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 question attempts to find the distribution of the stochastic term in a Heston-style asset process by substituting an integral expression for the variance process, taking a square root, and applying Brownian differential identities. The response…
The document derives a static option replication for a payoff equal to the square of a non-dividend-paying stock’s terminal price. It rewrites the square as twice the integral, over strikes from zero to infinity, of the call payoff at each strike. This…
The document considers statistical tools for testing whether historical changes in a market variable appear random, with the aim of identifying unusual patterns and consistency over time. It points to the NIST test suite, a collection originally used to…
The document addresses how a T-forward measure can be used when a directly observed zero-coupon bond for the required maturity is unavailable. It explains that market curves are built from instruments spanning different horizons: spot borrowing rates anchor…
The document derives the finite-horizon change in log price when log price, after subtracting a linear time trend, follows an Ornstein–Uhlenbeck process. The drift pulls the detrended log price toward zero at a rate set by the mean-reversion parameter, while…
The document explains why a minimum-risk portfolio optimization with fully equal active weights can return the benchmark itself. It defines each active weight as the difference between a portfolio weight and its benchmark weight. If both the portfolio and…
The document examines whether the conditional covariance of two assets at a later date can be decomposed by conditioning on their forward values at an intermediate expiration. It starts from the usual expression for covariance as the conditional expectation…
The document presents an apparent contradiction from three two-sided t-tests: one fails to reject a zero mean for a company’s monthly stock returns, another rejects a zero mean for the market index, and a third fails to reject equal means for the stock and…
The document explains why the term duration can refer to different bond measures and how compounding conventions affect their relationship. Macaulay duration is described as a time-weighted measure of fixed cash flows, while modified duration measures the…
The document asks whether the minimum-variance hedge ratio derived from returns also minimizes variance at the price level. It highlights that a hedge ratio chosen to reduce variation in changes or returns may not optimize variation in price levels,…
The document explains why the expected return, or drift, of a stock does not appear in the standard Black–Scholes option pricing formula. Under the model's assumptions, a replicating hedge removes the underlying's risk, so no-arbitrage pricing values the…
The document asks why the Feynman–Kac theorem can be applied to a derivatives valuation partial differential equation that includes minimum and maximum terms. The displayed equation models an adjustment value with bilateral counterparty default intensities,…
The document addresses how to represent directional change between adjacent market bars when discrete price observations do not provide a directly meaningful geometric angle. It clarifies that angles are defined between vectors, rather than isolated points,…
The document explains how to interpret a VaR limit specified over a fraction of a year and convert the volatility implied over that horizon into annualized volatility. For a horizon represented as one of m equal yearly intervals, first identify the VaR…
The document asks whether a European call is worth more when volatility is random with a given mean than when volatility is fixed at that mean. It introduces Jensen’s inequality as the key tool: if the option price is a convex function of volatility, then…
The document investigates the integrated value of a call option across maturities, defined as the time integral of its price, and asks why its leverage ratio becomes hump-shaped near the at-the-money spot level under low initial variance in the Heston model.…
The document asks which terminal stock-price distributions can serve as risk-neutral pricing distributions in a frictionless market with a constant interest rate and a non-dividend-paying stock. No-arbitrage fixes the distribution’s expected value to the…
The document addresses how to obtain approximate critical values for the Johansen cointegration trace test when the system has more variables than commonly tabulated references cover. It proposes an approximation based on a correction factor that depends on…
The document examines whether a dividend discount model can explain the long-run rise of broad stock markets. It starts with the constant-growth model, in which price depends on dividends, the required return, and dividend growth. It then considers adding…
This note considers a stock whose daily return is equally likely to be a gain or loss of the same percentage, with no dividends or interest. It asks whether buying or selling has an expected-value advantage under those assumptions. Starting from one unit of…
This note explains how to derive the ordinary Marshall–Olkin copula from its survival-copula form. It starts with two exponentially distributed lifetimes that may be affected by a shared failure event, then gives the survival copula in terms of parameters…
The document evaluates a calculation that uses a mean-reverting short-rate process to explain a particular future long-term Treasury yield. It confirms that the displayed closed-form solution applies to a Vasicek process with a constant long-run mean. It…
The document explains how a LIBOR market model can be used to estimate the future value-at-risk of a portfolio of interest rate swaps. The proposed workflow starts by choosing a time grid and collecting current market rates, then bootstrapping or…
The document addresses a Black–Scholes derivation involving the standard normal density evaluated at d1 and d2. It clarifies that the notation N′(x) refers to the normal probability density function, not to differentiating the expression with respect to…