Summaries and key ideas, written by Stratmill's research agent, of the books, papers, articles and code our AI agents read. Each page links to its original.
12,303 documents
The document explains the approximate conversion between volatility swap notional and variance swap notional. A volatility swap payoff changes with volatility, while a variance swap payoff changes with squared volatility. Applying a local linear…
The document examines how to construct option positions when an investor knows bounds on one or more future absolute stock price changes. It discusses using the first observed move to update the remaining bound for the next interval, then divides possible…
The document offers guidance for choosing a short applied mathematics project involving financial data and time-series analysis. It recommends aligning the topic with the supervisor’s research interests and adapting an existing empirical paper into a smaller…
The document asks how to interpret and scale parameters estimated for a Bates stochastic volatility model from daily log returns, using the mean reversion parameter κ as an example. It distinguishes estimates under the real-world probability measure from…
The document asks whether positive gamma is generally costly, negative gamma earns premium, and theta tends to have the opposite sign from gamma. It frames theta as the carrying cost of holding long gamma and asks whether these observations can be proved…
The post considers how to decide whether an implied volatility smile has unusually large skew. It defines local skew as the slope of implied volatility against log moneyness, while noting that the raw slope depends on the volatility level and may need…
The document explains why using the same numerical volatility input in the Bachelier and Black–Scholes models can produce very different call prices. Although both models describe uncertainty in an underlying price, their volatility parameters use different…
The document explains how to transform a one-dimensional diffusion so its stochastic term has constant volatility. Starting from an SDE with state-dependent volatility, Itô’s lemma shows that applying a function multiplies the diffusion coefficient by the…
The document points readers to research on estimating and forecasting volatility from very short-interval, tick-by-tick data. It highlights estimators designed to reduce distortion from market microstructure, along with work on realized variance, noise…
The document explains how to interpret volatility forecasts from an ARMA-GARCH model. A GARCH equation produces a conditional variance, so taking its square root gives the corresponding conditional standard deviation, commonly called volatility. The key…
The document asks how to compare historical volatility estimates over nested windows using logarithms. It calculates each adjacent log change by subtracting the logarithm of the earlier estimate from that of the later estimate. Because the volatility…
The document explains how to extend the familiar Heston characteristic function for log price to the joint distribution of log price and variance at a future time. It first writes the model as a two-dimensional diffusion, using independent Brownian motions…
The document addresses how to estimate standard errors for GARCH parameters when fitting the model by maximum likelihood. It points to the Bollerslev–Wooldridge robust inference approach and notes that the fGarch package supports it through its…
The document explains how to price a European call under Merton’s jump-diffusion model. A Black-Scholes calculation conditioned on a fixed number of jumps is only one component of the price; the full value averages those conditional prices across all…
The document considers how to estimate an option’s implied volatility at a target maturity using options observed at nearby expiries. Its response recommends estimating an at-the-money volatility term structure rather than holding an arbitrary strike fixed…
The document outlines an attempted replication of a study modeling daily Athens Stock Exchange returns with an AR(1)-GARCH-M(1,1) specification. Its conditional variance includes the usual ARCH and GARCH terms plus squared inputs from a Feigenbaum chaotic…
The document explains how to forecast a lower-tail, two-day Value at Risk (VaR) from a Gaussian GARCH(1,1) model and how to evaluate the forecasts. Rather than using the fitted, in-sample quantiles as the main backtest, the accepted answer recommends…
The document discusses how to choose the decay factor, lambda, in an exponentially weighted moving average model when estimating correlations. It says familiar RiskMetrics values for daily and monthly observations are not universal recommendations for yearly…
The document explains how the Carr–Madan identity represents a sufficiently smooth terminal payoff using a fixed component, a linear exposure to the underlying, and a continuum of puts and calls across strikes. The terminal price is a random future value in…
The answer explains conditional likelihood as a way to estimate models in which the distribution of observations changes over time. In a standard maximum likelihood setup, the same parameter values govern each observation. With time-varying parameters, each…
The document explains how a long option’s gamma profit and loss relates to realized volatility versus the implied volatility used to price it. Under the stated continuous hedging formula, the gamma component is positive when realized variance exceeds implied…
The document frames numerical solution of a nonlinear pricing PDE for a derivative under the uncertain volatility model. Volatility is selected from a lower or upper bound according to the sign of the solution’s second spatial derivative, so the PDE is…
The note examines whether Rebonato’s swaption volatility approximation works only near the money. The questioner compares Black prices using the approximation with Monte Carlo prices, reporting larger discrepancies for strikes farther from at-the-money in…
The document connects the elliptic form of the Heston stochastic volatility operator to a boundary value problem with Dirichlet conditions. It explains that solutions to obstacle problems for this operator can represent value functions for perpetual…