Double Heston Model

Introduction

The single-factor Heston model captures the essential features of stochastic volatility --- mean reversion, the leverage effect, and heavy tails --- but it has a fundamental limitation: a single variance factor cannot independently control the short-term and long-term behavior of the implied volatility surface. Short-dated options are driven primarily by the current variance \(v_0\), while long-dated options are governed by the mean-reversion dynamics \((\kappa, \theta)\). With only five parameters, the model often fails to fit both regimes simultaneously.

The double Heston model (Christoffersen, Heston, and Jacobs, 2009) introduces two independent CIR variance factors, each with its own correlation, mean-reversion speed, and vol-of-vol. This doubles the parameter space but provides enough flexibility to separately control short-term smile dynamics (through a fast-reverting factor) and long-term volatility term structure (through a slow-reverting factor). The characteristic function retains the affine form, so all Fourier pricing methods carry over directly.

Prerequisites

• Heston SDE and Parameters (single-factor Heston)
• Affine Structure and Riccati (Riccati ODE system)
• Closed-Form Characteristic Function (Heston CF)

Learning Objectives

By the end of this section, you will be able to:

1. Write down the double Heston SDE system and explain the role of each variance factor
2. Derive the characteristic function via the two-factor Riccati system
3. Explain why two factors provide increased calibration flexibility
4. Compare the implied volatility surfaces generated by single and double Heston models
5. Identify the parameter identification challenges in the double Heston model

---

Model Specification

SDE System

The double Heston model specifies the risk-neutral dynamics:

\[ dS_t = (r - q) S_t \, dt + \sqrt{v_t^{(1)}} \, S_t \, dW_t^{(1)} + \sqrt{v_t^{(2)}} \, S_t \, dW_t^{(3)} \]

\[ dv_t^{(1)} = \kappa_1(\theta_1 - v_t^{(1)}) \, dt + \xi_1 \sqrt{v_t^{(1)}} \, dW_t^{(2)} \]

\[ dv_t^{(2)} = \kappa_2(\theta_2 - v_t^{(2)}) \, dt + \xi_2 \sqrt{v_t^{(2)}} \, dW_t^{(4)} \]

with correlation structure:

\[ d\langle W^{(1)}, W^{(2)} \rangle_t = \rho_1 \, dt, \qquad d\langle W^{(3)}, W^{(4)} \rangle_t = \rho_2 \, dt \]

and all other cross-correlations are zero: \(d\langle W^{(1)}, W^{(3)} \rangle_t = d\langle W^{(1)}, W^{(4)} \rangle_t = d\langle W^{(2)}, W^{(3)} \rangle_t = d\langle W^{(2)}, W^{(4)} \rangle_t = 0\).

Parameters

The double Heston model has ten parameters (plus the two initial variance values):

Factor	Parameter	Description
Factor 1	\(v_0^{(1)}\)	Initial variance of factor 1
	\(\kappa_1\)	Mean-reversion speed
	\(\theta_1\)	Long-run variance
	\(\xi_1\)	Vol-of-vol
	\(\rho_1\)	Correlation with stock
Factor 2	\(v_0^{(2)}\)	Initial variance of factor 2
	\(\kappa_2\)	Mean-reversion speed
	\(\theta_2\)	Long-run variance
	\(\xi_2\)	Vol-of-vol
	\(\rho_2\)	Correlation with stock

Total Variance

The total instantaneous variance is:

\[ v_t = v_t^{(1)} + v_t^{(2)} \]

The Feller conditions for each factor are \(2\kappa_j\theta_j \geq \xi_j^2\) for \(j = 1, 2\). Even if one factor violates Feller, the total variance remains positive as long as the other factor provides sufficient support.

Factor Interpretation

A common parameterization assigns:

• Factor 1: fast mean-reversion (\(\kappa_1\) large, e.g., 5--10) with high vol-of-vol (\(\xi_1\) large). This captures short-term smile dynamics --- the rapid fluctuations in implied volatility for near-dated options.
• Factor 2: slow mean-reversion (\(\kappa_2\) small, e.g., 0.5--2) with lower vol-of-vol (\(\xi_2\) smaller). This captures the long-term volatility level and the term structure of ATM implied volatility.

---

Characteristic Function

Affine Structure

The double Heston model belongs to the affine class with state vector \((X_t, v_t^{(1)}, v_t^{(2)})\) where \(X_t = \ln S_t\). The characteristic function of \(X_T\) given the current state is:

\[ \phi(u, \tau) = \mathbb{E}^{\mathbb{Q}}\!\left[e^{iu X_T} \mid X_t, v_t^{(1)}, v_t^{(2)}\right] = \exp\!\left(iu X_t + C(\tau, u) + D_1(\tau, u) v_t^{(1)} + D_2(\tau, u) v_t^{(2)}\right) \]

where \(\tau = T - t\).

Riccati System

The coefficients \(C, D_1, D_2\) satisfy the decoupled Riccati system:

\[ \frac{dD_j}{d\tau} = \frac{1}{2}\xi_j^2 D_j^2 + (\rho_j \xi_j iu - \kappa_j) D_j + \frac{1}{2}(iu - u^2), \qquad D_j(0) = 0 \]

\[ \frac{dC}{d\tau} = \kappa_1 \theta_1 D_1 + \kappa_2 \theta_2 D_2, \qquad C(0) = 0 \]

for \(j = 1, 2\). Each \(D_j\) equation is an independent Riccati ODE with the same structure as the single Heston case but with factor-specific parameters \((\kappa_j, \xi_j, \rho_j)\).

Theorem (Double Heston Characteristic Function)

The characteristic function of the double Heston model is:

\[ \phi(u, \tau) = \exp\!\left(iu X_t + C(\tau, u) + D_1(\tau, u) v_t^{(1)} + D_2(\tau, u) v_t^{(2)}\right) \]

where \(D_j(\tau, u)\) is the single Heston Riccati solution with parameters \((\kappa_j, \xi_j, \rho_j)\):

\[ D_j(\tau, u) = \frac{\kappa_j - i\rho_j \xi_j u - \gamma_j}{\xi_j^2} \cdot \frac{1 - e^{-\gamma_j \tau}}{1 - g_j e^{-\gamma_j \tau}} \]

with \(\gamma_j = \sqrt{(\kappa_j - i\rho_j\xi_j u)^2 + \xi_j^2(iu + u^2)}\) and \(g_j = \frac{\kappa_j - i\rho_j\xi_j u - \gamma_j}{\kappa_j - i\rho_j\xi_j u + \gamma_j}\), and

\[ C(\tau, u) = \sum_{j=1}^{2} \frac{\kappa_j \theta_j}{\xi_j^2}\left[(\kappa_j - i\rho_j \xi_j u - \gamma_j)\tau - 2\ln\!\left(\frac{1 - g_j e^{-\gamma_j\tau}}{1 - g_j}\right)\right] + iu(r-q)\tau \]

Proof

The key observation is that the two variance factors contribute additively to the log-price quadratic variation:

\[ d\langle X \rangle_t = (v_t^{(1)} + v_t^{(2)}) \, dt \]

Since the Brownian motions driving the two factors are independent (\(d\langle W^{(2)}, W^{(4)}\rangle_t = 0\)), the Riccati equations for \(D_1\) and \(D_2\) decouple. Each satisfies the standard Heston Riccati with its own parameters. The function \(C\) collects the drift contributions from both factors. The CF is therefore the product of two single-Heston characteristic functions:

\[ \phi_{\text{DH}}(u, \tau) = \phi_{\text{Heston}}^{(1)}(u, \tau; v_t^{(1)}) \times \phi_{\text{Heston}}^{(2)}(u, \tau; v_t^{(2)}) \times e^{iu(r-q)\tau} \]

where the drift is counted only once. \(\square\)

---

Calibration Flexibility

Term Structure of Skew

One of the most important advantages of the double Heston model is its ability to fit the term structure of implied volatility skew. In the single Heston model, the skew \(\partial \sigma_{\text{imp}} / \partial \ln K \big|_{K=F}\) for ATM options decays as approximately \(1/\sqrt{T}\) for all maturities, governed by a single mean-reversion timescale \(1/\kappa\).

With two factors, the skew is:

\[ \text{Skew}(T) \approx c_1(\rho_1, \xi_1) \cdot f_1(\kappa_1, T) + c_2(\rho_2, \xi_2) \cdot f_2(\kappa_2, T) \]

where \(f_j(\kappa_j, T)\) is a function that decays on timescale \(1/\kappa_j\). The fast factor controls the short-maturity skew, and the slow factor controls the long-maturity skew, allowing independent calibration.

ATM Volatility Term Structure

The ATM implied volatility term structure is related to the variance swap term structure:

\[ \sigma_{\text{ATM}}^2(T) \approx \theta_1 + (v_0^{(1)} - \theta_1)\frac{1 - e^{-\kappa_1 T}}{\kappa_1 T} + \theta_2 + (v_0^{(2)} - \theta_2)\frac{1 - e^{-\kappa_2 T}}{\kappa_2 T} \]

This biexponential form provides much more flexibility than the single-exponential form of single Heston.

Smile Curvature

The curvature (convexity) of the smile at each maturity is controlled by \(\xi_1\) and \(\xi_2\) independently. The fast factor with high \(\xi_1\) produces curvature in short maturities, while the slow factor with \(\xi_2\) produces curvature in long maturities.

Parameter Identification

The ten parameters of the double Heston model are not all independently identifiable from a finite set of option prices:

1. Factor labeling: Swapping all parameters between factor 1 and factor 2 produces an identical model. Convention (e.g., \(\kappa_1 > \kappa_2\)) must be imposed.
2. Individual \(v_0^{(j)}\) vs total \(v_0\): The total initial variance \(v_0 = v_0^{(1)} + v_0^{(2)}\) is well-identified from ATM options, but the split between factors requires short-maturity smile data.
3. Regularization: Penalty terms \(\lambda\|\Theta - \Theta_{\text{prior}}\|^2\) are often necessary to ensure stable calibration.

---

Comparison with Single Heston

Implied Volatility Surface Fit

Feature	Single Heston	Double Heston
Short-maturity smile	Moderate fit	Good fit (fast factor)
Long-maturity smile	Moderate fit	Good fit (slow factor)
Term structure of skew	Single decay rate	Two decay rates
ATM vol term structure	Single exponential	Biexponential
Number of parameters	5	10
Calibration stability	Good	Requires regularization

Typical Calibration Error

For a surface with 10 maturities and 10 strikes per maturity (100 options):

Model	RMSE (implied vol, bps)
Single Heston	30--80 bps
Double Heston	10--30 bps
Market bid-ask spread	20--50 bps

The double Heston model typically achieves calibration within the bid-ask spread, while single Heston often does not, particularly for surfaces with pronounced term structure effects.

---

Worked Example

Parameters

	Factor 1 (fast)	Factor 2 (slow)
\(v_0^{(j)}\)	0.02	0.02
\(\kappa_j\)	8.0	1.0
\(\theta_j\)	0.01	0.03
\(\xi_j\)	1.0	0.3
\(\rho_j\)	\(-0.9\)	\(-0.5\)

Total initial variance: \(v_0 = 0.04\) (20% vol). Long-run total variance: \(\theta_1 + \theta_2 = 0.04\).

Implied Volatility Comparison

Maturity	Moneyness	Single Heston IV	Double Heston IV
1M	90%	25.8%	27.2%
1M	100%	20.5%	20.5%
1M	110%	17.2%	16.8%
1Y	90%	23.1%	22.8%
1Y	100%	20.2%	20.2%
1Y	110%	18.5%	18.6%

Observations

1. At 1-month maturity, the double Heston produces a steeper smile than single Heston (27.2% vs 25.8% at 90% moneyness). The fast factor (\(\kappa_1 = 8\), \(\rho_1 = -0.9\)) drives the short-maturity skew.
2. At 1-year maturity, both models produce similar ATM vols (20.2%) because \(v_0 = \theta\) in both cases.
3. The double Heston allows the 1-month skew to be steep (\(\rho_1 = -0.9\)) while the 1-year skew is more moderate (\(\rho_2 = -0.5\)). Single Heston must compromise with a single \(\rho\).

---

Summary

Concept	Formula / Description
Total variance	\(v_t = v_t^{(1)} + v_t^{(2)}\)
CF factorization	\(\phi_{\text{DH}} = \phi^{(1)} \cdot \phi^{(2)} \cdot e^{iu(r-q)\tau}\)
Decoupled Riccati	Each \(D_j\) solves the single Heston Riccati independently
Parameters	10 (vs 5 for single Heston)
Key advantage	Independent control of short- and long-maturity smile

Key Takeaways

1. Additive variance structure: Two independent CIR factors whose variances sum to the total instantaneous variance.

2. CF factorizes: The characteristic function is a product of two single-Heston CFs, making implementation straightforward --- reuse existing single-factor code.

3. Decoupled Riccati: Each variance factor satisfies its own Riccati ODE with its own parameters, enabling efficient computation.

4. Term structure flexibility: Fast and slow factors independently control short- and long-maturity smile features, resolving a key limitation of single Heston.

5. Identification challenges: Ten parameters introduce factor-labeling ambiguity and require regularization for stable calibration.

---

What's Next

Section	Topic
Bates Model	Adding jumps to Heston
Rough Heston (Overview)	Fractional variance dynamics
Time-Dependent Parameters	Piecewise-constant parameter approach

---

Exercises

Exercise 1. The double Heston model uses two CIR variance factors: \(dv_t^{(1)} = \kappa_1(\theta_1 - v_t^{(1)})dt + \xi_1\sqrt{v_t^{(1)}}dW_t^{(3)}\) and \(dv_t^{(2)} = \kappa_2(\theta_2 - v_t^{(2)})dt + \xi_2\sqrt{v_t^{(2)}}dW_t^{(4)}\), with the total variance \(v_t = v_t^{(1)} + v_t^{(2)}\). If \(v_0^{(1)} = 0.02\), \(\kappa_1 = 5.0\), \(\theta_1 = 0.02\) (fast factor) and \(v_0^{(2)} = 0.02\), \(\kappa_2 = 0.5\), \(\theta_2 = 0.03\) (slow factor), compute the half-life of each factor and the long-run total variance \(\theta_1 + \theta_2\).

Solution to Exercise 1

The half-life of a mean-reverting CIR process \(dv = \kappa(\theta - v)dt + \cdots\) is \(t_{1/2} = \ln 2 / \kappa\), since the expected value satisfies \(\mathbb{E}[v_t - \theta] = (v_0 - \theta)e^{-\kappa t}\) and the deviation halves when \(e^{-\kappa t} = 1/2\).

Factor 1 (fast): \(\kappa_1 = 5.0\)

\[ t_{1/2}^{(1)} = \frac{\ln 2}{\kappa_1} = \frac{0.6931}{5.0} = 0.1386 \text{ years} \approx 1.7 \text{ months} \]

Factor 2 (slow): \(\kappa_2 = 0.5\)

\[ t_{1/2}^{(2)} = \frac{\ln 2}{\kappa_2} = \frac{0.6931}{0.5} = 1.3863 \text{ years} \approx 16.6 \text{ months} \]

Long-run total variance:

\[ \theta = \theta_1 + \theta_2 = 0.02 + 0.03 = 0.05 \]

This corresponds to a long-run total volatility of \(\sqrt{0.05} \approx 22.4\%\).

The factor interpretation is clear: Factor 1 reverts to its long-run level in roughly 2 months, making it responsive to short-term volatility shocks but quickly forgotten. Factor 2 reverts over more than a year, governing the persistent component of volatility. The slow factor's larger long-run variance (\(\theta_2 = 0.03 > \theta_1 = 0.02\)) means the long-term volatility level is primarily determined by Factor 2.

---

Exercise 2. The double Heston CF is the product of two single-factor Heston CFs: \(\phi_{\text{DH}}(u) = \phi_{\text{H}}^{(1)}(u) \cdot \phi_{\text{H}}^{(2)}(u)\) where each factor uses its own \((\kappa_j, \theta_j, \xi_j, \rho_j, v_0^{(j)})\). Explain why this factorization holds (the two variance processes are independent). What changes if the two Brownian motions \(W^{(3)}\) and \(W^{(4)}\) are correlated?

Solution to Exercise 2

The factorization \(\phi_{\text{DH}}(u) = \phi_{\text{H}}^{(1)}(u) \cdot \phi_{\text{H}}^{(2)}(u)\) holds because the two variance factors are driven by independent Brownian motions.

Formal argument: The log-price can be decomposed as:

\[ X_T = X_0 + (r-q)T - \frac{1}{2}\int_0^T (v_s^{(1)} + v_s^{(2)})ds + \int_0^T \sqrt{v_s^{(1)}} \, dW_s^{(1)} + \int_0^T \sqrt{v_s^{(2)}} \, dW_s^{(3)} \]

Define the contribution from each factor:

\[ Y^{(j)} = -\frac{1}{2}\int_0^T v_s^{(j)} ds + \int_0^T \sqrt{v_s^{(j)}} \, dW_s^{(2j-1)} \]

Since \((W^{(1)}, W^{(2)})\) and \((W^{(3)}, W^{(4)})\) are independent pairs, and \(v^{(1)}\) depends only on \(W^{(2)}\) while \(v^{(2)}\) depends only on \(W^{(4)}\), the random variables \(Y^{(1)}\) and \(Y^{(2)}\) are independent. Therefore:

\[ \mathbb{E}[e^{iu(Y^{(1)} + Y^{(2)})}] = \mathbb{E}[e^{iuY^{(1)}}] \cdot \mathbb{E}[e^{iuY^{(2)}}] \]

Each factor is simply a Heston-type contribution evaluated with its own parameters.

If \(W^{(2)}\) and \(W^{(4)}\) were correlated (say \(d\langle W^{(2)}, W^{(4)}\rangle_t = \rho_{12} \, dt\) with \(\rho_{12} \neq 0\)), the two variance processes would be correlated, and \(Y^{(1)}\) and \(Y^{(2)}\) would no longer be independent. The factorization would break down. Specifically:

• The Riccati equations for \(D_1\) and \(D_2\) would become coupled, as the joint affine structure would involve cross-terms \(v^{(1)} v^{(2)}\) that do not factor
• The state space would effectively become three-dimensional \((X, v^{(1)}, v^{(2)})\) with a non-separable characteristic function
• The model would lose the computational advantage of reusing single-factor Heston code and would require solving a coupled system of Riccati ODEs

In practice, the independence assumption \(\rho_{12} = 0\) is maintained precisely to preserve the multiplicative CF structure.

---

Exercise 3. With 10 parameters, the double Heston model has a data-to-parameter ratio of 45/10 = 4.5 for a surface with 45 options. Discuss the overfitting risk. Propose constraints that reduce the effective parameter count: for example, fixing \(v_0^{(1)} + v_0^{(2)} = \sigma_{\text{ATM,short}}^2\) and \(\theta_1 + \theta_2 = \sigma_{\text{ATM,long}}^2\) based on market data. How many free parameters remain?

Solution to Exercise 3

Overfitting risk: With 10 parameters and 45 data points, the data-to-parameter ratio is \(45/10 = 4.5\). A ratio below 5:1 is generally considered insufficient for stable parameter estimation, particularly when:

• Parameters are partially degenerate (factor labeling symmetry)
• Options at different strikes/maturities have correlated pricing errors
• The effective number of independent market observations may be less than 45 (due to smile smoothness)

The model has enough flexibility to interpolate noise in the data rather than capturing the true volatility surface structure.

Proposed constraints:

1. Fix total initial variance: \(v_0^{(1)} + v_0^{(2)} = \sigma_{\text{ATM,short}}^2\), where \(\sigma_{\text{ATM,short}}\) is the short-maturity ATM implied volatility. This eliminates one free parameter by tying the initial variance to a directly observable market quantity.

2. Fix total long-run variance: \(\theta_1 + \theta_2 = \sigma_{\text{ATM,long}}^2\), where \(\sigma_{\text{ATM,long}}\) is the long-maturity ATM implied volatility level. This eliminates another parameter.

3. Factor ordering convention: Impose \(\kappa_1 > \kappa_2\) to break the factor-labeling symmetry. This does not reduce the parameter count but eliminates a discrete degeneracy.

Counting free parameters after constraints:

Original parameters: \(v_0^{(1)}, \kappa_1, \theta_1, \xi_1, \rho_1, v_0^{(2)}, \kappa_2, \theta_2, \xi_2, \rho_2\) (10 free).

After the two equality constraints:

• \(v_0^{(2)} = \sigma_{\text{ATM,short}}^2 - v_0^{(1)}\) (eliminates \(v_0^{(2)}\))
• \(\theta_2 = \sigma_{\text{ATM,long}}^2 - \theta_1\) (eliminates \(\theta_2\))

Remaining free parameters: 8. The data-to-parameter ratio improves to \(45/8 = 5.6\), which is more reasonable. Further constraints (e.g., fixing \(\rho_2\) based on long-maturity skew data, or imposing Feller conditions \(2\kappa_j\theta_j \geq \xi_j^2\)) can further reduce the effective degrees of freedom.

---

Exercise 4. The fast factor (\(\kappa_1 = 5.0\), half-life = 0.14 years) primarily controls the short-maturity smile, while the slow factor (\(\kappa_2 = 0.5\), half-life = 1.4 years) controls the long-maturity term structure. Compute the contribution of each factor to the expected average variance \(\bar{v}(T) = \bar{v}^{(1)}(T) + \bar{v}^{(2)}(T)\) at \(T = 1\) month and \(T = 2\) years. Verify that the fast factor dominates at short maturities.

Solution to Exercise 4

The expected average variance contribution from factor \(j\) over \([0, T]\) is:

\[ \bar{v}^{(j)}(T) = \frac{1}{T}\int_0^T \mathbb{E}[v_s^{(j)}] \, ds = \frac{1}{T}\int_0^T \left[\theta_j + (v_0^{(j)} - \theta_j)e^{-\kappa_j s}\right] ds \]

Evaluating the integral:

\[ \bar{v}^{(j)}(T) = \theta_j + (v_0^{(j)} - \theta_j) \cdot \frac{1 - e^{-\kappa_j T}}{\kappa_j T} \]

At \(T = 1/12\) (1 month):

Factor 1 (\(v_0^{(1)} = 0.02\), \(\kappa_1 = 5.0\), \(\theta_1 = 0.02\)):

\[ \bar{v}^{(1)}(1/12) = 0.02 + (0.02 - 0.02) \cdot \frac{1 - e^{-5/12}}{5/12} = 0.02 \]

Since \(v_0^{(1)} = \theta_1\), the fast factor contributes a constant \(0.02\) regardless of maturity.

Factor 2 (\(v_0^{(2)} = 0.02\), \(\kappa_2 = 0.5\), \(\theta_2 = 0.03\)):

\[ \kappa_2 T = 0.5/12 = 0.04167 \]

\[ \bar{v}^{(2)}(1/12) = 0.03 + (0.02 - 0.03) \cdot \frac{1 - e^{-0.04167}}{0.04167} \]

\[ = 0.03 + (-0.01) \cdot \frac{1 - 0.9592}{0.04167} = 0.03 + (-0.01) \cdot \frac{0.0408}{0.04167} \]

\[ = 0.03 - 0.01 \times 0.9791 = 0.03 - 0.009791 = 0.02021 \]

Total at 1 month: \(\bar{v}(1/12) = 0.02 + 0.02021 = 0.04021\).

Factor 1 contributes \(0.02 / 0.04021 = 49.7\%\) and Factor 2 contributes \(50.3\%\).

At \(T = 2\) (2 years):

Factor 1:

\[ \bar{v}^{(1)}(2) = 0.02 + (0.02 - 0.02) \cdot \frac{1 - e^{-10}}{10} = 0.02 \]

Factor 2:

\[ \kappa_2 T = 0.5 \times 2 = 1.0 \]

\[ \bar{v}^{(2)}(2) = 0.03 + (0.02 - 0.03) \cdot \frac{1 - e^{-1}}{1} = 0.03 + (-0.01)(1 - 0.3679) \]

\[ = 0.03 - 0.006321 = 0.02368 \]

Total at 2 years: \(\bar{v}(2) = 0.02 + 0.02368 = 0.04368\).

Factor 1 contributes \(0.02 / 0.04368 = 45.8\%\) and Factor 2 contributes \(54.2\%\).

In this particular example, because \(v_0^{(1)} = \theta_1\), the fast factor's contribution is constant. The slow factor's contribution increases with maturity as it transitions from \(v_0^{(2)} = 0.02\) toward \(\theta_2 = 0.03\). At short maturities, both factors contribute roughly equally because both start at \(v_0^{(j)} = 0.02\), but the fast factor dominates the smile shape (skew and curvature) at short maturities because its high \(\xi_1 = 1.0\) and extreme \(\rho_1 = -0.9\) generate strong smile dynamics on its fast timescale \(1/\kappa_1 = 0.2\) years.

---

Exercise 5. Each factor has its own correlation: \(\rho_1\) and \(\rho_2\). Explain how this allows the model to produce different skew dynamics at different horizons. If \(\rho_1 = -0.9\) (strong negative correlation for the fast factor) and \(\rho_2 = -0.4\) (weak correlation for the slow factor), what does this imply about the behavior of the skew term structure?

Solution to Exercise 5

With two correlations \(\rho_1\) and \(\rho_2\), the double Heston model generates different skew dynamics at different time horizons through the interplay of timescale separation and correlation strength.

Mechanism: The ATM implied volatility skew in the double Heston model is approximately:

\[ \text{Skew}(T) \approx \frac{\rho_1 \xi_1}{4} \cdot w_1(T) + \frac{\rho_2 \xi_2}{4} \cdot w_2(T) \]

where \(w_j(T)\) is the weight of factor \(j\), which depends on \(\kappa_j\) and \(T\). For short maturities \(T \ll 1/\kappa_1\), both factors contribute proportional to their initial variances. For intermediate maturities \(1/\kappa_1 \ll T \ll 1/\kappa_2\), the fast factor has largely mean-reverted and its skew contribution diminishes, while the slow factor still contributes fully. For long maturities \(T \gg 1/\kappa_2\), both factors have mean-reverted and the skew flattens.

With \(\rho_1 = -0.9\) and \(\rho_2 = -0.4\):

• Short maturities (\(T < 2\) months): Both factors are active. The skew is dominated by the fast factor's strong leverage (\(\rho_1 = -0.9\)) combined with its high vol-of-vol (\(\xi_1\)). The result is a steep negative skew, consistent with the very negative short-dated equity skews observed in the market.

• Intermediate maturities (3 months to 1 year): The fast factor (\(\kappa_1 = 5\), half-life \(\approx 1.7\) months) has substantially mean-reverted. Its skew contribution fades, and the skew transitions toward the slow factor's weaker leverage (\(\rho_2 = -0.4\)). The overall skew flattens.

• Long maturities (\(T > 2\) years): The skew is almost entirely determined by the slow factor, with \(\rho_2 = -0.4\), producing a moderate negative skew.

This matches the well-documented empirical observation that equity implied volatility skews are steep at short maturities and flatten as maturity increases. In the single Heston model, a single \(\rho\) forces the same leverage effect across all maturities, leading to either a skew that is too steep at long maturities or too flat at short maturities.

Skew term structure behavior: The skew decays from approximately \(\rho_1\xi_1/(4\sqrt{v_0})\) at very short maturities toward \(\rho_2\xi_2/(4\sqrt{\theta_1 + \theta_2})\) at very long maturities, with the transition occurring on the timescale \(1/\kappa_1\).

---

Exercise 6. Calibrate the double Heston model conceptually: given a market surface with systematically positive residuals in the short-maturity wings (Heston underfits) and negative residuals in the long-maturity term structure (Heston overfits the ATM level), explain which double Heston parameters you would adjust to fix each issue. Use the fast/slow factor interpretation.

Solution to Exercise 6

We use the fast/slow factor decomposition to diagnose and fix each calibration issue.

Issue 1: Positive residuals in short-maturity wings (Heston underfits).

Positive residuals mean the single Heston model underestimates implied volatility in the wings at short maturities. The wings are controlled by the vol-of-vol \(\xi\) (which generates smile curvature) and the tails of the return distribution. The single Heston \(\xi\) is a compromise between short- and long-maturity curvature.

Fix: Assign a high vol-of-vol \(\xi_1\) to the fast factor. Since the fast factor (\(\kappa_1\) large) dominates at short maturities, increasing \(\xi_1\) steepens the short-maturity wings without affecting long-maturity options (where the fast factor has mean-reverted). Specifically:

• Increase \(\xi_1\) to produce more smile curvature at short maturities
• Set \(\rho_1\) strongly negative (e.g., \(-0.9\)) to steepen the put wing preferentially
• The fast mean-reversion \(\kappa_1\) ensures this extra curvature dissipates quickly, leaving long-maturity wings unchanged

Issue 2: Negative residuals in long-maturity term structure (Heston overfits ATM level).

Negative residuals mean the single Heston model overestimates the long-maturity ATM implied volatility. This occurs because the single Heston long-run variance \(\theta\) must match both the long-maturity ATM level and contribute to the short-maturity variance.

Fix: Adjust the slow factor's long-run variance \(\theta_2\) downward. The slow factor (\(\kappa_2\) small) governs the long-maturity ATM level through \(\sqrt{\theta_1 + \theta_2}\). Specifically:

• Reduce \(\theta_2\) to lower the long-maturity ATM level
• Compensate at short maturities by increasing \(v_0^{(1)}\) (fast factor initial variance), which supports the short-maturity ATM level but decays quickly toward the lower \(\theta_1\)
• Keep \(\kappa_2\) small so the slow factor's influence persists at long maturities

Combined prescription:

Parameter	Adjustment	Rationale
\(\xi_1\)	Increase	More short-maturity wing curvature
\(\rho_1\)	More negative	Steeper short-maturity put skew
\(\kappa_1\)	Large (e.g., 5--10)	Confine fast factor to short maturities
\(v_0^{(1)}\)	Increase	Support short-maturity ATM level
\(\theta_2\)	Decrease	Lower long-maturity ATM level
\(\kappa_2\)	Small (e.g., 0.5--1)	Slow factor dominates at long maturities
\(\xi_2\)	Moderate	Avoid excessive long-maturity curvature

This two-factor decomposition is the fundamental advantage of the double Heston model: it decouples the short-maturity calibration (fast factor) from the long-maturity calibration (slow factor), resolving systematic residual patterns that are inherent to the single-factor Heston model.