The Hidden Cost of Volatility Drag

A 10-Year Backtest Reveals Insights into No-Arbitrage Models

Interest Rate Volatility and No-Arbitrage Affine Term Structure Models∗ Scott Joslin† Anh Le‡ This draft: April 3, 2016 Abstract An important aspect of any dynamic model of volatility is the requirement that volatility be positive. We show that for no-arbitrage affine term structure models, this admissibility constraint gives rise to a tension in simultaneous fitting of the physical and risk-neutral yields forecasts. In resolving this tension, the risk-neutral dynamics is typically given more priority, thanks to its superior identification. Consequently, the time-series dynamics are derived partly from the cross-sectional information; thus, time-series yields forecasts are strongly influenced by the no-arbitrage constraints. We find that this feature in turn underlies the well-known failure of these models with stochastic volatility to explain the deviations from the Expectations Hypothesis observed in the data.

The No-Arbitrage Structure and Volatility Instruments

The tension between matching physical and risk-neutral yields forecasts arises because volatility must be a positive process. This requires that forecasts of volatility must also be positive. This introduces a tension between first and second moments under the historical distribution. In particular, the presence of stochastic volatility induces a tension between matching first moments under the historical distribution (M1(P)) and the risk-neutral distribution (M1(Q)). This tension accentuates the difficulty in matching first and second moments under the historical distribution.

A 10-Year Backtest Reveals Insights into No-Arbitrage Models

Consider a three-factor no-arbitrage model with m = 0, 1, or 2 factors driving volatility. We estimate these models using historical data from the US Treasury bond market. The results show that when risk premia are constant, the coefficients φn should be uniformly equal to one across all maturities. However, in reality, the empirical φn coefficients are all negative and increasingly so with maturity.

The Impact of No-Arbitrage Models on Time-Series Dynamics

The presence of stochastic volatility in these models creates a tension between matching physical and risk-neutral yields forecasts. To resolve this tension, we need to derive time-series dynamics that are partly from the cross-sectional information. This is achieved by estimating the Q dynamics using the estimated historical term structure model.

A Practical Convenience: Generating Good Starting Points

One practical convenience of our approach is that we can use Gaussian models to generate very good starting points for these Am(N) models. In our estimation, these starting values take only a few minutes to converge to their global estimates. This makes it easier to compare the results between different models and to identify any differences.

The Importance of No-Arbitrage Restrictions in Stochastic Volatility Models

Our findings help clarify the nature of the relationship between the no-arbitrage structure and volatility instruments extracted from the cross-section of bond yields documented by several recent studies. For example, we show that for the A1(N) class of models (an N factor model with a single factor driving volatility), the cross-section of bonds will reveal up to N linear combinations of yields, given by the N left eigenvectors of the risk-neutral feedback matrix (KQ 1).

Invariance of Risk-Neutral Forecasts under Volatility Considerations

The estimates of KQ 1 are very strongly identified and essentially invariant to volatility considerations. This invariance implies that a Gaussian term structure model can reveal which instruments would be admissible for a stochastic volatility model.

Conclusion: No-Arbitrage Restrictions Are Not Completely Irrelevant

Our results help identify aspects of model specifications that may or may not have any signifficant bearing on the model implied volatility outputs. For example, within the A1(N) class of models, different specifications of the market prices of risks are unlikely to signifficantly affect the identification of the volatility factor.

Additional Insights from Our Study

Our study provides further evidence that no arbitrage restrictions are not completely or nearly irrelevant for the estimation of Gaussian dynamic term structure models. The answer to this question is surprising, given the existing evidence regarding Gaussian DTSMs. However, our results suggest that the "first moments" tension essentially provides a channel through which relatively more precise Q information will spill over and influence the estimation of the P dynamics.

Practical Takeaways

In conclusion, our study highlights the importance of considering no-arbitrage restrictions when estimating term structure models with stochastic volatility. By understanding the role of risk-neutral forecasts in matching physical and risk-neutral yields forecasts, we can gain insights into the limitations of these models. This knowledge can inform investment decisions and help investors better understand the dynamics of interest rates and bonds.

YES /10 (moderate interest, but with some novelty)