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 the time increment in a simulated Heston model relates to the number of steps per year: once the time unit is fixed, choosing one determines the other. Its main example is the Euler–Maruyama discretization of the stock and variance…
The document concerns adapting the Avellaneda–Stoikov market-making model to a longer time horizon. Its question centers on the reservation-price expression, where inventory, risk aversion, volatility, and remaining time shift the indifference price away…
The discussion examines when at-the-money (ATM) implied volatility can approximate the risk-neutral expectation of average integrated variance. For very short maturities, it motivates the approximation by noting that instantaneous volatility near the start…
The discussion asks whether implied volatility (IV) produces better option prices than historical volatility (HV), noting that the common forward-looking argument for IV is not enough by itself to establish better pricing performance. It cites an experiment…
The document explains why daily volatility is commonly scaled by the square root of the number of trading days in a year. Under constant daily variance and uncorrelated returns, variances add across trading days, so annual volatility equals daily volatility…
The document asks whether the profit or loss of a Black-Scholes delta-hedged option depends only on realized volatility at expiry or also on its path. It references a hedge-error expression involving gamma exposure and the difference between implied variance…
The document examines a density that appears when expressing forward implied variance as an average of local variance. It derives the weighting from the risk-neutral distribution of the underlying and a factor involving spot squared and Black-Scholes gamma.…
The document considers whether high and low prices can improve correlation estimates from OHLC bars. Its central point is that open and close observations have known timing, while the relative timing of each asset’s high and low is unknown. A joint estimator…
The document clarifies that an option-adjusted spread is not a return that a hedge will realize on every interest-rate path. OAS is found by projecting an MBS’s cash flows across modeled rate paths, discounting them with the yield curve plus a spread, and…
The document explains two special cases of the SABR model by conditioning on a realized volatility path. With beta equal to one, the asset follows a multiplicative diffusion, so its logarithm has a normal distribution and the asset level is lognormally…
The document clarifies that normal and lognormal implied volatility refer to different option pricing conventions, not necessarily to the distribution assumed for the underlying in the SABR model. A normal implied volatility is the value used with Bachelier…
The document explains why a Double Heston model uses two stochastic variance processes. A second factor adds flexibility to the relationship between asset returns and volatility, allowing that correlation to change as the factors’ relative contributions…
The document discusses implicit equity trading costs and the market conditions that shape them. A central explanation is adverse selection: market makers facing informed traders risk trading at prices that will soon move against them. They may respond by…
The document asks how to define option vega when volatility varies with the underlying asset price, using the CEV form in which local volatility is proportional to a power of the asset price. It contrasts this setting with Black–Scholes, where volatility is…
The document asks how to assess whether a GARCH(1,1) volatility forecast is significantly above realized volatility. It describes a proposed approach: apply an F-test to the ratio of forecasted and realized volatility, then determine the degrees of freedom…
The discussion compares possible ways to track changes in option volatility skew relative to at-the-money volatility. One answer proposes measuring the implied-volatility difference between the strikes where Black–Scholes d2 equals zero and d1 equals zero.…
The note asks how to differentiate a Carr–Madan Fourier integral for option prices with respect to a model parameter. Since the integration limits do not depend on that parameter, the proposed application of differentiation under the integral sign leaves the…
The note explains why option prices generated by a stochastic volatility model can imply a volatility smile when those prices are converted using the Black–Scholes formula. Under Black–Scholes pricing with one fixed volatility, the prices of calls across…
The document describes a comparison between intraday VIX-style implied volatility and realized variance computed from one-minute prices aggregated into hourly intervals. The implied measure is calculated from option prices for two maturities around 30 days,…
The document derives the variance of a weighted sum of two correlated Brownian increments. For coefficients applied to each increment, the variance includes both individual variance terms and a covariance term determined by their correlation. Since each…
The document shows how to estimate geometric Brownian motion parameters from a series of daily prices. It first maps prices to log returns, which follow a normal process with drift equal to the price process’s drift minus half the variance rate. When the…
This note applies Itô’s lemma to the logarithm of a price following geometric Brownian motion. Starting from a stochastic differential equation with drift proportional to the price and volatility proportional to the price, it differentiates the logarithm…
The document addresses a common confusion in stochastic calculus: an Itô process can have zero quadratic variation while still having nonzero variance at a fixed time. In the example, integrating Brownian motion with respect to time yields a process whose…
The document explains convertible bond arbitrage as buying a convertible bond while shorting the issuer’s stock to create a delta-hedged, long-volatility position. The strategy seeks to benefit when realized volatility exceeds the bond’s implied volatility,…