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 note derives the conditional volatility of a stock in a one-step binomial model where its price moves up or down by one unit with equal probability. With the stated definition, the conditional mean of the next price is the current price, while the…
The document works through the variance of the time integral of squared standard Brownian motion. An initial attempt applies Itô’s lemma to a time-weighted square of the process, then computes variances of resulting terms as if they were independent. That…
The document asks how maximum drawdown varies with the mean and standard deviation of daily returns and the length of the return series. It frames the problem with simulated paths: draw daily observations from a normal distribution, calculate each path’s…
The document describes a simplified way to represent an active long volatility allocation discussed in a portfolio research paper. The proposed rule monitors the equity market over a rolling three-month window and buys options in the direction of a…
The document raises a question about replicating a variance swap fair strike from options when implied volatility is flat. The author expects the strike to equal implied volatility squared, but reports that a replication using constant volatility returns a…
The document presents an attempt to infer implied volatility by minimizing the difference between observed option prices and Black–Scholes prices. The calculation uses spot, strike, time to expiry, interest rates, and dividend yield, with rates interpolated…
The document explains why a stock used as numeraire changes the stock’s drift under the associated pricing measure. In a Black–Scholes setting with constant volatility, the drift becomes the risk-free rate plus the variance rate. The central idea is that…
The document considers whether a practical option model can produce better delta hedges than Black–Scholes for equity options. One response says the Heston model is relatively quick to implement analytically, but may still miss important features of…
The document raises a question about a statement in a paper on variance swaps: a stock call option’s vega is greatest when the option is at the money. The author wonders whether this is because vanilla options are nearly linear in volatility at that point,…
The document poses a question about how expected asset prices can be equal under the physical probability measure, while the expected value of the more volatile asset is lower under the risk-neutral measure. It asks whether this relationship can be proved or…
The document presents a workflow for forecasting stock volatility with a GARCH(1,1) model through R’s rugarch package called from Python with rpy2. It calculates daily log returns, fits a Student-t model to returns scaled as percentages, reserves a final…
The document asks why formulas connecting local volatility to implied volatility appear to give zero volatility when the underlying price equals the strike. That apparent result would make the equations seem unusable for at-the-money options, and values near…
The document separates three related ideas that are often conflated in financial modeling. Standard deviation describes the spread of outcomes around a mean, while volatility commonly denotes the diffusion parameter in a geometric Brownian motion model and…
The document explains why a standard dynamic conditional correlation (DCC) GARCH model does not allow unrestricted cross effects in the conditional variance equations. In the described two asset setup, each asset's conditional variance is fitted with a…
The discussion considers why option-pricing models commonly use stock-return processes with finite variance, even when heavier tails are desired. It gives several explanations: the historical influence of Black–Scholes, mathematical and computational…
The discussion asks whether historical stock returns can be used to estimate volatility skew across option strikes. It distinguishes estimating at-the-money volatility from determining how implied volatility varies for out-of-the-money puts and calls, and…
The document discusses the reported finding that stocks with greater sensitivity to innovations in aggregate volatility have lower average returns, and asks why such exposure might act as a hedge against market downside risk. It does not settle the puzzle;…
The document asks how to price options quickly during model calibration, comparing the Carr–Madan and COS methods. It highlights Variance Gamma models with stochastic arrivals and related models where cumulants may not be available in closed form. The…
The document investigates why a VIX-like volatility estimate calculated from option prices can fall below the volatility used to generate synthetic options. The example uses Black–Scholes prices with constant volatility, zero interest rate, and zero cost of…
The document discusses numerical pricing accuracy in the Heston stochastic volatility model, especially for very far out-of-the-money options near expiry. It contrasts a reported Gaussian quadrature error with option prices smaller than that error, raising…
The document compares the time values of equally deep in-the-money European calls and puts with no dividends, using strikes the same dollar distance from spot. It applies put-call parity to split the difference in spot-based time values into a discounting…
The document explains why a Black–Scholes theta estimate can exceed an option’s current value without implying that the option will actually become negative. The closed-form theta is an instantaneous derivative, commonly stated per year; multiplying it by a…
The document considers how to use parameters estimated from hourly Ornstein–Uhlenbeck (OU) data in a daily simulation. It offers two practical routes: aggregate the observed data to daily frequency and re-estimate the parameters, or simulate at hourly…
The document answers a question about how far an asset price might move over a three-month horizon given its annualized volatility. The response treats the estimate as a static one-standard-deviation range, scaling volatility by the square root of time and…