Modeling Market Geometry

An Engineering Approach to Implied Volatility in Options

F. J. Penagos · Duke University, Pratt School of Engineering · December 2025

Read the paper (PDF, 11 pages)

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Abstract

The implied-volatility surface is usually treated as an input to option pricing. This project treats it as a signal. Each day's option chain is fit with a penalized cubic spline, the day-over-day deformations of the surface are decomposed with principal component analysis into level, skew, and curvature, and the momentum of those components drives a delta-neutral volatility strategy: long at-the-money straddles when the surface is expanding, short out-of-the-money strangles when it is compressing.

The approach grew out of a stochastic-search optimization framework built for a trading problem in CEE 251L, and was then extended to options and validated in QuantConnect across in-sample, out-of-sample, stress, and live periods.

At a glance

Universe	Options on AAPL, TSLA, NVDA, AMZN, and SPY; end-of-day chains, 10–45 days to expiry, moneyness within ±25%
Signal	Daily changes in the first three principal components (level, skew, smile), ATM curvature change, and the implied–realized volatility spread
Positions	Long ATM straddle or short 5% OTM strangle, delta-hedged whenever |Δ| > 0.25
Exits	+20% profit, −14% loss, or 15 trading days, whichever comes first
Risk limits	At most three concurrent positions, 0.5% of capital per trade, curvature and liquidity filters
Platform	QuantConnect research and backtesting environments

Method

1. Surface construction. For each trading day, quotes with positive spreads and open interest are projected onto a smooth surface over moneyness $m = K/S_t - 1$ and days to expiry $\tau$:

$$ IV_t(m,\tau) = \arg\min_{f \in C^2} \sum_{i=1}^{N_t} \big(IV_i - f(m_i,\tau_i)\big)^2 + \lambda \iint \left(\frac{\partial^2 f}{\partial m^2} + \frac{\partial^2 f}{\partial \tau^2}\right)^2 dm, d\tau $$

2. Dimensionality reduction. Fixed-maturity slices $S_t(m)$ are differenced and stacked, and PCA is applied to the deformations $\Delta S_t$. The first three modes explain over 90% of the variance and map to level, skew, and smile:

$$ PC_i^t = e_i^\top \Delta S_t, \qquad \Delta PC_i^t = PC_i^t - PC_i^{t-1} $$

3. Features. The PCA deltas are combined with a three-day rolling level change, the change in ATM curvature $\kappa_t$, and the volatility risk premium $IV^{ATM}_t - RV_t$.

4. Thresholds. Entry thresholds come from a randomized quantile search that maximizes the joint Sharpe ratio of all trades; the optimal quantiles fall between the 54th and 67th percentiles. A trade fires only when all four surface features agree.

Results

Period	Length	Ann. return	Sharpe	Max drawdown	Win rate
In-sample	4 years	22.74%	1.23	39.4%	56%
Out-of-sample A	6 months	−3.35%	−0.71	17.5%	46%
Out-of-sample B	1 year	130.27%	1.87	9.8%	43%
Out-of-sample C	1 year	24.82%	1.17	16.6%	46%
Stress: COVID-19	3 months	−0.41%	−0.17	17.3%	48%
Stress: 2008	8 months	1.23%	0.74	4.21%	50%
Live trading	1 week	1.0%	—	0.1%	—

The strategy does best in sustained volatility regimes; single-day shocks remain hard to capture. Backtests fill at mid-quotes with frictionless hedging, and Section 6 of the paper discusses how spreads and hedging frequency would reduce these numbers.

_{Out-of-sample period C in QuantConnect (July 2023 – July 2024).}

Origins: the quality equation

The project began as a stock-trading optimization in CEE 251L. Each stock's "quality" was modeled as a weighted sum of smoothed momentum, acceleration, and local volatility,

$$ Q_{d,s} = q_1,\dot{\bar p}_{d,s} + q_2,\ddot{\bar p}_{d,s} + q_3,v_{d,s}, $$

and the weights, thresholds, and smoothing window were tuned with Optimized Step-Size Random Search (ORSopt) under drawdown and turnover penalties. Because the objective landscape was rugged, the parameter surfaces were Gaussian-smoothed to pick broad, stable optima over sharp, overfit peaks. That emphasis on robustness carried directly into the volatility-surface model.

_{Buy and sell thresholds tightening and relaxing over the 200-day training window.}

Repository contents

research_geometric_markets.pdf   the paper
images/                          figures from the paper and this page

The strategy itself was implemented in QuantConnect's research and algorithm environments.

Acknowledgments

Research conducted under the academic supervision of Prof. Henri P. Gavin (Civil and Environmental Engineering, Duke) and Prof. David Ye (Mathematics, Duke). The QuantConnect implementation was co-developed with Shiya Ye, Tianai Wang, and Ben Greene in MATH 585.

Citation

@misc{penagos2025marketgeometry,
  author       = {Penagos, F. J.},
  title        = {Modeling Market Geometry: An Engineering Approach to Implied Volatility in Options},
  year         = {2025},
  institution  = {Duke University, Pratt School of Engineering},
  howpublished = {\url{https://github.com/felipejpenagos/Capstone-Research-Modeling-Market-Geometry}}
}