LLM-chats

Links to my LLM chats

Equations of the fully parameterized bivariate GARCH(1,1) model? (ChatGPT o3-mini-high)


Tail behavior of the lognormal and log-Laplace distributions (ChatGPT o1) 2025-03-26


Kurtosis of the symmetric hyperbolic and Champernowne distributions (ChatGPT o3-mini-high) 2025-03-26 codes here and here


Setting compiler-specific options with the Fortran Package Manager (untested) (Grok) 2025-03-26


Random correlation matrices with specified average off-diagonal correlation (ChatGPT o3-mini-high) 2025-04-14


GARCH models with open-high-low-close data (ChatGPT o3) 2025-04-18


Chernoff (MGF) tail bounds and the Vysochanskiĭ–Petunin bounds for probability distributions (ChatGPT 04-mini) 2025-04-22, code here. John D. Cook blogged about the VP bounds.


Reseach on intraday futures trading strategies (ChatGPT 4.5) 2025-05-01


Inflation-aware asset allocation (OpenAI Deep Research) 2025-05-25


Applications of the leadz and trailz Fortran intrinsic functions (ChatGPT o3) 2025-05-27


Applications of the leadz and trailz Fortran intrinsic functions (Grok) 2025-05-27


Investing in global bond markets using carry (ChatGPT o3) 2025-06-23


Tensor Decompositions for Multi-dimensional Forecasting (Grok) 2025-06-06


How to model stationary integer time series that can be positive, zero, or negative (ChatGPT o3) 2025-06-06


How do you test for changes in the autocorrelations of a univariate time series? (ChatGPT o3) 2025-06-06


Stochastic models of stock returns (ChatGPT o3) 2025-06-12


Stochastic volatility models with 1/2 (Heston), 1 (lognormal), and 3/2 variance exponents (ChatGPT o3 deep research) 2025-07-29


Heston vs. Hull-White (lognormal) stochastic volatility models (ChatGPT 5) 2025-09-25


How does the earnings yield of the S&P 500 depend on 10-year Treasury bond yields and inflation (ChatGPT o3 deep research) 2025-06-21


Machine learning approaches to predicting both the conditional means and covariances of multivariate continuous data such as asset returns (ChatGPT o3 deep research) 2025-07-01


Machine learning algorithms for trading (ChatGPT 5 2025-10-27)


Empirical performance of volatility estimators using OHLC data (ChatGPT 5.1 deep research) 2025-11-24


Forward implied vol in the presence of a smile (ChatGPT o3) 2025-07-17


Forward implied vol in the presence of a smile (ChatGPT o3 deep research) 2025-07-17


Stock price level and volatility (ChatGPT o3 deep research) 2025-08-01


Option pricing with modified log Student's t distributions (ChatGPT o3 deep research) 2025-08-04


Tempered Student's t distribution (ChatGPT 5 2025-10-09)


R packages for continuous non-normal time series, including mixture models (ChatGPT o3 deep research) 2025-08-05


Mixture autoregressive models (ChatGPT 5 deep research) 2025-08-18


Non-normal probability distributions to fit empirical stock market returns and option-implied distributions, with links to R and Python packages (ChatGPT o3 deep research) 2025-08-05


Software to backtest option strategies (ChatGPT o3 deep research) 2025-08-06


How the implied volatility surface of stock index options responds to changes in spot, covering the sticky strike, sticky delta, and other models (ChatGPT o3 deep research) 2025-08-06


Alpha and tail risk hedging strategies in VIX futures and VIX options (ChatGPT o3 deep research) 2025-08-06


Carry strategies in U.S. commodity futures (ChatGPT o3 deep research) 2025-08-06


News impact curves of asymmetric GARCH models of S&P 500 returns (ChatGPT o3 deep research) 2025-08-06


GARCH models with announcement days (ChatGPT o3 deep research light) 2025-08-06


GARCH models with long memory (ChatGPT o3 deep research light) 2025-08-06


GARCH models with non-normal conditional distributions, including finite mixtures.
Markov-switching aka regime-switching GARCH models.
GARCH models with lags higher than (1,1).
GARCH-in-mean (GARCH-M) models.
GARCH models using the daily range.
How long a sample should be used to fit GARCH models and testing for parameter stability.
Panel data GARCH estimation.
GARCH models of seasonal volatility and the Samuelson effect for commodity futures returns.
GARCH models for multi-period volatility forecasts.
What is the best power of volatility to model? (APARCH)
(ChatGPT o3 deep research light) 2025-08-06


GARCH models with finite mixture innovations for stock indices (ChatGPT 2025-12-11)


Multi-period return distributions of GARCH processes (ChatGPT 5 2025-10-28)


Best GARCH model to fit a lognormal autoregressive asymmetric stochastic volatility process, as in stock index returns (ChatGPT 5 deep research 2025-08-20)


Smooth transition GARCH models (ChatGPT 5 deep research 2025-10-14 and 2025-12-17)


Stochastic volatility estimation using a Laplace filter (ChatGPT 5 2025-10-07)


Tail behavior of the Normal Inverse Gaussian (NIG) and Variance Gamma (VG) distributions, and their simulation and fitting (ChatGPT 5 2025-08-12), later (ChagGPT 2025-10-14)


Probability distribution with polynomial left tail and exponential right tail -- CGMY aka tempered stable (ChatGPT 5 202508)


Classical GARCH vs. Dynamic Conditional Score (DCS) or Generalized Autoregressive Score (GAS) models (ChatGPT 5 deep research 2025-08-16). The equations of the two types of models are given here


Combining alpha signals with different horizons (ChatGPT deep research 2025-08-19)


Function that is quadratic in the center and linear in the tails (ChatGPT 5 2025-09-08)


Probability distributions: hyperbolic secant, logistic, Laplace, and others, with log versions (ChatGPT 5 2025-09-14)


Option pricing with log-sech, log-logistic, and log-Champernowne distributions (ChatGPT 5 2025-09-15 and ChatGPT 5.2 2026-01-07))


Dagum and generalized logistic distributions (ChatGPT 5 2025-10-01)


Option pricing with the log generalized hyperbolic distribution (ChatGPT 5 2025-09-17)


Generalized Student t distribution of McDonald and Newey (ChatGPT 5.1 2025-12-01)


Indexed annuities (ChatGPT 5 2025-10-01)


Volatility risk premium as a directional signal in stock indices and individual stocks (ChatGPT 5 2025-10-01)


Black-Scholes formula with a general distribution (ChatGPT 5 2025-10-02)


Which financial time series can be modeled as Fractional Brownian Motion (ChatGPT 5 2025-10-03)


Smoothly tapering the left tail of the normal and other distributions (ChatGPT 5 2025-10-03)


Continuous positive distributions with an unbounded right tail (ChatGPT 5 2025-10-03)


Asset growth anomaly (ChatGPT 5 2025-10-06)


Exponentially weighted moving average (EWMA/RiskMetrics) volatility forecasting (ChatGPT 5 2025-10-06)


Johnson SU distribution (ChatGPT 5 2025-10-06)


Distributions that are transformations of a single normal variate (ChatGPT 5 2025-10-06)


Transformation and symmetrization to get rid of skew (ChatGPT 5 2025-10-08)


Measures of statistical skewness (ChatGPT 5 2025-10-10)


Are multivariate return distributions elliptical? (ChatGPT 5 2025-10-08)


Option positioning and future returns and volatility (ChatGPT 5 2025-10-08)


Non-linear time series models building on Doyne Farmer's local linear models (ChatGPT 5 2025-10-14)

also ChatGPT Deep Research 2025-10-14


International small-cap and all-cap value stock funds (ChatGPT 5 2025-10-14)


Multivariate normal mixture distributions for stock returns, and extensions to model changing covariance (ChatGPT 5 2025-10-15)


Filtering stocks to use in momentum and mean-reversion strategies (ChatGPT 5 deep research 2025-10-18); also


Option pricing using intraday realized volatility (ChatGPT 5 deep research 2025-10-18)


Multi-step ahead predictive distribution of asset prices using GARCH and other methods (ChatGPT 5 deep research 2025-10-19)


Theoretical multi-step return distributions for GARCH and log stochastic volatility models (ChatGPT 2025-10-19)


FIGARCH (ChatGPT 5 deep research 2025-10-20)


Random walk stochastic volatility models (ChatGPT 5 2025-10-20)


Multivariate GARCH (ChatGPT 5 2025-10-20)


Riskmetrics forecasts of volatilities and correlation with mean reversion, and asymmetric multivariate GARCH models (ChatGPT 5 2025-10-20)


Arithmetic Brownian motion to model stocks and stock indices (ChatGPT 5 deep research 2025-10-22)


Distribution of sample correlations, covariances, and covariances of normally distributed data (ChatGPT 5 2025-10-22)


Cointegration of gold and silver prices and futures trading strategies to exploit cointegration (ChatGPT 2025-10-22)


Cointegration of commodity futures prices (ChatGPT 2025-10-22)


Cointegration of implied volatility and realized volatility (ChatGPT 2025-10-27)


Optimal portfolio rebalancing with proportional transaction costs (ChatGPT 5 2025-10-27)


Hawkes process for asset returns (ChatGPT 5 2025-10-28)


Cryptocurrency strategies (ChatGPT 2025-10-28)


Heston-Nandi GARCH model (ChatGPT 5 2025-10-29)


GARCH option pricing (ChatGPT 5 2025-10-29), also here


GARCH option pricing for multiple expirations (ChatGPT 5.1 2025-11-21)


Edgeworth expansion probability density (ChatGPT 5 2025-10-29)


Distribution of index log returns when underlying stock log returns are normal (ChatGPT 5 2025-10-29)


Distribution of portfolio returns when assets returns follow a multivariate Student's t distribution (ChatGPT 5 2025-10-30)


Extracting data from a Bank of America (BoA) stock report (ChatGPT 5 2025-10-31)


Yield curve and U.S. stock market returns (ChatGPT 5 2025-11-02)


When OPEC announces oil production changes (ChatGPT 5 2025-11-02)


Measures of call skew and put skew and how skew varies with option tenor (ChatGPT 5 2025-11-03) Later (ChatGPT 5.1 2025-12-11)


Python packages to get fundamental stock data from Yahoo Finance (ChatGPT 5 2025-11-04)


Pricing American options (ChatGPT 5 2025-11-06)


Option hedging with daily return autocorrelations (ChatGPT 5 2025-11-06)


Optimal option portfolio (ChatGPT 5 2025-11-06)


Stock price falls on ex-dividend dates (ChatGPT 5 2025-11-06)


Volatility forecast metrics (ChatGPT 5 2025-11-09)


Vector Autogressive (VAR) and other models for high-dimensional multivariate time series (ChatGPT 5 2025-11-10)


Forecasting temporal aggregates of time series (ChatGPT 5 2025-11-10)


Stochastic volatility models with non-normal innovations (ChatGPT 5 2025-11-10)


Implications of asymmetric marginal distributions of a multivariate distribution (ChatGPT 5 2025-11-10)


Exponential smoothing combinations (ChatGPT 2025-11-11)


Volatility spillovers (ChatGPT 2025-11-11)


Adjusting an element of a correlation matrix (ChatGPT 2025-11-12)


Deferred vs. immediate income annuity (ChatGPT 5.1 2025-11-12)


Binomial and trinomial trees for pricing options (ChatGGPT 5.1 2025-11-16) and here


GARCH option pricing using trees (ChatGPT 2025-11-16)


Portfolio optimization with non-normal returns and general utility functions (ChatGPT 5.1 2025-11-18)


Software for pricing options using the implied tree local volatility method of Derman and Kani (ChatGPT 5.1 2025-11-20)
also here


Loss functions for volatility forecasts (ChatGPT 5.1 2025-11-22)


Jurik Moving Average (JMA) and other smoothing methods for trend-following (ChatGPT 5.1 2025-11-24)


Volatility cone with GARCH (ChatGPT 5.1 2025-11-28)


Portfolio optimization under multivariate Student t, finite mixtures of normals or Student t returns, skewed Student t returns, and using mean-variance-skew-kurtosis (ChatGPT 5.1 2025-11-29)


Bitcoin options and volatility (ChatGPT 5.1 2025-11-28)


Bivariate Student t distribution (ChatGPT 5.1 2025-11-29)


Multivariate skewed student t distributions (ChatGPT 5.1 2025-11-30)


Mean-variance-skew-kurtosis (MVSK) portfolio optimization under the multivariate skewed student t distribution (ChatGPT 2025-12-04)


Testing the assumption of equal kurtosis of the multivariate Student t distribution (ChatGPT 5.1 2025-11-30)


Downward bias of historical volatility estimated from square root of variance (ChatGPT 5.1 2025-12-02)


Lp norm (absolute, squared, and other powers) and robust estimation of standard deviation for normal, Student's t, and other distributions (ChatGPT 5.1 2025-12-02)


Robust estimation of covariances (ChatGPT 5.1 2025-12-02)


Size-weighted mid-market price (market microstructure) (ChatGPT 5.1 2025-12-04)


Calculation of realized volatility from intraday and overnight returns (ChatGPT 5.1 2025-12-04)


Option calendar spread strategies (ChatGPT 2025-12-05)


Student t copula vs. multivariate Student t to model return distribution (ChatGPT 2025-12-05)


Correlation of squared deviations for bivariate Student t distribution (ChatGPT 5.1 2025-12-07)


Correlations of squared deviations for the bivariate Laplace, generalized error, and other elliptical distributions (ChatGPT 5.1 2025-12-07)


Multivariate hyperbolic distribution for asset returns (ChatGPT 5.1 2025-12-07)


Multivariate Laplace and generalized error distributions for asset returns (ChatGPT 5.1 2025-12-07)


Regularized generalized error distribution (GED) (ChatGPT 5.1 2025-12-07)


Predicting monthly volatility, covariance, and correlation from monthly and higher-frequency returns (ChatGPT 5.1 2025-12-08)


Price/earnings ratio vs. history as an indicator (ChatGPT 5.1 2025-12-09)


Variable income annuities and dividend-paying annuities (ChatGPT 5.1 2025-12-09)


Return on Equity (ROE) factor (ChatGPT 5.1 2025-12-09)


NAGARCH and GJR-GARCH asymmetric GARCH models (ChatGPT 5.1 2025-12-10)


Hitting time and the inverse Gaussian distribution (ChatGPT 5.1 2025-12-10)


Azzalini and Fernandez-Steel skewed distributions (ChatGPT 5.1 2025-12-11)


Moments of the hyperbolic secant distribution (ChatGPT 5.1 2025-12-11)


Long-horizon regressions (ChatGPT 2025-12-12)


Fitting univariate normal mixtures from moments (ChatGPT 2025-12-12) Also here


Simulate returns using machine learning (ChatGPT 5.2 deep research 2025-12-14)


Fitting finite non-normal mixtures in Python (ChatGPT 5.2 2025-12-14)


GARCH models with generalized hyperbolic innovations (ChatGPT 5.2 2025-12-14)


C++ codes for Monte Carlo option pricing (ChatGPT 5.2 2025-12-14)


inverse portfolio optimization / reverse optimization (ChatGPT 5.2 2025-12-15)


summary of paper "Growth of Income Funds and Death of Volatility" (ChatGPT 5.2 2025-12-15)


GARCH models with time-varying conditional skewness and kurtosis, and ARCD models (ChatGPT 5.2 2025-12-15)


Joint distribution of minimum, maximum, and terminal values of a Brownian motion (ChatGPT 5.2 2025-12-15)


Autocorrelations (ACF) of the differences of a Fractional Brownian Motion (FBM) process (ChatGPT 5.2 2025-12-16)


Financial econometrics books using R and Python (ChatGPT 5.2 2025-12-16)


Rough volatility (ChatGPT 5.2 2025-12-18)


Find papers that use the rough fractional stochastic volatility model to forecast realized volatility. (Google Scholar 2025-12-19)


Multivariate logistic distributions and copula (ChatGPT 5.2 2025-12-19)


Transforming data to achieve normality (ChatGPT 5.2 2025-12-19)


Pricing options under the Hull-White stochastic volatility model (ChatGPT 5.2 2025-12-21)


Fisher information for normal and Student's t distribution (ChatGPT 5.2 2025-12-21)


Data snooping corrections for backtesting multiple systems (ChatGPT 5.2 2025-12-21)


Representing densities as scale mixtures of normals (ChatGPT 5.2 2025-12-21)


Normal-lognormal (lognormal variance mixture) distribution (ChatGPT 5.2 2025-12-22)


Software for intraday stochastic volatility models (ChatGPT 5.2 2025-12-22)


Discussion of the paper "Machine Learning Meets Markowitz" and multi-period portfolio optimization (ChatGPT 5.2 2025-12-23)


Simulate estimators of mean for skewed distributions (ChatGPT 5.2 2025-12-24)


Optimal trimming of trimmed mean for Student t distribution for various degrees of freedom (ChatGPT 5.2 2025-12-23)


Brownian motion, first hitting time, and the inverse Gaussian distribution (ChatGPT 5.2 2025-12-25)


Dividend yield as a predictor of stock volatility (ChatGPT 5.2 2025-12-27)


Machine learning and volatility forecasting (ChatGPT 5.2 deep research 2025-12-29)


Tyler's M-estimator and Orthogonalized Gnanadesikan-Kettenring (OGK) for covariance matrices, its use in portfolio optimization, and how it differs from Student's t covariance estimation


Option structures to maximize utility for risk-averse investors (ChatGPT 5.2 2025-01-02)


Comparative performance of asymmetric GARCH models such as GJR GARCH and NAGARCH in predicting stock index volatility (ChatGPT 5.2 2025-01-02)


Effects of annual commodity futures index rebalancing (ChatGPT 5.2 2025-01-02)


ARCH models with parameter constraints (ChatGPT 5.2 2025-01-03)


Training neural networks with a prior on R^2 (Bayesian neural networks) (ChatGPT 5.2 2026-01-05)


Autocorrelations of squared and absolute returns of GARCH, NAGARCH, GJR-GARCH, EGARCH, FIGARCH, and log ARSV processes (ChatGPT 5.2 2026-01-06)


Regularized t distribution with all moments finite (ChatGPT 5.2 2026-01-06)


Monotonic transformations from (-inf, inf) to (0, inf) (ChatGPT 5.2 2026-01-08)


Calibration of SABR model to SPX index options, and differences between SABR and expOU SV models (ChatGPT 5.2 2026-01-12)


Packages for the correlation matrix completion and nearest correlation matrix problems (ChatGPT 5.2 2026-01-15)


Machine learning models similar to decision trees but which are continuous everywhere (ChatGPT 5.2 2026-01-15)


Functional time series models and packages (ChatGPT 5.2 2026-1-16)


Simple models and software for the profit of an option portfolio as a function of spot move (ChatGPT 5.2 2026-01-16)


Python and R packages that implement GARCH-M (ChatGPT 5.2 2026-01-16)


Vectorized vs. event-based trading systems in Python (ChatGPT 5.2 2026-01-18)


Stochastic processes with power law terminal distribution such as Student t (ChatGPT 5.2 2026-01-20)


Terminal distribution of a constant elasticity of variance (CEV) process (ChatGPT 5.2 2026-01-20)


Tail behavior modeling in R (ChatGPT 5.2 2026-01-21)


Weather forecasts and natural gas futures prices (ChatGPT 5.2 2026-01-24)


Fitting the degrees-of-freedom parameter in Student t mixtures (ChatGPT 5.2 2026-01-24)


Univariate nonparametric density estimation methods (ChatGPT 5.2 2026-01-26)


Computing European Black-Scholes implied vols from market data (ChatGPT 4.2 2026-01-28)


Estimation of Merton jump-diffusion model from empirical returns (ChatGPT 5.2 2026-01-28)


Universal Density Approximation though finite mixtures (ChatGPT 5.2 2026-01-29)


Fitting successively larger finite mixture models (ChatGPT 5.2 2026-01-29)


Smoothing Bachlier implied vol curves (ChatGPT 5.2 2026-10-31)


Transfer function time series models vs. VAR (vector autoregression) and VARMA (ChatGPT 5.2 2026-02-02)


Fractional Difference Estimation (ARFIMA) for time series (ChatGPT 5.2 2026-02-02)


Continuous Threshold Autoregressive (TAR) time series models (ChatGPT 5.2 2026-02-05)


Root cancellation in ARMA time series models (ChatGPT 5.2 2026-02-06)


Stationarity and unit-root tests for time series (ChatGPT 5.2 2026-02-08)


Fitting cointegration models to bivariate time series (ChatGPT 5.2 2026-02-08)


Time series models of serially correlated regression errors (ChatGPT 5.2 2026-02-08)


Robust regression methods (ChatGPT 5.2 2026-02-08)


Fortran interfaces to C functions for a transpiler from C to Fortran (ChatGPT 5.2 2026-02-19)


Solving the retirement spending problem (decumulation) using dynamic programming (Bellman equation) (ChatGPT 5.2 2026-02-28), updated 2026-03-02


Simple and advanced NumPy functions with examples (ChatGPT 5.2 2026-02-28)