SRMFIV is still a significant predictor of future realized volatility. The F statistic is large, ranging from 4.58 to 83.36, indicating that model-free implied volatility is a biased forecast of future realized exchange rate volatility.
Panel B reports the results when the lagged realized volatility and conditional volatility estimated from a GARCH model are included in the regression. Except for the case of 1-year maturity swap rate and 3-month forecast horizon and the 6-month forecast horizon case, where the lagged realized volatility shows weak predictive power, neither lagged realized volatility nor conditional volatility estimated from a GARCH model process much additional predictive power. Although both lagged realized volatility and conditional volatility estimated from a GARCH model help reduce the RMSEs compared to the benchmark model, the decrease in RMSEs is largest for Model 1 when SRMFIV is used as a predictor.
Panel D examines whether there are additional contributions from the lagged realized volatility or conditional volatility estimated from a GARCH model. Similar to the previous section, we find that adding either the lagged realized volatility or conditional volatility estimated from a GARCH model in addition to the model-free implied volatility does little to improve out-of-sample performance. Overall, we find that the superior performance of the model-free implied volatility in predicting future realized volatility persists with longer forecast horizons.
Conclusion
Finally, the superior out-of-sample performance of Model 1 declines as the forecast horizon becomes longer. This figure plots the in-sample and out-of-sample forecast performance with different market conditions for different models: “Model 1” is with model-free implied volatility as the predictor; "Model 2" is with lagged realized volatility as a predictor and "Model 3" is with the GARCH model estimated conditional volatility as a predictor. Panels A and B show the in-sample and out-of-sample forecast performance with NBER-defined business cycles (recession vs. expansion).
Panels C and D show the in-sample and out-of-sample forecast performance with monetary policy uncertainty states (high vs. low). This table provides the summary statistics for the realized volatility (RV) and exchange rate model-free implied volatility (SRMFIV) for exchange rates with tenors: 3-month, 6-month, 1-year, 2-year, 5-year, 10-year, 20-year and 30 years. Summary for realized volatility is reported in Panel A and summary for model-free implied volatility is reported in Panel B.
This table presents the correlation for realized volatility and model-free implied volatility for exchange rates with periods: 3-month, 6-month, 1-year, 2-year, 5-year, 10-year, 20-year and 30-year . The results for realized volatility are reported in Panel A and the results for model-free implied volatility in Panel B. Panel B reports the RMSE for the three models: “Model 1” is with model-free implied volatility as a predictor;
Model 2" is with lagged realized volatility as predictor and "Model 3" is with GARCH model estimated conditional volatility as predictor. Panel D presents the RMSEs for different multiple regression models: "Model 4" is the model when both SRMFIV and lagged realized volatility is included in the predictive regression from equation(8). Model 6” is the model when SRMFIV, lagged realized volatility and conditional volatility estimated from the GARCH model are all included in the predictive regression from equation(11).
This table reports the in-sample and out-of-sample R2s conditional on market conditions: NBER-dated business cycle. This table reports the in-sample and out-of-sample R2s conditional on market conditions: monetary policy uncertainty.
