Shubham Kotal, MFE

Stochastic Interest Rate Models in Quant Finance 📊📊 In fixed income modeling, interest rates aren’t constants — they’re random variables evolving through time. 🎳🎳 That’s where stochastic interest rate models come into play. Here are some of the most widely used models in the industry 📚📚 👉 Vasicek Model – Mean-reverting, simple, and analytically tractable. But allows for negative interest rates. 👉 Cox-Ingersoll-Ross (CIR) Model – Also mean-reverting but ensures rates stay positive. Popular for bond and credit models. 👉 Hull-White Model – A time-dependent extension of Vasicek. Highly flexible and widely used in practice. 👉 Black-Karasinski Model – Lognormal version of Hull-White. Ensures positive rates with log transformation. 👉 Ho-Lee Model – Arbitrage-free model used for bootstrapping term structures. Good for calibration but can allow negative rates. 👉 Libor Market Model (LMM / BGM) – Models the evolution of forward LIBOR rates. Key in pricing caps, swaptions, and other IR derivatives. 👉 Heath-Jarrow-Morton (HJM) Framework – Models the entire forward rate curve directly. Very general and foundational to LMM. 👉 Affine Term Structure Models – A family that includes Vasicek, CIR, and others. Useful for tractability and closed-form bond pricing. These models are core to: 👉 Bond pricing 👉 Option-adjusted spread analysis 👉 Interest rate derivatives (swaptions, caps/floors) 👉 ALM and duration-convexity risk 👉 Regulatory models for capital & stress testing Interest rates shape the entire yield curve — modeling them correctly is essential for robust financial engineering. #QuantFinance #FixedIncome #InterestRateModels #Vasicek #CIR #HullWhite #LMM #HJM #FinancialEngineering #ModelValidation #BondPricing #InterestRateDerivatives

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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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📊 Time Series Modeling – A Must-Have Skill in Risk and Forecasting! 🔍📉 Whether you’re working in quant finance, risk management, or regulatory modeling like CCAR and DFAST, mastering time series techniques is non-negotiable. Over the past few years, time series models have played a central role in stress testing, forecasting credit losses, and capital planning. 👉 So what are the key concepts you should be confident in? 👉 Stationarity and Differencing 👉 ACF & PACF Plots 👉 AR, MA, ARMA, ARIMA Models 👉 Seasonality & Trend Decomposition 👉 SARIMA for seasonal data 👉 Vector Autoregression (VAR) 👉 Cointegration & Error Correction Models 👉 Forecast Evaluation (RMSE, MAPE, AIC/BIC) 👉 Rolling Forecasts & Backtesting 👉 Volatility Models like GARCH 👉 Machine Learning for Time Series (e.g., XGBoost, LSTM) 💼 Applications in CCAR & DFAST: 👉 Forecasting Pre-Provision Net Revenue (PPNR) 👉 Modeling Credit Losses over time 👉 Macroeconomic variable projections (GDP, unemployment, HPI) 👉 Scenario-based stress testing over multi-quarter horizons Time series isn’t just a math-heavy subject — it’s a powerful tool used by banks to navigate regulatory requirements and protect against future downturns. Enroll in our Quant Finance Bootcamp 📊 📊 https://lnkd.in/gUD9iJwV Enroll in our Python Bootcamp for Risk & ML📊 📊 https://lnkd.in/gtbE6E9j Enroll in our Linear Algebra Bootcamp (live batch) 📊📊 https://lnkd.in/g2YdaanA #TimeSeries #CCAR #DFAST #QuantFinance #StressTesting #RiskModeling #FinancialForecasting #ARIMA #GARCH #Econometrics #QuantitativeFinance

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Expected credit loss with PD-LGD correlation adjustment The traditional Basel (IRB) formula for ECL (“Expected Credit Loss”), also used for IFRS 9 provisions, is the following: ECL = PD*EAD*LGD (Of course, considering present value of EAD and forward-looking calibrations). This formula is based on the assumptions of parameters’ independence - in particular, no correlation between PD and LGD. An alternative version of this formula is the second-order estimation (from a Taylor series formalism) of the expected value of the credit loss, considering constant exposures. This version accounts for the covariance between PD and LGD. Have you ever seen such formulation? References in the comments section.

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📢 Portfolio Optimization Using Advanced ML Techniques I’m pleased to share a research-backed portfolio optimization model that integrates core financial theory with modern machine learning to enhance investment decision-making. 🔍 Overview of the Approach:The framework combines two key methodologies: 1️⃣ Mean-Variance Optimization▪ Calculates Sharpe Ratios to identify the most efficient asset combinations within the same class▪ Helps construct portfolios that optimize risk-adjusted returns 2️⃣ K-Means Clustering▪ Detects price trend patterns by grouping assets with similar historical behavior▪ Recommends dynamic weight allocations based on current market trends for improved diversification ✅ This approach provides a data-driven method to optimize portfolios not just by statistical performance, but also by adaptive trend analysis. I had the opportunity to present this work to my corporate mentor, Bishwajit Singh , whose feedback on the depth and practical relevance of the project was greatly encouraging. 🔗 Full project and codebase available here:👉 https://lnkd.in/gjptyxz9 📌 Tools Used: Python • Excel • Scikit-learn • Pandas • Matplotlib • NumPy I look forward to feedback from professionals in the finance and analytics domain, and am open to discussions or collaborations on related topics. #PortfolioOptimization #SharpeRatio #MeanVariance #KMeans #MachineLearning #QuantitativeFinance #PythonForFinance #InvestmentStrategy #FinTech #DataScience #GitHub

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🌟Types of Long-Only Quant Approaches: Single/Multi-Factor & Quantamental🌟 Speakers: - Aditya Sharma, CFA (S&P Global Market Intelligence)   - Gaurav Jain, CFA, CFTe, EPAT (Fidelity International)   - Sandip Santdasani, CFA (ICICI Prudential AMC Ltd)  Moderator - Kshitij Anand CFA (ICFAI), MS Finance, ETMarkets The Economic Times At #IIQC2025, organized by LAQSA, leading experts shared their views on the evolving landscape of quant methodologies in India. The discussion covered the nuances between single and multi-factor models, as well as the rise of quantamental thinking. Insights focused on how systematic techniques are shaping portfolio construction and driving innovation in the Indian ecosystem. Rishi Kohli Pankaj Mani(Jha) Nuvama Asset Services BSEIndia Nubra Nishith Desai Associates S&P Global Market Intelligence ICICI Prudential AMC Ltd AlphaGrep Investment Management Khaitan & Co MethodTech #IIQC2025 #LAQSA2025 #QuantModels #Quantamental #SystematicThinking #SingleFactor #MultiFactor #PortfolioTrends #IndexApproach #StrategyDebate

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Here’s how machine learning can give you an edge. It starts with just 1 dataset and a laptop. This free PDF download describes how: The study uses weekly data for the historical constituents of the S&P 500 between January 1999 and March 2021. The paper shows out-performance of machine learning-based stock selection models compared to an equally weighted benchmark. Here's the link: https://lnkd.in/gNZYTEv ~~~ Looking to start using Python for systematic trading? Here's a free Crash Course with everything you need to get started. Join the 1,000s of people who finally started with Python after reading it: https://lnkd.in/eaayJEfj

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