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 local volatility and jump diffusion models can each produce skew and kurtosis in Black–Scholes implied volatilities. It clarifies that “jump volatility” can refer to randomly varying jump sizes, which are parameters within a general…
The document asks how to fit a cubic spline to observed call option prices for a single maturity while enforcing conditions intended to prevent arbitrage. It outlines a least-squares objective with a smoothing penalty, piecewise cubic segments, continuity…
The discussion compares ways to estimate a Sharpe ratio from historical returns, focusing on simple versus log returns, arithmetic versus geometric treatment, and annualization. One answer recommends calculating volatility from log returns, scaling it across…
The discussion distinguishes volatility measured over a return horizon from volatility annualized from that horizon. Under independent returns with additive variances, a 30-day return volatility can be converted to a 2-day return volatility by multiplying by…
The document asks how to derive an approximation that maps at-the-money implied volatility to volatility at another strike under a lognormal distribution. The relationship is framed as a strike-dependent scaling rule, using the ratio of the at-the-money…
The document asks why two stated formulas for option vanna appear to differ by a factor of one hundred. It gives expressions involving spot price, volatility, time to maturity, the Black-Scholes variables, and vega, then reports that substituting a…
The document explains how counterparties manage exposure on OTC variance swaps when initial or maintenance margin is not posted. It says that many OTC swaps are governed by a credit support annex, but arrangements vary; without margin, counterparties rely…
The note explains why the effect of implied volatility on a vertical credit spread depends on the net Greeks of its two option legs. A credit spread sells one option and buys another with the same expiration but a different strike, receiving a net premium at…
The document compares two ways to forecast daily variance with a GARCH(1,1) model when five-minute returns are available. In the first approach, the model is fit or applied at the daily horizon, using a realized-variance estimate and a prior daily variance…
The discussion examines why replacing a vanilla option at one strike with a strip of options across nearby strikes can reduce the sharp gamma concentration near expiry. It offers two related explanations: a strip distributes exposure across strikes, and as…
The document considers how to define carry for a USD payer swaption with a six-month expiry and a ten-year underlying swap. It questions a simple calculation based on subtracting a short-term LIBOR fixing from a ten-year yield, and whether that spread alone…
The note explains the CBOE Put Protection Index as a tail-protection strategy: it holds exposure to the S&P 500 and buys a monthly put struck out of the money. The put premium reduces returns in ordinary markets, while the option can cushion losses when the…
The document challenges the idea that fitting a convex curve in strike-implied-volatility space guarantees an arbitrage-free option surface. Implied volatility need not itself be convex for market prices to be free of arbitrage, and convexity of implied…
The document presents two ways to price a forward-start call whose payoff depends on the asset price at the start date as well as at expiry. One approach conditions on the information at the forward start date and rewrites the value using the price of an…
The document distinguishes estimating volatility with a GARCH model from forecasting future volatility. It also considers realized volatility calculated from high-frequency price observations as another way to quantify volatility. Its central point is that…
The document outlines checks for evaluating an ARMA-GARCH model fit. Before using GARCH, test whether the data show ARCH effects; without conditional heteroskedasticity, a GARCH model may not be needed. After fitting, inspect standardized residuals for…
The document examines the Black–Scholes prices of digital calls and puts as volatility approaches infinity, assuming a zero interest rate. From the model’s formulas, it identifies limiting prices of zero for the call and one for the put, independent of…
The document explains how daily leverage and compounding shape leveraged ETF returns. An ETF resets its exposure at each rebalance, so its multi-day result is the product of daily leveraged returns, adjusted for fees. This means a 3× fund does not simply…
The document raises a conceptual question about volatility skew: options at different strikes imply different volatilities, even though implied volatility is often treated as a measure of expected future risk. It asks whether those differences represent…
The discussion explains how Black-Scholes is used in over-the-counter FX vanilla options. Market quotes commonly express implied volatility in delta space; traders convert those quotes into prices using the relevant market inputs, such as spot, rates, and…
The document asks how to select the optimal bandwidth for a realized-kernel volatility estimate when the desired output uses one-minute bins. It gives the bandwidth formula from the cited reference and notes that its noise-to-signal input uses integrated…
The document asks whether a flat implied-volatility skew implies a normal risk-neutral distribution when option prices are converted into a density using the Breeden–Litzenberger relationship. It notes that the density inferred from its stated formula…
The document seeks to interpret a paper's implied-gamma method for fitting option prices across strikes. In the described approach, gamma is a modeler-selected parameter defining a return uncertainty set. For each in-sample strike, the parameter is…
The document works through the conditional expectation of the squared value of a Brownian motion at a later time, given its filtration at an earlier time. It expands the later value into the known earlier value plus an independent increment, then asks why…