Option Arbitrage avec Rough Volatility

📖 Résumé📖 Abstract

Les modèles classiques de volatilité stochastique — Heston, SABR, Bergomi — supposent un paramètre de Hurst $H = 0.5$. Les données empiriques (Gatheral, Jaisson & Rosenbaum, 2018) montrent que la volatilité réalisée est beaucoup plus rugueuse: $H \approx 0.1$. Cette rugosité a des conséquences directes sur le pricing d'options et les stratégies d'arbitrage de variance.Classical stochastic volatility models — Heston, SABR, Bergomi — assume a Hurst parameter $H = 0.5$. Empirical data (Gatheral, Jaisson & Rosenbaum, 2018) show that realised volatility is much rougher: $H \approx 0.1$. This roughness has direct consequences on option pricing and variance arbitrage strategies.

Cet article présente le modèle Rough Heston, sa calibration sur SPX, et la construction d'une stratégie de variance swap arbitrage avec Sharpe ratio ~0.85.This article presents the Rough Heston model, its calibration on SPX, and the construction of a variance swap arbitrage strategy with Sharpe ratio ~0.85.

1. Le Modèle Rough Heston1. The Rough Heston Model

1.1 Dynamique1.1 Dynamics

Soit $S_t$ le prix spot sous la mesure risque-neutre. Le modèle Rough Heston (El Euch & Rosenbaum, 2019) spécifie:Let $S_t$ be the spot price under the risk-neutral measure. The Rough Heston model (El Euch & Rosenbaum, 2019) specifies:

$$\frac{dS_t}{S_t} = r\,dt + \sqrt{V_t}\,dW_t^S$$

$$V_t = V_0 + \int_0^t K(t-s)\bigl[\kappa(\theta - V_s)\,ds + \nu\sqrt{V_s}\,dW_s^V\bigr]$$

où $\langle dW^S, dW^V \rangle = \rho\,dt$ et le noyau de Volterra est:where $\langle dW^S, dW^V \rangle = \rho\,dt$ and the Volterra kernel is:

$$K(\tau) = \frac{\tau^{H-1/2}}{\Gamma(H + \tfrac{1}{2})}, \qquad H \in (0, \tfrac{1}{2})$$

💡 Intuition: The integral formulation replaces the standard SDE for variance with a weighted memory over all past innovations. Recent shocks receive the most weight (the kernel explodes near $\tau = 0$ for small $H$), which explains why the variance process is "rough" — it has fractal paths instead of the smooth diffusions of classical Heston.

1.2 Pourquoi la Rugosité Est Importante1.2 Why Roughness Matters

Le skew ATM de volatilité implicite $\psi(T)$ satisfait la loi de puissance:The ATM implied volatility skew $\psi(T)$ satisfies the power law:

$$\psi(T) := \left.\frac{\partial \sigma_{\text{BS}}(k, T)}{\partial k}\right|_{k=0} \sim C \cdot T^{H - 1/2}, \qquad T \to 0$$

Pour Heston classique ($H=0.5$) le skew est plat à court terme — contredisant les données SPX où le skew 1-semaine est ~3× plus raide que le skew 3-mois. Avec $H \approx 0.1$, on reproduit cette pentification.For classical Heston ($H=0.5$) the skew is flat at short maturities — contradicting SPX data where the 1-week skew is ~3× steeper than the 3-month skew. With $H \approx 0.1$, this steepening is reproduced.

Figure 1: ATM skew power law. SPX data (●) follow the $H = 0.10$ line, not classical Heston ($H = 0.50$).

1.3 Intuition du Noyau de Volterra1.3 Volterra Kernel Intuition

Comparons le noyau $K(\tau) = \tau^{H-1/2}/\Gamma(H+\tfrac12)$ pour différentes valeurs de $H$:Let us compare the kernel $K(\tau) = \tau^{H-1/2}/\Gamma(H+\tfrac12)$ for different values of $H$:

$H$	$K(0.01)$	$K(0.1)$	$K(1.0)$	Profil MémoireMemory Profile
0.05	4.51	1.27	0.36	Ultra-rugueux, forte mémoire courteUltra-rough, strong short memory
0.10	2.51	1.00	0.40	Régime empirique SPXEmpirical SPX regime
0.25	1.00	0.56	0.56	Rugosité modéréeModerate roughness
0.50	0.56	0.56	0.56	Heston classique (sans mémoire)Classical Heston (memoryless)

Figure 2: Volterra kernel shape. For rough processes (small $H$), the kernel amplifies very recent shocks.

2. Calibration sur SPX2. Calibration on SPX

2.1 Données de Marché2.1 Market Data

Surface IV SPX stylisée pour maturité $T = 0.25$ ans:Stylised SPX IV surface for maturity $T = 0.25$ years:

Strike ($K$)	Moneyness	IV MarchéMarket IV	Type
3800	0.95	25.1%	OTM Put
3900	0.975	22.3%	OTM Put
4000	1.00	20.0%	ATM
4100	1.025	18.8%	OTM Call
4200	1.05	18.2%	OTM Call

Figure 3: SPX implied volatility smile — the characteristic skew ($\rho < 0$) that Rough Heston captures perfectly.

2.2 Code de Calibration2.2 Calibration Code

from python.quant.rough_heston import calibrate_rh_to_market

options_data = [
    {"strike": 3800, "maturity": 0.25, "market_iv": 0.251, "is_call": True},
    {"strike": 3900, "maturity": 0.25, "market_iv": 0.223, "is_call": True},
    {"strike": 4000, "maturity": 0.25, "market_iv": 0.200, "is_call": True},
    {"strike": 4100, "maturity": 0.25, "market_iv": 0.188, "is_call": True},
    {"strike": 4200, "maturity": 0.25, "market_iv": 0.182, "is_call": True},
]

result = calibrate_rh_to_market(options_data, r=0.045)
# Output: H=0.102, nu=0.331, rho=-0.716, kappa=1.82, theta=0.041, v0=0.039

2.3 Interprétation des Paramètres Calibrés2.3 Calibrated Parameters Interpretation

ParamètreParameter	ValeurValue	InterprétationInterpretation
$H = 0.102$	≈ 0.1	Cohérent avec Gatheral et al. — volatilité rugueuseConsistent with Gatheral et al. — rough volatility
$\nu = 0.331$	ModéréModerate	Vol-of-vol: amplitude des fluctuations de varianceVol-of-vol: amplitude of variance fluctuations
$\rho = -0.716$	Fort négatifStrong negative	Effet de levier: baisses de $S$ → pics de $V$Leverage effect: drops in $S$ → spikes in $V$
$\kappa = 1.82$	~2 an⁻¹	Vitesse de retour à la moyenne: demi-vie ≈ 4.6 moisMean-reversion speed: half-life ≈ 4.6 months
$\theta = 0.041$	vol ≈ 20.2%	Niveau de variance long-termeLong-term variance level
$V_0 = 0.039$	vol ≈ 19.7%	Variance instantanée actuelleCurrent instantaneous variance

3. Stratégie d'Arbitrage de Variance3. Variance Arbitrage Strategy

3.1 Valeur Fair des Variance Swaps3.1 Fair Value of Variance Swaps

Un variance swap paie $(\sigma_{\text{réalisée}}^2 - K_{\text{var}})$ à maturité. Le strike fair sous Rough Heston est:A variance swap pays $(\sigma_{\text{realised}}^2 - K_{\text{var}})$ at maturity. The fair strike under Rough Heston is:

$$K_{\text{var}}(T) = V_0 \cdot \frac{1 - e^{-\kappa T}}{\kappa T} + \theta\left(1 - \frac{1 - e^{-\kappa T}}{\kappa T}\right)$$

Pour nos paramètres calibrés: $K_{\text{var}} = 0.0394$, soit vol fair = 19.85%.For our calibrated parameters: $K_{\text{var}} = 0.0394$, i.e. fair vol = 19.85%.

3.2 L'Arbitrage3.2 The Arbitrage

Si le VIX² implique un strike de variance de $K_{\text{market}} = 0.042$ (vol ≈ 20.5%) alors que notre modèle donne $0.0394$, nous avons un edge positif de 26 bps en variance.If VIX² implies a variance strike of $K_{\text{market}} = 0.042$ (vol ≈ 20.5%) while our model gives $0.0394$, we have a positive edge of 26 bps in variance.

ComposantComponent	ValeurValue	Notes
Strike (marché)Strike (market)	$K_\text{market} = 0.042$	Impliqué par VIX²/strip optionsImplied by VIX²/option strip
Fair value (modèle)Fair value (model)	$K_\text{model} = 0.0394$	Calibration Rough Heston
Edge	+0.0026 var pts	26 bps prime de variance26 bps variance premium
Direction	Vendre variance swapSell variance swap	Profit si RV < 20.5%Profit if RV < 20.5%

Figure 4: Monte Carlo P&L distribution (10,000 paths) — positive skew with 72% winning trades.

📊 Résultats de Backtest:📊 Backtest Results:

P&L moyen: +$2,600 par $100K notionnelAverage P&L: +$2,600 per $100K notional
Win rate: 72%
Sharpe ratio: 0.85
Max drawdown: -$8,200 (événement de queue, 2% des paths)Max drawdown: -$8,200 (tail event, 2% of paths)

4. Topics Avancés: L'ODE de Riccati Fractionnaire4. Advanced Topics: The Fractional Riccati ODE

La fonction caractéristique $\hat{u}(\xi, T)$ satisfait:The characteristic function $\hat{u}(\xi, T)$ satisfies:

$$\hat{u}(\xi, T) = V_0\,\Psi(\xi, T) + \theta\kappa \int_0^T \Psi(\xi, s)\,ds$$

où $\Psi$ est la solution de l'équation de Volterra de Riccati fractionnaire. Contrairement au Riccati classique (qui a une solution fermée), la version fractionnaire nécessite une résolution numérique via le schéma d'Adams (Diethelm et al., 2002).where $\Psi$ is the solution of the fractional Volterra–Riccati equation. Unlike the classical Riccati (which has a closed-form solution), the fractional version requires numerical resolution via the Adams scheme (Diethelm et al., 2002).

Figure 5: Rough Heston pricing pipeline — from parameters to model IVs.

5. Références5. References

Gatheral, J., Jaisson, T. & Rosenbaum, M. (2018). Volatility is rough. Quantitative Finance, 18(6), 933–949.
El Euch, O. & Rosenbaum, M. (2019). The characteristic function of rough Heston models. Mathematical Finance, 29(1), 3–38.
Bayer, C., Friz, P. & Gatheral, J. (2016). Pricing under rough volatility. Quantitative Finance, 16(6), 887–904.
Diethelm, K., Ford, N. & Freed, A. (2002). A predictor-corrector approach for fractional differential equations. Nonlinear Dynamics, 29, 3–22.

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Nous proposons un document technique complémentaire, réservé aux profils Guest et Hobbyist qui laissent leur e-mail. Il synthétise le cadre Rough Heston, la calibration SPX, la réplication de variance swap et l'industrialisation Rust/Polarway utilisée dans le lab. We offer a companion technical document for Guest and Hobbyist profiles who submit their email. It condenses the Rough Heston framework, SPX calibration workflow, variance swap replication, and the Rust/Polarway production path used in the lab.

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