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.
8,116 documents
The document raises a practical problem in measuring historical volatility when an instrument’s prices fluctuate around zero and may be negative. Log returns are undefined for nonpositive prices, while simple percentage changes can become extremely large…
The discussion distinguishes physical-measure forecasts from risk-neutral valuation for an exchange rate quoted as domestic currency per unit of foreign currency. Under the domestic money-market numeraire, the Garman–Kohlhagen model gives a risk-neutral…
The document asks why a European contingent claim with a convex payoff in the underlying should itself be convex in the underlying, which corresponds to nonnegative gamma when the value is twice differentiable. The attempted Jensen argument is confused: it…
The discussion asks whether warrants carry higher implied volatility than comparable exchange-traded options, whether the difference can be exploited, and whether warrant volatility can help analyze structured products. The answers explain that an apparent…
The note discusses whether studying Itô and Malliavin calculus is useful for someone pursuing algorithmic trading. Its central distinction is between quantitative researchers focused on forecasting and risk from market data, and researchers who build or…
The document considers a European call in a three-period recombining binomial tree and asks how to price it efficiently by hand. It presents the risk-neutral valuation principle: discount the expected terminal payoff, using the binomial probability of each…
The note explains why FX option market makers may describe positions in terms of long or short strikes rather than whether the option is a put or a call. With a delta hedge included, put-call parity makes a put position equivalent to a call position, so the…
The discussion presents four idealized ways an implied-volatility surface may respond when spot moves: sticky delta, sticky strike, sticky local volatility, and jumpy volatility. Each implies a different relationship between spot and implied volatility.…
This brief exchange asks which volatility to use when pricing and hedging a European option, given conflicting daily and annual volatility figures. It mentions that Black–Scholes conventionally uses annualized volatility and asks whether a different measure…
The document asks why historical volatility is commonly estimated over a rolling window, such as a short run of daily observations, and then annualized. It contrasts this practice with annualized historical return, which the author understands as being…
The document describes a numerical experiment comparing an explicit finite-difference solution for a European call option with the analytical Black–Scholes price. The author fixes the time step and plots pricing error against the price-grid spacing,…
The document asks how an option’s time decay relates to gamma, focusing on the gain from delta-hedged exposure to small underlying price moves. It describes the intuition that, when realized volatility matches the volatility implied in the option price, the…
The answer clarifies why option pricing models are often calibrated against a market implied-volatility surface that has a smile or skew, rather than against a flat surface. A correctly specified model's own volatility structure may be flat where its…
The answer addresses whether gamma is positive for American calls and puts under Black–Scholes, including settings with continuous or discrete dividends. Its reasoning is that American option values retain homogeneity of degree one in spot and strike because…
The document defines static arbitrage by whether positions need to be changed after the initial trade. A static arbitrage is established with a portfolio that requires no subsequent rebalancing; an example is trading different-sized futures contracts in…
The document derives a simplified expression for the risk-neutral up-move probability in a standard forward binomial tree. The tree uses up and down factors built from the risk-free rate, dividend yield, volatility, and period length. Starting from the usual…
The document discusses how often financial institutions recalibrate the SABR model when pricing FX options. Its practical guidance is that calibration is typically performed daily, with additional intraday recalibration sometimes needed on especially…
The document considers a derivative payoff that equals the underlying’s initial price above a threshold and the underlying’s maturity value at or below it. Because the payoff jumps at the threshold, the author asks how to approximate it using a constant, a…
The document considers whether a volatility exposure can be approximated by buying at-the-money S&P 500 straddles and delta-hedging them, potentially as a volatility position or a spread against VIX futures. It also describes the William’s VIX Fix, a…
The document explains how to identify candidate pairs for trading relative moves in equity-option implied volatility, even when the individual volatility series are already stationary. Cointegration is not essential: the relevant question is whether a spread…
The document outlines a method for formulating perpetual American option valuation as a free-boundary problem. In the continuation region, where holding the option is more valuable than exercising, the value satisfies the perpetual Black–Scholes differential…
The document addresses why a standard perpetual-option PDE solution for an American put does not transfer directly to a call. In the no-dividend setting, the continuation-region solution for the call has a linear form, and the question is how to determine…
Option pricing models can support trading in several ways. A market maker can use a model to estimate option sensitivities, hedge exposures, and seek returns from the bid–ask spread. A risk taker can compare model values with market prices and trade options…
The discussion distinguishes two quantities often called forward implied volatility. One is calculated from the change in implied variance between two maturities for vanilla options at a fixed strike. It represents a forward measure inferred from today’s…