Artur Jaworski

Options traders: Here’s how to build your own volatility estimator (with code). This post explains how to build options volatility estimates. ARCH models are statistical tools that model time-varying volatility using historical financial data. They help predict periods of high or low market risk essential for accurate equity options pricing. Get the code: https://lnkd.in/eDeRPJ_s

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Pricing barriers in the partial differential equation framework Pricing barriers within this framework involve handling exotic options with features such as knock-in or knock-out conditions, which are activated when the underlying asset's price reaches certain levels (barriers). Here's a detailed look at how this can be approached: 1. Setting Up the Problem Consider a European option with a barrier feature. The underlying asset's price S(t)S(t)S(t) follows a stochastic process, typically modelled by a geometric Brownian motion: dS(t)=μS(t)dt+σS(t)dW(t)dS(t) where: μ is the drift rate, σ is the volatility, W(t) is a Wiener process (standard Brownian motion). For an option with a payoff function f(S(T)) at maturity T, and a barrier level B, the goal is to determine the option price V(S,t) at any time t. 2. Deriving the PDE Using Itô's lemma and the no-arbitrage condition, we derive the Black-Scholes PDE for the option price V(S,t): 3. Incorporating Barrier Conditions For barrier options, boundary conditions are critical. Suppose we have an up-and-out call option, which becomes worthless if the asset price hits barrier B. The conditions are: Initial Condition: At maturity t=Tt = Tt=T, V(S,T)=max(S(T)−K,0) for S(T)<B V(S,T)=0 for S(T)≥B Barrier Condition: For all t V(S,t)=0 for S≥B Boundary Conditions: For S→0S and S→∞S As S→0,V(S,t)→0 As S→∞, V(S,t)→S−Ke^−r(T−t) 4. Numerical Methods for Solving the PDE Analytical solutions to the PDE with barriers are rare, especially for complex payoffs. Therefore, numerical methods are often employed: Finite Difference Methods (FDM): Explicit Method: Simple but conditionally stable. Implicit Method: Unconditionally stable but requires solving a system of linear equations. Crank-Nicolson Method: A blend of explicit and implicit methods, offering stability and accuracy. Monte Carlo Simulations: Simulate a large number of paths for S(t). Apply the payoff and barrier conditions to each path. Average the discounted payoffs to estimate the option price. Binomial and Trinomial Tree Methods: Construct a discrete lattice of possible price paths. Incorporate the barrier condition by setting option value to zero for nodes beyond the barrier. Kickstart your Quant Interview Prep Check out Quant Insider Stack - https://lnkd.in/gcfdUEfg A Bundle of Interview Byte (500+ Interview questions) and Quant Insider Project Handbook Interview Byte’ contains 500+ Interview questions (https://lnkd.in/gkqcrrKf) Quant Insider Project Handbook has 21 industry-oriented projectsprojects(https://lnkd.in/gWBEn78U) Check out our Machine Learning for Finance course- Designed by Industry Veterans Hariom Tatsat, CQF, FRM with years of working on Wall Street - https://lnkd.in/gtJDWcus Use the Coupon code - "EARLYBIRD20" for 20% off on the ML for Finance course Algorithmic Trading Certificate : A Practitioner’s Guide, taught by Ex-Citadel Director - https://lnkd.in/gkMfXVP2 Join our community - https://lnkd.in/g7TaxyPA

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Hull-White Model:- Register for Quantitative Finance Cohort’25 now:- https://lnkd.in/g7BQXdfD The Hull-White model is a short-rate interest rate model extending the Vasicek model, allowing time-dependent parameters. It describes how interest rates evolve stochastically over time. By incorporating mean reversion and volatility, it provides flexibility in fitting the current term structure of interest rates. The Hull-White model is widely used for pricing interest rate derivatives, bond options, and swaptions. It helps in risk management and yield curve modeling by simulating future interest rate movements. Its analytical tractability makes it ideal for Monte Carlo simulations and calibration to market data. Register for Quantitative Finance Cohort’25 now:- https://lnkd.in/g7BQXdfD 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!! #HullWhiteModel #InterestRates #Derivatives #FinancialModeling #QuantitativeFinance

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While XGBoost and LightGBM get most of the attention, CatBoost quietly solves some of the toughest tabular problems. In this new Medium article, I share lessons learned from using CatBoost in real-world risk modeling projects: • Explainability • Built-in feature statistics and selection • Text & embeddings support • CatBoost with MLflow in SageMaker 📘 Read it here → https://lnkd.in/dCnwZ3tj  #DataScience #MachineLearning #CatBoost #Python #ExplainableAI

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4th Econometric Model – S&P500 Daily Volatility (Period: 11/08/2020 – 11/08/2025): Model: GARCH(1,1) with Dummy Variable Why Use a Dummy Variable? Markets occasionally undergo abrupt regime shifts—such as policy resolutions, crisis peaks, or structural reforms—that standard GARCH cannot capture alone. A structural‐break dummy allows the model to adjust its baseline variance precisely at a known turning point, improving both interpretability and volatility forecasting. Model Highlights Mean Equation: Returns are unpredictable (AR(1) term insignificant), consistent with efficient markets. Volatility Clustering: ARCH = 0.135, GARCH = 0.831 (both p<0.0001) confirm that shocks and past volatility drive future risk. Structural Break Effect: Dummy = –0.000452 (p = 0.0014) marks 15 September 2022 as the point where baseline volatility declined by 4.52 bp in our GARCH(1,1) model—an estimate consistent with U.S. inflation turning down after its summer peak and the Federal Reserve’s clear policy guidance around that time. This interpretation is based on the model’s estimates and aligns with observed CPI data and FOMC communications. Practical Takeaways: Anticipate regime shifts: Incorporate event‐driven dummies to detect discrete changes in risk. Balance complexity and parsimony: Retain only significant breaks to avoid overfitting. Enhance forecasts and hedging: Use regime‐aware GARCH models to improve VaR estimates and dynamic hedging strategies around major market events. This GARCH(1,1)+dummy approach offers a concise yet powerful framework for understanding how key events permanently reshape market volatility regimes Note: In situations where negative shocks have a different impact than positive shocks (the ‘leverage effect’), an EGARCH(1,1) model may be more appropriate. EGARCH captures asymmetric responses directly and ensures positivity of conditional variance without non‐negativity constraints on parameters. Source of Data: Federal Reserve Bank of St. Louis

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Reducing Transaction Costs in Volatility-Managed Portfolios Volatility targeting requires frequent rebalancing. The paper investigates the volatility targeting technique under the constraint of transaction costs. Specifically, it introduces a rebalancing boundary, which triggers adjustments only when certain conditions are met. Findings -The study introduces a rebalancing boundary setup into the volatility target asset allocation mechanism to reduce transaction costs over an investment horizon. -The optimal rebalancing boundary level is determined to maximize a portfolio return measure while controlling a portfolio risk measure within a constrained optimization framework. -The Omega ratio is used as the portfolio return measure, and mean volatility deviation is used as the portfolio risk measure. -Performance is evaluated by comparing the optimized target volatility portfolio with the rebalancing boundary (“Portfolio E”) to a traditional target volatility portfolio without rebalancing boundaries (“Portfolio S”). -Comparative analysis is conducted under different market scenarios. -Portfolio E generates significantly lower transaction costs and higher returns than Portfolio S. -Portfolio E exhibits lower downside risk, evidenced by a higher Value-at-Risk (VaR) level than Portfolio S. -The addition of the rebalancing boundary level does not compromise portfolio risk control. Reference: Zefeng Bai, Dessislava Pachamanova, Victoria Steblovskaya, Kai Wallbaum, Target volatility strategies: optimal rebalancing boundary for transaction cost minimization, Financial Markets and Portfolio Management, 2025 Join a community of 6,000+ quants—subscribe to the newsletter! Link in profile #portfoliomanagement #volatility #quantitativefinance Abstract Transaction costs are a major factor affecting portfolio returns in asset management. We propose an enhanced target volatility strategy to reduce the deleterious effect of transaction costs by adding rebalancing boundaries to the target volatility asset allocation mechanism. We formulate a constrained optimization problem to determine the optimal rebalancing boundary level. Based on simulations using an overlapping block bootstrap approach, we find that the extended target volatility portfolio with rebalancing boundary levels can provide better investment outcomes (higher portfolio returns and reduced transaction costs) without losing the ability to control portfolio risk under a pre-determined threshold. Further computational analysis on different real market scenarios confirms these findings and allows us to summarize insights on the appropriate boundaries to use under different market conditions and transaction cost magnitudes. Our findings have important practical implications given the popularity of the target volatility investment strategy as well as other asset management concepts with a dynamic asset allocation mechanism.

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Risk - how does correlation impact probability densities? Lets say we have a random variable Z = X+Y, the additive sum of two random variables. Suppose X and Y are correlated, by correlation parameter rho. Then, the mean or expected value of Z, is just the sum of the individual averages. E(Z) = E(X) + E(Y). The variance of Z, however, is not just the sum but it has a covariance / correlation term, which can be worked out by the sum of squares of X+Y. Var(Z) = Var(X) + Var (Y) + 2*rho*StDev(X)*StDev(Y). This only specifies the mean and variance. Let's assume X and Y are standard normal. Then this specifies the probability density of Z (sum of two normal is still normal). How does Z vary with correlation? We see as rho approaches 1.0 (perfect positive correlation), the normal density of Z flattens out, with heavier tails (not exactly platykurtic). Some intuition: - When X and Y are positively correlated, whenever the quantile X is negative then Y tends to be negative too. Similarly when X is positive. X and Y move together, thus the sums are "amplified". - Conversely, then X and Y are negatively correlated, whenever X is negative, Y tends to be positive. The sum lands at some in between number. Thus, more density around the mean. This has implications in the risk context. Using normal distribution is probably a bad idea in reality. But for discussion sake, suppose X and Y are losses. Losses are amplified when assets become positively correlated. Much like how the Global Financial Crisis 2008 evolved. Nearly all assets became correlated. Correlation can significantly impact density functions. Quantifying correlation risk is an age old problem. However, even qualitatively, it can't be ignored.

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