Seasonal Stochastic Volatility and Correlation together with the Samuelson Effect in Commodity Futures Markets

Abstract: We introduce a multi-factor stochastic volatility model based on the CIR/Heston volatility process that incorporates seasonality and the Samuelson effect. First, we give conditions on the seasonal term under which the corresponding volatility factor is well-defined. These conditions appear to be rather mild. Second, we calculate the joint characteristic function of two futures prices for different maturities in the proposed model. This characteristic function is analytic. Finally, we provide numerical illustrations in terms of implied volatility and correlation produced by the proposed model with five different specifications of the seasonality pattern. The model is found to be able to produce volatility smiles at the same time as a volatility term-structure that exhibits the Samuelson effect with a seasonal component. Correlation, instantaneous or implied from calendar spread option prices via a Gaussian copula, is also found to be seasonal.

• The paper presents a multi-factor stochastic volatility model that extends the CIR/Heston process by incorporating seasonality and the Samuelson effect.
• The model derives closed-form expressions for seasonality functions and a joint characteristic function for two futures prices, aiding option pricing.
• Numerical results show that seasonality significantly impacts implied volatility smiles, term structures, and correlations, affecting risk management strategies.

Seasonal Stochastic Volatility and Correlation in Commodity Futures Markets

This paper introduces a multi-factor stochastic volatility model for commodity futures markets, incorporating seasonality and the Samuelson effect. The model builds upon the CIR/Heston volatility process, extending it to include a seasonal component in the volatility dynamics. The authors derive conditions for the well-definedness of the seasonal volatility factor and calculate the joint characteristic function of two futures prices with different maturities. Numerical examples illustrate the model's ability to generate volatility smiles, a volatility term structure exhibiting the Samuelson effect, and seasonal correlation patterns.

CIR Process with Time-Dependent Drift

The paper extends the CIR model by allowing the mean-reversion level, $\theta$, to be time-dependent, while keeping the mean-reversion rate, $\kappa$, and volatility, $\sigma$, constant. The authors establish conditions on the seasonality function $\theta(t)$ to ensure the existence and uniqueness of a strong solution to the SDE describing the variance process: $$dv(t) = \kappa (\theta(t) - v(t))dt + \sigma \sqrt{v(t)}dB(t)$$. These conditions require $\theta(t)$ to be piecewise continuous and bounded by positive constants. Under these conditions, the solution remains strictly positive if the Feller condition, $\sigma^2 < 2\kappa\theta_{min}$, is satisfied.

Seasonal Stochastic Volatility Model

The authors present a futures-based model where the futures price $F(t, T_m)$ follows the SDE: $$dF(t, T_m) = F(t, T_m) \sum_{j=1}^n e^{-\lambda_j(T_m - t)} \sqrt{v_j(t)} dB_j(t)$$, where $v_j(t)$ are stochastic variance processes with seasonal mean-reversion levels: $$dv_j(t) = \kappa_j (\theta_j(t) - v_j(t))dt + \sigma_j \sqrt{v_j(t)} dB_{n+j}(t)$$. The correlation between the Brownian motions is given by $\langle dB_j(t), dB_{n+j}(t) \rangle = \rho_j dt$. The joint characteristic function $\phi$ of two log-returns $X_1(T)$ and $X_2(T)$ is derived, which is crucial for option pricing.

Seasonality Functions

The paper explores several parametric forms for the seasonality function $\theta(t)$, including sinusoidal, exponential-sinusoidal, sawtooth, triangle, and spiked patterns. These functions are characterized by parameters $a$, $b$, and $t_0$, representing the volatility level, seasonality magnitude, and peak volatility timing, respectively. Closed-form expressions for the transform $$\hat{\theta}_T(\lambda) = \int_0^T \theta(t) e^{\lambda t} dt$$ are provided for the sinusoidal, sawtooth, and triangle patterns, facilitating efficient numerical implementation.

Implied Volatility

Numerical illustrations demonstrate the model's ability to reproduce implied volatility smiles and term structures with seasonal patterns. The term structures exhibit both the Samuelson effect and seasonality, while the strike structures display smiled shapes. The choice of seasonality pattern has a limited impact on the shape of the volatility smile, but significantly affects the term structure.

Seasonal Stochastic Correlation

The paper investigates the impact of seasonality on the correlation between futures contracts in a multi-factor model. The instantaneous correlation $\rho(t)$ is derived as a function of the variance processes $v_1(t)$ and $v_2(t)$, and the Samuelson effect damping factors $\lambda_1$ and $\lambda_2$. The authors observe that the seasonality in the variance processes induces a seasonal pattern in the correlation. The numerical examples show that the relative magnitudes of $\lambda_1$ and $\lambda_2$ influence the direction of the effect of seasonality on the correlation. Calendar spread option prices are also analyzed, revealing that seasonality affects both the prices and implied correlations.

Conclusion

This paper presents a comprehensive framework for modeling seasonal stochastic volatility and correlation in commodity futures markets. The model's ability to accommodate various seasonality patterns and reproduce key empirical features makes it a valuable tool for practitioners and researchers. The finding that the instantaneous correlation between futures contracts is both stochastic and seasonal has important implications for risk management and hedging strategies. Further research could explore the relationship between Samuelson damping factors and the effect of seasonality functions on correlation.