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The document considers how to assemble historical intraday data for penny stocks that gap up by at least 10%, using a definition based on low share price and market capitalization. Its suggested workflow is to identify qualifying gap-up events with a stock…

The document describes a factor model in which factor returns are estimated by weighted least squares from asset returns and known exposures. It derives the corresponding factor mimicking portfolio weights as a matrix involving the exposure matrix and the…

The document presents a derivation of Hamada's formula, which relates a firm's levered equity beta to its unlevered beta, debt-to-equity ratio, and tax rate. It starts from beta as covariance with market returns divided by market variance, then expresses…

The document investigates why a QuantLib USD overnight indexed swap curve built from Bloomberg par rates reproduces Bloomberg discount factors at short maturities but diverges slightly at longer tenors. The example constructs OIS rate helpers and a…

The note asks why modified duration equals Macaulay duration divided by one plus the yield, seeking an intuition beyond the derivative-based derivation. The response frames Macaulay duration as the present-value-weighted average time at which a bond’s cash…

The note explains how the option portfolio’s gamma contribution to price change follows from the second-order term in Itô’s lemma. It distinguishes the instantaneous variance term in the pricing equation from realized movement in the underlying price. Using…

The document gives intuition for the variable changes used to transform the Black–Scholes partial differential equation into the heat equation. The response breaks the spatial transformation into two steps: first express the underlying price in…

The document explains that mean-variance portfolio optimization does not require normally distributed returns; it requires finite second moments. For returns with finite variance, including the stated Fréchet example, mean-variance allocation remains…

The document asks how to calibrate the market price of risk in the Black–Karasinski short-rate model when zero-coupon bond prices lack a closed-form expression. The proposed Monte Carlo approach simulates short-rate paths, discounts cash flows along them,…

The document discusses how to include swaps, FX forwards, and other non-equity derivatives in Monte Carlo Value at Risk simulations. It contrasts sensitivity-based approximations with full revaluation under simulated market scenarios. Sensitivity methods…

The document presents a fixed-income futures position and asks why its realized price-based loss differs from a loss estimated using the change in the underlying Treasury note’s yield. The trader reports buying a US two-year note futures contract, later…

The document clarifies an ambiguous claim that spot price and variance have correlation near negative one. One response explains that the leverage effect concerns negative association between stock returns and volatility, rather than between the price level…

The document describes an attempt to build implied volatility smiles for European options from American option chain data. The author calculates implied volatilities with Black–Scholes, selects out-of-the-money options to reduce early-exercise concerns, and…

The note frames CAPM beta as a regression coefficient linking an asset’s excess returns to market excess returns, with residual returns representing variation not explained by the market factor. It explains that asking whether beta differs significantly from…

The note derives a way to divide a total fund among people who retire at different times so that each allocation grows to the same value at that person’s retirement. It assumes a constant return and annual compounding. Each person’s share is set in…

The document raises two questions about a volatility swap under a flat one-year volatility assumption with no skew: whether the fair volatility equals the quoted volatility when the number of fixings matches the stated annualization factor, and why the fair…

The note explains why an American put on a non-dividend-paying stock has the same value as its European counterpart when interest rates are non-positive. Its key comparison is between the European put’s lower bound, the discounted strike minus the stock…

The question asks whether limited capital can produce a larger payoff by requiring several stocks to meet a return threshold, rather than holding equal investments in each. The response proposes a different approach: if expected price moves occur at…

The discussion explains why a regression intercept used in rough volatility analysis can be interpreted as volatility of volatility. Researchers measure increments in log volatility across sampling intervals, calculate their absolute moments, and regress the…

The document identifies a source for Swiss zero-coupon bond yield curve data: the Swiss National Bank’s data portal. The original question explains that the data is sought for research and that the asker could not access a Bloomberg terminal. The reply…

The document points readers to a practical paper on calibrating the SABR model and to foundational references for interpreting its parameters. It explains that beta is commonly fixed in advance to represent a chosen dynamic, rather than estimated alongside…

The document asks under what conditions subtracting the time integral of a diffusion's infinitesimal generator from a function of the process yields a martingale. It applies Itô's formula to a function of an Itô process and identifies the remaining…

The document considers how to convert monthly Jensen's alpha estimates for individual stocks into quarterly or yearly measures. It describes two approaches: aggregate returns to the desired observation period and rerun the regression, or combine the monthly…

The document asks why a drift term obtained after applying Itô's formula to a discounted European option price must vanish in a pricing partial integro-differential equation. The setup models the asset with a Lévy process, so the discounted option value has…