The Discrete-Time 4-Factor Path-Dependent Volatility Model: Calibration under P and Q
Authors present a discrete time path dependent volatility model with fat tailed shocks that fits SPX and VIX. Under Q it matches VIX with r squared 0.96 in sample and 0.87 out of sample and captures V shaped SPX smiles plus a wing. Under P it beats benchmarks and a three factor wins out of sample. P and Q align. A mixed loss stabilizes memory by 1.5 to 3 times, leaving smile and futures gaps.
What it examines
The paper studies a discrete-time 4-factor path-dependent volatility model and how to calibrate it under both the real-world (P) and risk‑neutral (Q) measures. Using a Student‑t--like Gaussian mixture for returns, it fits VIX, SPX volatility surfaces, and joint SPX/VIX smiles, estimates parameters by maximum likelihood, and proposes mixed P+Q calibration.
What it concludes
The model matches VIX and SPX/VIX smiles, with fat tails capturing the right wing. MLE under P outperforms benchmarks, and P and Q dynamics are close, supporting stable, path‑dependent volatility. A mixed P+Q calibration improves parameter stability with small smile-fit cost. Uses: pricing, SPX/VIX joint modeling, hedging, risk. Limits: symmetric shocks; future: skewed innovations.
Evidence objects
New 4-factor path-dependent parsimonious model with fat-tailed three-Gaussian shocks approximating Students t to fit SPX and VIX, reproducing V-shaped SPX smiles without ad hoc terms; VIX fit achieves $r^2$ 0.96/0.87.
key_findings bullet 1 · key_findings · validation V0
Under $P$, MLE on SPX returns beats diagonal QARCH, ZHawkes, quadratic rough Heston, EWMA Heston by likelihood and AIC; a 3-factor variant (trend plus two memories) simpler and wins out-of-sample.
key_findings bullet 2 · key_findings · validation V0
P and Q calibrations align: P-estimated parameters generate VIX paths ($r^2$ 0.92/0.78) and SPX/VIX smiles across 2019--2024; mixed-loss blending stabilizes kernels 1.5--3x; noted ATM VIX-smile, futures fit, skew, subjective-weight limitations.
key_findings bullet 3 · key_findings · validation V0
This paper discretizes a 4-factor path-dependent volatility model, allowing non-Gaussian innovations and practical hedging alignment, and links $P$ and $Q$ calibration. It proposes a Gaussian-mixture matched via Hellinger to Student-$t$, and a hybrid $P$--$Q$ estimation. Strong SPX/VIX fits, including joint smiles, and superior $P$-measure MLE deliver practitioner-relevant value, plus robustness.
key_findings bullet 4 · key_findings · validation V0
Raw abstract and provenance
This research benefited from the financial support of the BNP Paribas chair Futures of Quantitative Finance. Quantitative Finance , volume 17 , issue 2
Source row: 1930 · abstract type: snippet