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.
21,023 documents
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 question sets up a discretized forward-rate model in which each forward is lognormal under its associated zero-coupon bond measure. It derives the resulting normal distribution for the log forward at its fixing date and uses that distribution to motivate…
The note contrasts pricing in complete and incomplete financial models. In a complete model, it describes a unique martingale measure and a derivative price given by the discounted expectation under that measure. In an incomplete model, multiple martingale…
The note concerns a measurability step in proving that the first time an adapted càdlàg process enters an open set is a stopping time. The stated argument expresses the event that the hitting time is strictly before a fixed time as a union over rational…
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 document derives the futures price for an asset whose price process and short rate are driven partly by the same Brownian motion. Since the futures price is the expected future spot price, the solution first expresses the integrated short rate as a…
The document raises a replication question about estimating a monthly factor regression for European equity portfolios, with particular interest in the intercept as abnormal risk-adjusted performance. It describes calculating monthly portfolio returns from…
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 distinguishes a bid or ask tax from a transaction tax: the charge applies when a limit order is submitted, even if the order never trades. It describes two laboratory double-auction studies in which participants traded against experimenter-set…
The document considers the probability that one of two correlated geometric Brownian motion prices finishes above the other, while the lower price also exceeds a threshold. It rewrites the event using shifted Brownian variables as two simultaneous…
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…
The document explains how derivatives can change a borrower’s practical liquidity and recovery profile even when derivative claims rank equally with senior debt in bankruptcy. Under collateral agreements, a firm may have to post cash or securities as market…
The document frames a feature-selection problem using order book snapshots with five bid and ask price levels, corresponding displayed quantities, and a midprice defined from the best bid and ask. The proposed target is the change in midprice across a few…
The document asks how to extend a CAPM certainty-equivalent present-value formula to several priced risk factors. The proposed expression subtracts each factor’s price of risk multiplied by the cash flow’s covariance with that factor, then discounts 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 asks how to decide whether a portfolio consideration belongs inside an existing objective function or should be optimized as a separate objective. It starts with a conventional formulation that seeks higher expected return while controlling…
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 asks how to derive risk-neutral dynamics for a dividend-paying asset when the bank account is the numeraire. It outlines the martingale condition for the discounted asset plus accumulated discounted dividends, then applies Itô’s formula and…
The document shows why subtracting calendar dates does not reliably locate observations in daily financial data. Weekends and market holidays create gaps, so the number of elapsed days can differ from the number of rows between two trading dates. The example…
The document explains XIRR as the annualized rate that equates a dated series of irregular cash flows with an equivalent constant-rate investment. Its examples contrast a simple loan with interest paid at different dates and a short holding-period trade,…
The document describes a credit-risk data preparation workflow for estimating probability of default. The example defines a default event from arrears, term-expiry, or bankruptcy indicators, recodes the outcome for modeling convenience, and uses stratified…
The document derives a rough out-of-sample observation length for testing whether a strategy’s mean return exceeds zero. Assuming independent, identically distributed returns, it models the sample mean’s uncertainty as the return standard deviation divided…
The document asks whether solutions to a linear stochastic differential equation become differentiable when Brownian noise is filtered through the system’s dynamics. The question contrasts Brownian motion’s irregular paths with the intuition that a colored…