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 asks whether the low-volatility anomaly—the empirical pattern that lower-volatility stocks can earn higher returns than riskier stocks—could be explained by the difference between arithmetic and geometric average returns. It points to the…
The discussion cautions against assuming that VIX futures will reliably offset losses in a long-only equity portfolio. VIX measures expected volatility, and it does not invariably rise when stocks fall; consequently, an inverse relationship with stock…
The note explains how to interpret the instantaneous forward variance quantity in a model of volatility smiles. It distinguishes that quantity from a forward-starting variance swap, whose fair strike is constructed from the variances of two maturities. The…
This discussion asks how to use volatility from an implied-volatility surface in Reiner-Rubinstein formulas for an up-and-out call, expressed as a vanilla call minus an up-and-in call. The vanilla component may reference the strike while another component…
This example compares delta-gamma and delta-vega hedging for a short call using another call and shares. It gives Black-Scholes Greeks for an in-the-money call and an at-the-money call, then solves for the option quantity that offsets gamma or vega and the…
The document considers whether some equity markets still have nearly flat implied volatility smiles, in contrast with the negative skew often associated with major indices after the 1987 crash. One response reports that options on many Korean equities, and…
The document discusses how to fit a Black-Scholes implied volatility surface when options trade with bid-ask spreads. It explains why separate arbitrage-free bid and ask surfaces are generally not useful for practical hedging, since trades and hedges can…
The document explains why delta-hedged option profits should not be confused with recovering the premium as trading revenue. Under Black–Scholes assumptions of zero rates, continuous hedging, and realized volatility equal to implied volatility, gamma gains…
The document discusses modeling S&P 500 return volatility with GARCH models that use the VIX as an external regressor. The question concerns comparing GARCH forecasts with implied volatility forecasts, using data from 2020 through 2022, and asks whether to…
The document clarifies the role of the innovation term in ARCH and GARCH models. Returns are represented as a conditional mean plus the square root of conditional variance multiplied by a standardized white-noise shock. The variance equation updates over…
The document explains how to visualize Black–Scholes Vanna, the sensitivity of an option’s value to changes in both the underlying price and volatility. The central task is to evaluate Vanna over every pair drawn from two input ranges, rather than matching…
The document discusses difficulties fitting the Heston stochastic volatility model to vanilla option prices. Parameter dependence and flat regions in the calibration objective can make optimization unstable, especially for mean reversion. It considers…
The discussion considers how to select the orders of an EGARCH model for forecasting volatility associated with S&P 500 option portfolios. It distinguishes choosing a model for in-sample fit from choosing one for predictive performance. AIC, BIC, and…
The document introduces Emanuel Derman’s volatility approximation for a volatility swap under an assumed linear volatility skew. It describes the approximation as assuming a linear put skew and a flat call skew, with applicability focused on strikes…
The document explains why the relationship between the underlying price and future volatility can affect autocall valuation. Autocalls often include digital payoffs and barrier conditions, so their value depends on the chance that the underlying crosses a…
This exchange explains why SVXY, a daily inverse VIX futures exposure, did not lose all its value during the sharp volatility episode of February 2018. The key distinction is between the VIX index, which is calculated from options and is not directly…
This exchange diagnoses why a Newton–Raphson routine for Black–Scholes implied volatility diverges. The original code divides the option-price error by gamma, but implied volatility is the quantity being solved for, so the derivative needed for the update is…
The document asks whether an option can have a strike that varies randomly over time and what such products are used for. The answers point to several related structures: an Asian option uses an average of the underlying over the contract period in its…
The document considers how to simulate a variance process that can turn negative under a discretized stochastic volatility model with a variance-gamma jump term. Rather than forcing the jump component itself to be nonnegative, the answer presents ways to…
The document addresses how changes in a short-term interest rate’s yield volatility relate to price volatility, while questioning whether a proposed bond formula applies to a short-term interest rate option. The response gives the standard first-order…
The document asks whether GARCH(1,1) parameters can be updated as each new price arrives, without rerunning the full estimation. Its answer is that maximum likelihood estimates depend on the full likelihood across observations. Because the GARCH recursion…
The document introduces the maximum diversification portfolio objective, defined as the weighted sum of asset volatilities divided by portfolio volatility. The numerator uses asset weights and individual volatilities; the denominator depends on the…
The document explains how to simulate a down-and-in barrier call under a local volatility model. At each simulation step, the volatility should be evaluated at the current time and stock level, rather than at the option’s fixed strike and maturity. The…
The note distinguishes the roles of the ARMA and GARCH components in a return model used for directional trading. The ARMA conditional mean supplies the expected return and therefore drives a rule that goes long or short according to the forecast sign. GARCH…