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 compares local-volatility and stochastic-volatility valuations for a double no-touch option that pays a fixed amount if spot remains between two barriers until maturity. It assumes both models are calibrated to the same implied volatility…
The document explains why a GARCH fit that reports failure to converge should be treated cautiously. Numerical estimation methods work iteratively from initial parameter guesses and can return the latest coefficients even when they have not met their…
The document distinguishes a historical quantile estimate from a parametric Value-at-Risk forecast based on Student’s t returns. It describes fitting a t-distribution to log returns, then estimating a one-day-ahead VaR by taking a lower-tail quantile from…
The document explains how to relate an option on a delivery-date futures contract to a Black-76 valuation using a different futures contract associated with the option's expiry. Under deterministic rates and basis, it expresses the delivery-date futures…
The document explains how to interpret the sample count in the two-scale realized variance estimator (TSRV) and why the estimate can be negative. It treats n as the number of intraday observations over the interval being studied, rather than as a fixed count…
The document discusses extreme forward-rate paths in a lognormal Libor Market Model simulated under the spot measure. It notes that reducing the time step or switching from ordinary Euler to log-Euler discretization may fail to eliminate the problem. One…
The document asks how to interpret a condition that arises in Gatheral’s relationship between local variance and the implied total variance surface. In that formula, the local variance is proportional to the maturity derivative of implied total variance…
The discussion concerns how to infer implied volatility from options on VSTOXX. It describes an attempt using Black–Scholes with the corresponding VSTOXX futures as the underlying and an assumption about rates and yield, which produced an unsatisfactory gap…
A portfolio’s dollar volatility is unchanged when every position is reversed: variance is sign-insensitive, so a short portfolio with the same absolute value as a long portfolio has the same standard deviation. The discussion distinguishes this total risk…
The discussion explains why a commodity spread option may use variability measured in absolute price changes rather than return volatility. For a spread defined as the difference between two asset prices, the value can cross zero. That makes a geometric…
This note discusses selecting the tuning parameters for the Two-Scale Realized Volatility estimator. It describes setting the lag parameter to one in the cited extension for dependent noise, and choosing the subsampling parameter as a constant times the…
The document presents an implementation question about numerically solving a Dupire partial differential equation to price vanilla options in log-moneyness and time coordinates. The author uses a fully implicit finite-difference scheme and reports difficulty…
The document explains how a cash-or-nothing digital call relates to a vanilla call. Its price can be represented as the negative derivative of the call price with respect to strike, and a narrow call spread gives a practical approximation to the digital…
The document asks how to estimate annualized historical volatility when price observations arrive at unequal time intervals. Its baseline method for evenly spaced data calculates returns over a fixed window, takes their sample standard deviation, and scales…
The answer challenges the idea that implied volatility (IV) is simply an unbiased forecast of future realized volatility. An option’s IV varies by strike and maturity, forming a smile or skew, while realized volatility is a single measure for a period. IV…
The document describes a problem in out-of-sample modeling of GARCH-filtered residuals using a semiparametric distribution with a generalized Pareto tail. When the GPD shape parameter is negative, the tail has a finite endpoint. The inverse CDF can therefore…
The document explains why fitting an autoregressive model can produce fewer usable residuals than observations, and how the likelihood method affects that count. Conditional maximum likelihood starts the recursion after the model's lag order, leaving the…
This discussion separates the theoretical SABR model from formulas used to approximate its implied volatility. It examines a case with very high vol-of-vol where prices calculated by inserting Hagan implied volatilities into Black–Scholes appear to violate…
The document asks how to reduce Monte Carlo estimation error when pricing vanilla options under the Heston stochastic volatility model. The questioner reports that Black–Scholes controls using implied, average, or long-term volatility have not worked well,…
The document introduces the Black–Scholes derivative of an option’s value with respect to its strike, often called dual delta. It gives the call and put formula and asks how to interpret the measure in practice when an option’s strike is fixed during the…
The document examines how to simulate a local volatility model calibrated from European options using Dupire’s formula. It presents the Euler–Maruyama step, which evaluates volatility at the current time and state and holds it over the next interval. The…
The document asks how exchanges obtain implied volatilities and Greeks for American-style stock options, where early exercise makes direct use of standard European option formulas unsuitable. It suggests that a computationally manageable tree or…
The document poses a stochastic modeling question: how to represent a geometric Brownian motion and an Ornstein–Uhlenbeck process whose innovations have a specified correlation. The first process models a quantity with proportional drift and diffusion, while…
The document explains why a trial-and-error Black–Scholes calculation can return a range of implied volatility estimates that appear to reproduce the same option price. The cause is limited numerical precision: the search stops when the difference between…