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The document discusses narrowing a large universe of candidate equity pairs after an Augmented Dickey–Fuller test and half-life calculation leave many candidates. Suggested filters include liquidity, trading costs, backtesting with cross-validation, and…
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 argues that risk management can sit beside an ML or AI alpha model rather than trying to interpret its internal features. A separate portfolio controller can translate forecasts or directional signals into feasible positions, accounting for…
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 surveys alternatives to Kelly sizing, focusing on how statistical arbitrage portfolios balance positions. Sector neutrality groups assets by industry, while risk neutrality uses a model of shared factor exposures to help an optimizer construct a…
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 discusses execution choices for an investor trading illiquid small-cap, micro-cap, or over-the-counter securities with holding periods of months or longer. The response recommends judging execution against the trading model’s intended fills and…
This note outlines ways a Europe-based investor can hedge the currency exposure of a USD bond. One approach is to align a cross-currency swap with the bond’s payment schedule, paying USD fixed to match the bond coupons and receiving either fixed or floating…
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…
The document asks for the mean and variance of a mean-reverting process whose noise is proportional to the state. The answers first present an explicit-solution expression and moment formulas, but a later response flags that expression as incorrect and gives…
The document explains basic short-selling mechanics and asks how short interest relates to a possible squeeze, whether the price at which most short positions were opened can be identified, and why a trader might short shares instead of buying puts. The…
The document presents a linear parabolic Cauchy problem with a terminal condition and gives its Feynman–Kac representation in terms of an Itô diffusion, a potential term, and a source term. It then asks how to formulate the corresponding representation for a…
The document asks how to estimate the time-varying tail-dependence functions proposed by Liu, Ji, and Fan. It distinguishes four dependence directions: upper–upper, lower–lower, lower–upper, and upper–lower, noting that the latter two extend the conventional…