Amine L.

This concept underpins virtually every quantitative trading model by providing the tools to represent, manipulate, and extract structure from high-dimensional financial data. Linear Algebra Matrices and vectors compactly encode asset returns and risk exposures, while operations like eigen decomposition and singular value decomposition (SVD) reveal hidden factors and directions of maximal variance. These techniques enable portfolio optimization, risk factor identification (e.g., yield-curve “level,” “slope,” “curvature”), dimension reduction for faster backtesting, and the core computations in machine-learning-based strategies. Without a solid grasp of linear algebra, building scalable, efficient, and robust quantitative trading systems is virtually impossible Applications in Quantitative Trading Mean–Variance Optimization: Formulate the quadratic program Constraints & Factor Models: Incorporate linear constraints (e.g., sector exposures) via augmented matrices; use factor models (Σ=BB⊤+D) to reduce dimensionality and improve stability Factor Modeling & PCA Principal Component Analysis (PCA): Decompose Σ=VΛV to extract orthogonal factors vi (eigenvectors) ranked by variance λi. Projecting returns onto the top kkk PCs reduces noise and captures key market drivers (e.g., “level,” “slope,” “curvature” in yield curves) Statistical Factor Portfolios: Construct uncorrelated factor portfolios using eigenvectors as weight vectors; use eigenvalues to allocate risk budgets and tilt exposures toward dominant factors Algorithmic Strategies & Machine Learning Signal Extraction: Represent feature matrices (e.g., price histories, order-book snapshots) and apply SVD or matrix factorization to denoise data and extract latent factors for signals . Regression & Classification: Solve β=(X⊤X)^−1 X⊤ via QR or SVD for robust linear regression in cross-sectional alpha models; kernel methods likewise rely on inner-product space Market Microstructure & Execution Order-Book Dynamics: Model the limit-order book as a high-dimensional state vector; use linear filters (e.g., Kalman filters, which employ Cholesky for covariance updates) to estimate hidden liquidity or imbalance signals If you are someone from a non-STEM background who faces challenges in studying and deciding on mathematics topics to get an initial start in Quant Finance, Machine Learning, or Data Science, especially for those coming from a non-STEM background. We are starting "Anybody Can Do Math – A Primer for Quantitative Finance, Data Science & Machine Learning", an online Live classes Cohort. Register Now - https://lnkd.in/gQeWjMRV Course brochure - https://lnkd.in/gUqJJZ9P Early Bird discount of 25%, use Coupon Code ABCD25Off, valid till 1st July 2025. Sample lecture - https://lnkd.in/geF4C7CJ Total Hours - 50 hours Accessible via our custom Web + Mobile App with lifetime access to: Recorded lectures Annotated notes For query abcdmaths@quantinsider.io +918349489231

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🧠 Can we model volatility and interest rates with memory – and still get closed-form pricing? Yes we can. 📄 The Volterra Stein-Stein model with stochastic interest rates ✍️ Eduardo Abi Jaber, Donatien Hainaut, Edouard Motte 📎 arXiv:2503.01716 ⸻ This paper introduces a hybrid equity–interest rate model where both volatility and short rates follow correlated Gaussian Volterra processes. ➡️ This means the model captures long memory, roughness, or Markovian dynamics — depending on the kernel — and still retains analytical tractability. ⸻ 🔍 Why is this important? ✅ General enough to include Stein–Stein, Hull–White, rough volatility, and shifted-fractional variants. ✅ Closed-form pricing for zero-coupon bonds and options using resolvent kernels. ✅ Semi-explicit characteristic function of the log-forward index (via Fredholm resolvents) enables fast Fourier pricing of equity options. ✅ Successfully calibrates both rate and equity vol surfaces: • Hump-shaped ATM cap/floor vols (with H > 0.5) • Concave ATM skew on S&P 500 in log-log scale ✅ Negative correlation between rates and index emerges naturally — aligned with empirical observations. ✅ Bridges to linear-quadratic Riccati systems, offering further analytical and numerical insights. ⸻ 🎯 Bottom line: This model is realistic, flexible, and efficient — making it a serious alternative to Heston–Hull–White for pricing and hedging hybrid products or building risk proxies with memory. ⸻ 📎 Paper: https://lnkd.in/eP8DYN_b

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Joint Calibration of SPX and VIX Options Using the Willow Tree Method The Willow Tree Method is a useful technique with many applications in derivative pricing. I have discussed how it can be used to determine the implied volatilities of American options (link in the comments) This paper proposed using the Willow tree method to build a model that describes the volatility dynamics of both SPX and VIX options concurrently. -The paper introduces a new nonparametric discrete-time model called the Joint Implied Willow Tree (JIWT) approach to address the joint calibration of SPX and VIX option prices. -The JIWT method avoids model-based simulations by deriving risk-neutral probabilities directly from observed SPX and VIX option prices. -One key contribution is the ability to extract conditional probability distributions between maturities using both SPX and VIX data, which is not possible when relying solely on SPX options. -This allows the model to capture the volatility smile in both SPX and VIX markets, particularly for short-term maturities. -The JIWT method is simpler, easier to implement, and more efficient across multiple maturities compared to existing methods. -The JIWT method is fully model-free and data-driven, as it requires no pre-specified mathematical model for SPX and instead extracts the entire risk-neutral probability distribution from market data. Reference: Bing Dong, Wei Xu, Zhenyu Cui, Joint Implied Willow Tree: An Approach for Joint S&P 500/VIX Calibration, Journal of Futures Markets, 2025; 1–22 Join a community of 5,000+ quants—subscribe to the newsletter! Link in profile. #vix #volatility #options ABSTRACT Since the inception of Volatility Index (VIX) options trading, academic literature has persistently sought accurate methods for jointly calibrating the prices of the S&P 500 index (SPX) and VIX options. This study introduces a novel nonparametric approach, called the joint implied willow tree (JIWT) method, aimed at resolving this joint calibration challenge. The resulting willow tree adheres to the martingale constraint for the SPX and ensures that the VIX is derived as the implied volatility of a 30-day log contract on the SPX. A notable advantage of our method is its ability to recover not only the unconditional probabilities for a fixed maturity but also the conditional probabilities across different maturities. Consequently, we reconstruct the entire term structure of the SPX, aligning it with market information from both SPX and VIX options. Numerical and empirical analyses demonstrate that the JIWT method excels in accurately capturing the volatility smile of SPX and VIX across various maturities.

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Some of the best models start with an annoying real-world constraint. This is one of those. Worth reading 𝙏𝙤 𝙃𝙚𝙙𝙜𝙚 𝙤𝙧 𝙉𝙤𝙩 𝙩𝙤 𝙃𝙚𝙙𝙜𝙚: 𝙊𝙥𝙩𝙞𝙢𝙖𝙡 𝙎𝙩𝙧𝙖𝙩𝙚𝙜𝙞𝙚𝙨 𝙛𝙤𝙧 𝙎𝙩𝙤𝙘𝙝𝙖𝙨𝙩𝙞𝙘 𝙏𝙧𝙖𝙙𝙚 𝙁𝙡𝙤𝙬 𝙈𝙖𝙣𝙖𝙜𝙚𝙢𝙚𝙣𝙩 by Philippe Bergault, Olivier Guéant and Hamza BODOR if you're managing flow with no skew — CRBs, agency desks, etc. The paper frames the internalise vs. hedge decision under stochastic inventory accumulation, using Almgren-Chriss with both CARA utility and mean-variance criteria. When execution costs are quadratic, matrix Riccati equations yield near-closed-form solutions. With general cost functions, the PDEs break — boundary sensitivity dominates. That’s where RL is used — not for novelty, but to bypass boundary assumptions entirely. The RL policy replicates key structures (like the no-trade zone under linear costs), and gracefully handles mixed cost terms. It aligns well with PDE benchmarks and handles large inventory states more robustly. I'm sharing this because flow uncertainty is rarely modelled head-on in execution. This one actually does — and offers a clean lens on whether to solve the control problem or learn it. https://lnkd.in/dvXEvQX6

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Regime Switching on NVDA with a Hidden Markov Model (HMM) Simple indicators like moving average crossovers are observables — useful, but limited. The real quant edge often comes from probabilistic regime inference. Here’s NVDA (2018–2025, Stooq EOD) modeled with a 3-state Gaussian HMM using: 1-day return 20-day realized volatility 20–50 SMA spread / price The model learns latent regimes directly from the data: 🟢 Bull: positive mean return, moderate volatility 🟠 Chop: near-zero mean, mixed volatility 🔴 Bear: negative mean return, higher volatility Policy (illustration only): long when P(Bull) > 0.60 CAGR: 42.35% Sharpe: 1.40 MaxDD: –31.34% 🔹 Takeaway Unlike hard rules, the HMM provides probabilities for each regime. This allows more flexible risk sizing — scaling back in Bear states, pressing exposure in Bull, and clipping leverage in volatile Chop. 🔹 Conclusion This isn’t a trading system yet — it’s a demonstration of how quants move beyond raw signals into regime-aware frameworks. The next steps are transaction costs, volatility targeting, and cross-asset features to make it robust. #QuantTrading #MachineLearning #HiddenMarkovModels #FinancialEngineering #TradingResearch

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Building a Volatility Surface - Mathematical Explanation A volatility surface provides a three-dimensional representation of implied volatilities across different strikes and maturities Dupire Local Volatility Model In the Dupire model, we focus on the local volatility as a function of spot price and maturity , not strike . The local volatility is derived from the call price using the following PDE: (Image attached below) The formulation emphasizes that strike is not directly considered but rather the spot price . Convexity constraints are crucial to maintaining no-arbitrage conditions:(Image attached below) Sticky Strike vs. Sticky Delta Sticky Strike: The volatility surface is parameterized based on strike levels. The implied volatilities remain fixed for specific strikes even as the underlying spot price changes. Sticky Delta: The volatility surface is parameterized based on delta levels. This approach is more dynamic, adjusting volatilities based on the relative position of the strike to the spot price. Heston Stochastic Volatility Model The Heston model introduces stochastic variance, defined as: (Image attached below) The third equation representing the correlation between the two Wiener processes is essential and must be included: (Image attached below) This correlation term significantly influences the skew and curvature of the implied volatility surface. SABR Model The SABR model operates in the forward price space, not the spot price. The dynamics are given by: (Image attached below) Since the underlying modeled is the forward price , there is no drift term in the asset price equation. The implied volatility approximation in the SABR framework is: (Image attached below) Imposing Arbitrage-Free Constraints Calendar Spread Arbitrage: Implied variance must increase with maturity, ensuring (Image attached below) Butterfly Spread Arbitrage: The second derivative of the call price with respect to strike must be non-negative: Image attached below) 4. Interpolation and Smoothing Techniques Smoothing methods include cubic splines and Gaussian kernel smoothing. Extrapolation techniques for extreme strikes or maturities involve asymptotic models or log-linear interpolation. The last two days left to register for our Commodities Options Trading and Market-making Masterclass with Kristopher Abdelmessih Kris brings 20+ years in options and volatility trading, having cut his teeth on SIG’s trading floors in market-making and risk management before managing a relative-value commodities volatility book at Parallax. Registration Now - https://lnkd.in/gnvykx3h (Use Coupon code - Options20 to get 20% off) Sunday, June 1st, 2025 Time: 8:30 PM IST / 8:00 AM PDT/ 3:00 PM GMT)

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Taylor series in Quantitative Finance: Taylor’s series plays a crucial role in approximating complex, non-linear pricing functions—like those in options pricing and risk management—by expressing them as infinite sums of polynomial terms based on derivatives. This approximation enables more tractable analytical solutions, sensitivity analysis (Greeks), and efficient numerical modeling, forming a backbone for many quantitative models in finance. Quantitative Finance Cohort’25 is the hub for deep-dive Quantitative Finance and expertise in areas like mathematical finance, derivatives pricing, volatility models, quant trading, interest rate models, risk, portfolio optimisation and quant development!! Register for Quantitative Finance Cohort’25 now:- https://lnkd.in/g7BQXdfD

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Mid-Frequency Quant Trading: The Overlooked Powerhouse High-frequency trading tends to steal the limelight. It’s fast, complex, and cloaked in secrecy — the Formula 1 of the financial world. But while HFT dazzles, it’s not where most researchers or funds live. Instead, many are quietly generating consistent alpha through mid-frequency trading — a space that balances sophistication with accessibility. Here’s why mid-frequency often makes more sense than chasing nanoseconds: • Easier Access – No need for cutting-edge latency infrastructure like colocation or specialized data links • More Transparent Signals – Built on stable statistical relationships rather than market noise • Scales More Easily – Strategies can extend across instruments, markets, and geographies • Handles Larger Capital – Less prone to self-trading issues or adverse market impact • Fewer Real-Time Pressures – Regulatory risks and operational burdens are lighter • Transferable Frameworks – Techniques often generalize well across macro, crypto, and equities While HFT grabs headlines, much of the sustainable alpha lives in the mid-frequency lane. #quanttrading #midfrequency #alpha #hedgefunds #quants #systematicinvesting #macro #crypto #equities #tradingedge

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There’s technical analysis and there’s fundamental analysis. But guess the most important kind? 𝗠𝗶𝗻𝗱𝘀𝗲𝘁 𝗮𝗻𝗮𝗹𝘆𝘀𝗶𝘀. This is one big lesson I learnt — with a lot of pain — during the recent market volatility. During the March–April volatility, we went fully risk-off. Our models flagged elevated risk, and we pulled the trigger — exited positions, raised cash, and watched from the sidelines. We thought we were being prudent. But in hindsight, we were being reactive. We ended up selling some incredible businesses — stocks our models had strong conviction in. The markets bounced back swiftly, and we missed the recovery. Those two months of underperformance weren’t about bad models. They were about a momentary lapse in conviction. “𝘛𝘩𝘦 𝘣𝘦𝘴𝘵 𝘵𝘳𝘢𝘥𝘦𝘳𝘴 𝘢𝘳𝘦𝘯'𝘵 𝘵𝘳𝘺𝘪𝘯𝘨 𝘵𝘰 𝘣𝘦 𝘳𝘪𝘨𝘩𝘵. 𝘛𝘩𝘦𝘺'𝘳𝘦 𝘵𝘳𝘺𝘪𝘯𝘨 𝘵𝘰 𝘦𝘹𝘦𝘤𝘶𝘵𝘦 𝘤𝘰𝘯𝘴𝘪𝘴𝘵𝘦𝘯𝘵𝘭𝘺.” 🧠 𝗠𝗶𝗻𝗱𝘀𝗲𝘁-𝗯𝗮𝘀𝗲𝗱 𝘁𝗿𝗮𝗱𝗶𝗻𝗴 𝗶𝘀 𝗮𝗯𝗼𝘂𝘁: • Trusting the process when the P&L looks red. Our signals are built on statistical evidence — not emotion — and staying consistent is the real alpha. • Holding winners when your gut says, “Take profits.” Even the best quant models can’t predict every rebound — but they can guide you to stay with strength. • Understanding that risk management doesn’t mean abandoning quality. The process must include a conviction overlay that respects great businesses, not just downside metrics. 🔁 𝗪𝗵𝗮𝘁 𝘄𝗲 𝗰𝗵𝗮𝗻𝗴𝗲𝗱: We’ve now introduced filters that reinforce conviction — ensuring we don’t let volatility shake us out of high-quality holdings. Our new rules aren’t just statistical — they’re psychological safeguards. Sticking to our guns we were able to outperform the market by a margin in the last 3 months on our factor fund. ✅ 𝗧𝗵𝗲 𝗿𝗲𝗮𝗹 𝗲𝗱𝗴𝗲 𝗶𝘀𝗻’𝘁 𝗷𝘂𝘀𝘁 𝗶𝗻 𝘁𝗵𝗲 𝗺𝗼𝗱𝗲𝗹. It’s in sticking with it when it’s hardest to. 🤔 𝗬𝗼𝘂𝗿 𝘁𝘂𝗿𝗻: Have you ever made a decision based on fear that cost you more than holding would have? How do you manage conviction vs caution in your process? #tradingpsychology #quantinvesting #investingmindset #riskmanagement #learningfrommistakes

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Price causality is critical for trading predictability.   We often see a strong correlation between multiple trading instruments. Correlation gives us a measure of statistical dependence, however it does not indicate predictability.   We evaluate the causality impact of key predictors affecting the daily forward predictability of TTF Gas in terms of convergence, strength and influence.   When a daily positive (negative) move in the instrument is expected following a positive (negative) move in the predictor, we say the predictor and the instrument are in convergence.   When a daily positive (negative) move in the instrument is expected following a negative (positive) move in the predictor, we say that the predictor and the instrument are in divergence.   Whether in convergence or divergence, we assess the strength (predictive ability) of each predictor.   Additionally, we indicate whether key predictors are gaining or losing influence.   Selected predictors meet proprietary conditions for both strength and influence.   Our causality analysis currently indicates:   - a strong divergence between Gas and CO2. This means that Gas is more likely to trade up (down) the next day if CO2 has traded down (up) the previous day.   Our causality analysis complements our fundamental, cyclical and technical analyses.   Register here https://lnkd.in/dWDX7-CV, or contact Joaquin Narro directly, to receive a free, no obligations trial of Alcazar´s reports.   Disclaimer: https://lnkd.in/dgi2mg8i   #gas #gasprices #powertrading #energytrading #alcazar #emissions #naturalgas #climatechange #climaterisk #voluntarycarbonmarket #carbontrading #carbonemissions #carbonoffsets #vcm #netzero #carbonneutral #carbonmarkets #coal #power #russia #trading #electricitymarkets #gasmarket #weather #weatherforecast #oil #oiltrading #hydrogen #hydrogeneconomy #electricitymarkets #co2

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