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 how to generalize first-passage results for geometric Brownian motion when the starting value and threshold are not normalized to convenient constants. Taking logarithms turns the process into Brownian motion with drift, so the relevant…
The document values a perpetual claim that pays one unit when a non-dividend-paying stock first falls to a lower barrier. It presents two equivalent routes: risk-neutral expectation of the discount factor at the hitting time, or an ordinary differential…
The document defines a forward volatility agreement as exposure to the at-the-money implied volatility observed at a future date for an option expiring later. It asks how such a contract is hedged in practice. The response mentions forward-start options or…
The document explains why a model-free measure built from option prices ideally requires a continuum of strikes, and how finite market listings change the problem. With observable prices across strikes and negligible bid-ask spreads, vanilla calls and puts…
The document asks how to interpret Rebonato’s approximation for swaption volatility when using a factor-reduced or multifactor Libor Market Model. It questions whether each factor should be represented by a separate volatility function tied to a tenor, with…
The document explains why a Heston model does not always produce lower implied volatility at longer maturities. Its variance process is mean reverting toward the long-run variance level, so the relationship between current variance and that target influences…
The document clarifies how to calculate a one-step-ahead density forecast from an estimated ARMA-GARCH model with Gaussian errors. At each forecast origin, estimate or update the model using information available so far, then obtain the forecast conditional…
The document explains how to model constant conditional skewness alongside an ARMA conditional mean. Its central point is that ARMA specifies the structure of the conditional mean, while the innovation distribution can be chosen to have skewness and…
The document addresses whether squared simple returns or squared log returns should be used when studying volatility and applying ARCH or GARCH models. It defines log returns as the difference between the logarithms of consecutive prices and says this is the…
The document explains why the European Black–Scholes relationship between vega and gamma generally does not carry over to American options. Early exercise creates a boundary that changes the option’s value and sensitivities. For a put with a positive…
The document explains why similar warrants from different issuers can imply distinct volatility surfaces. Issuers may add different markups to warrant prices, and those markups can depend on each issuer’s outstanding position in the product. As sales…
The document examines the relationship between required returns, variance, stock-index levels, and expected volatility measures. The response points to a decomposition of the market’s expected excess return that includes risk-neutral variance and additional…
The document asks how practitioners might extend a long-dated foreign exchange option hybrid model that captures volatility level and skew but may not reproduce the full smile. The questioner wants to retain a three-factor setup, avoid stochastic volatility,…
The document addresses whether individual-stock option volatility smiles become steeper after sharp declines. Its answer describes a common market pattern: implied volatility, especially for out-of-the-money puts, tends to rise after a stock price falls. A…
The document responds to an observation that volatile Bitcoin prices seem to form a Weierstrass-like pattern across candlestick time scales. It explains that rough, fractal-like paths are not unique to cryptocurrency: Brownian motion and related stochastic…
The document discusses choosing the ARCH and GARCH lag orders for a volatility model. One proposed approach estimates a range of candidate orders, checks whether coefficients are statistically significant, inspects residual autocorrelation, and uses…
The document distinguishes the Samuelson effect’s maturity-related volatility pattern from volatility measured directly on futures price levels. The effect concerns returns volatility: nearby commodity contracts are often more volatile than deferred…
The note discusses how options traders describe exposure to changes in implied volatility skew and volatility convexity. Skew refers to differences in implied volatility across strikes, such as between out-of-the-money puts and calls. Traders may…
The discussion asks whether traders can profit from the tendency of equity markets to fall quickly and recover more gradually. It explains that this asymmetry is reflected in option prices: downside puts often carry a premium relative to comparable calls. A…
The document frames option leverage as delta divided by option price and asks which European strike maximizes that ratio. It supplies Black–Scholes call and put prices and deltas, then presents a numerical exploration across strikes for selected fixed…
The document compares the classical LIBOR Market Model (LMM) with a stochastic-volatility extension called SABR-LMM. LMM models a family of forward rates jointly, allowing their correlation structure to inform the pricing of products that depend nonlinearly…
The document raises a theoretical question about a consistency property in the Vanna–Volga method. Starting from a price curve constructed by interpolation and extrapolation across strikes, the cited paper claims that selecting three different strikes on…
The document shows how to generate joint spot and variance paths from QuantLib's Heston process in Python. It constructs the process from initial variance, mean-reversion, long-run variance, volatility-of-variance, correlation, spot, and rate inputs, then…
The document asks whether information criteria or divergence measures can guide SABR calibration. The response describes a common rates-market workflow: preset beta, solve alpha to match the at-the-money volatility, then optimize rho and nu against smile…