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This document asks why a leveraged WTI exchange-traded note and its inverse counterpart appear to trace different slopes during large price moves. One note is described as providing leveraged long exposure and the other as leveraged inverse exposure. The…

This document asks whether a minimum-variance portfolio is more likely than a maximum-Sharpe, or tangency, portfolio to deliver out-of-sample performance close to its in-sample estimate. It defines disappointment as the gap between predicted in-sample…

This document poses a sequential elimination game between two teams of four gladiators. In each round, Alice chooses a fighter first and Bob responds with a matchup; each duel has no tie, and the stronger fighter’s advantage is expressed as a probability…

The document asks about geometric Brownian motion models whose drift or volatility depends on the stock’s past path, rather than only its current value and time. It gives examples of drift built from an accumulated function of earlier prices or from the…

The document proposes measuring how much a stock price moves within a day by summing absolute changes between selected observations, rounding each change down in units of a chosen price increment, and normalizing the total by the day’s range. It asks whether…

The document explains how ATR-based position sizing estimates a share count from account capital, a chosen risk fraction, and a stop distance expressed as a multiple of average true range. The resulting count represents shares sized to a target loss if the…

The document clarifies terminology for a trading account’s position at the end of a session. It treats overnight inventory, end-of-day holdings, and the end-of-day net position as referring to the inventory remaining at market close and carried into the…

The document answers whether the standard European put-call parity relation also applies when the underlying is modeled with Heston stochastic volatility. Its central point is that parity follows from the option contracts and the ability to construct…

The document addresses how to summarize implied volatility across an option chain rather than relying only on the at-the-money option. It describes a model-free variance-swap approximation built from option prices over a range of strikes: calls above the…

The document explains how covariance-matrix inversion affects the standard mean-variance portfolio solution. In an eigenvector decomposition, each direction’s contribution is scaled by the reciprocal of its covariance eigenvalue. Consequently, directions…

The document shows how to calculate a money-weighted rate of return for a pension fund with dated inflows and outflows. The method sets the future value equation to zero at the fund’s closing date, compounds the starting balance over twelve months, adjusts…

The document discusses how to evaluate residuals when comparing time-series forecasts, including volatility forecasts from models such as GARCH. It recommends examining autocorrelation: if errors contain predictable structure, the model may be missing a…

The document explains why the S&P 500 price index and its total return counterpart produce different performance charts. The price index reflects constituent share prices, so it falls when a company goes ex-dividend; it does not credit the dividend to the…

The document asks for the stochastic differential of an exponential built from an integral involving a diffusion process. The questioner applies Itô’s formula to the exponential, but assumes the integral has differential equal only to the integrand at the…

The document poses a conceptual question about the efficient market hypothesis (EMH): if prices reflect available information and investors are rational, why should prices change or investors expect returns? It considers three possibilities: no new…

The example asks how a company can exploit a forward contract offered at an unfavorable price. It assumes a non-dividend-paying stock, a continuously compounded interest rate, and a transfer of an existing long forward position for a fee before maturity. The…

The document outlines several ways to think about pricing options on a volatility index such as VIX. One approach starts from the option portfolio used to replicate variance and adds a convexity adjustment that accounts for the square root linking variance…

The document resolves a confusion between variance as a function of a probability parameter and portfolio variance as a function of asset weights. A Bernoulli variable’s variance can be concave in its probability parameter, but portfolio variance, expressed…

The note derives the range of a common off-diagonal correlation that makes a matrix with unit diagonal entries a valid covariance or correlation matrix. It corrects a variance calculation: when summing pairwise covariances in the variance of a sum, each…

The note asks why a CAPM generalized method of moments specification does not add a separate moment equating the factor’s sample mean to its expected value. It explains that the cross-sectional pricing moments compare each asset’s expected excess return with…

The document asks what changed in Monte Carlo methods for pricing American options between 2005 and 2016, following earlier machine-learning work on optimal stopping. One answer argues that the period brought few groundbreaking Monte Carlo advances and that…

The document explains a data choice for studying market expectations of Federal Reserve policy. Actual 30-day Fed Funds futures reference the average effective rate over a calendar month, so they do not offer a contract with a continuously rolling 30-day…

The document frames three ways to convert a fixed-rate bond position into floating-rate cash flows while seeking to isolate the issuer spread: a par-par asset swap, a same-notional payer swap, and a payer swap sized to match the bond’s DV01. It explains the…

The document derives a simple FX forward relationship from covered interest parity. For two currencies, it compares converting at spot and investing at the domestic rate with investing at the foreign rate and locking in the maturity conversion rate through a…