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 that the second derivative of an option’s value with respect to the underlying price is gamma. It measures how quickly delta, the option’s price sensitivity to the underlying, changes as the underlying moves. This curvature matters…
The document examines whether option-implied volatility can replace historical volatility in mean-variance portfolio optimization. It cautions that vanilla option implied volatility varies by strike and has a specific risk-neutral interpretation tied to…
The document clarifies the relationship between lognormal prices and normally distributed log returns. If a price or price ratio is lognormally distributed, its logarithm is normal; therefore, the log of a gross return is normal under that assumption. In…
The document derives a pricing equation for an option whose underlying price, volatility, and short rate all follow stochastic processes with correlated Brownian shocks. It first transforms the asset price to its logarithm, then represents the correlated…
The document explains how to interpret negative estimated coefficients in GARCH variance equations. A negative coefficient does not automatically make conditional variance negative, so the relevant checks are whether fitted variance becomes negative and…
The document outlines ways to allocate swaption bid-offer liquidity costs across SABR parameters. With beta held fixed, alpha represents the at-the-money volatility level, nu captures smile curvature, and rho captures skew. First estimate how the option…
Normal volatility is measured in the same units as the underlying changes used to estimate it. In the example, realized normal volatility is calculated as the standard deviation of absolute daily changes; for an index, the result is expressed in index…
The discussion addresses two practical inputs to equity option pricing: historical volatility and interest and dividend rates. It says there is no universally established industry standard for historical-volatility estimation, and recommends a consistent,…
The discussion outlines different levels of mathematical background for understanding geometric Brownian motion (GBM) in stock-price modeling. Basic probability and programming can support discrete-time simulation and intuition, while studying the…
The document explains why autoregressive conditional heteroskedasticity can be consistent with a version of the random walk hypothesis in which returns are uncorrelated but not statistically independent. In an ARCH(1) model, the return residual is white…
The document explains when a hedge can be established once and when it needs ongoing adjustment. Static hedging is possible when instruments can replicate a position’s payoff at inception; the payoff need not be linear. The example given is replicating a…
The document addresses the difficulty of obtaining historical data for volatility and variance swaps. The responses say that public historical bid and ask prices are generally unavailable, so researchers seeking exact market quotes may need to approach a…
The document explores why very long maturity interest rates can fall below rates at somewhat shorter maturities. Its answers emphasize convexity: longer bonds and forward-starting swaps generally have greater convexity, so their values respond more strongly…
The document raises a practical calibration problem for the Heston stochastic volatility model. The author fits the model in Matlab using Monte Carlo simulations, with 5,000 paths, two time steps per day, and a 360-day horizon. Calibration with the…
The document considers how to quantify the gap between a modeled implied volatility surface and market quotes, with the goal of deciding when recalibration is needed. It notes weaknesses in averaging signed percentage errors: positive and negative deviations…
The document presents a question about forecasting a differenced index series with an ARMA(2,2) mean model and a GARCH variance model. It gives the questioner’s proposed equations, a rugarch specification, and an output showing the forecast series and…
The document discusses the derivation of an option pricing equation for a stock with stochastic volatility. It asks what can serve as a second asset for hedging volatility exposure and how the volatility risk premium relates to the familiar Black–Scholes…
The document explains why comparing historical volatility (HV) with implied volatility (IV) requires matching their forecast horizons. IV is inferred from option prices and is forward-looking; HV is estimated from realized underlying returns and is…
The document asks how to query an implied volatility surface when simulating an option under a local volatility model. The surface is described in terms of log forward moneyness and expiry. For vanilla implied volatility, the questioner uses the expiry…
The discussion compares approaches to simulating the Heston stochastic volatility model, prompted by a moment-matching method that can fail when volatility of volatility is large and the Feller condition is strongly violated. In that method, equations for…
This note addresses apparent asymmetry between out-of-the-money put and call implied volatilities near at-the-money. It attributes the discontinuity in the displayed data to an implied-volatility calculation that uses the spot index in place of a…
This note establishes an upper bound for the realized volatility of a geometric basket of stocks. With nonnegative weights, the logarithmic return of the basket is a weighted sum of constituent log returns. Expanding its squared variation introduces cross…
The document explains how GARCH can be used to model changing error variance within a broader model for the relationship between an outcome and its predictors. Instead of assuming independent, identically distributed errors, a regression can allow its…
The document asks whether Heston model calibration requires first constructing a smoothed, arbitrage-free implied volatility surface, as is commonly needed in local volatility work. The accepted response describes Heston calibration as fitting model…