คลังความรู้

The document outlines checks for evaluating an ARMA-GARCH model fit. Before using GARCH, test whether the data show ARCH effects; without conditional heteroskedasticity, a GARCH model may not be needed. After fitting, inspect standardized residuals for…

In a binomial model, the risk-neutral probability is chosen so that the stock’s expected return under that pricing measure equals the risk-free rate. The document explains why a stock’s CAPM expected return is not substituted: option valuation uses…

Equity mutual funds may have different rules governing how much they can invest in particular industries. The document explains that investors can look for those rules in a fund’s prospectus. A broadly managed fund may have no hard industry limits stated,…

The document examines the Black–Scholes prices of digital calls and puts as volatility approaches infinity, assuming a zero interest rate. From the model’s formulas, it identifies limiting prices of zero for the call and one for the put, independent of…

The document explains how daily leverage and compounding shape leveraged ETF returns. An ETF resets its exposure at each rebalance, so its multi-day result is the product of daily leveraged returns, adjusted for fees. This means a 3× fund does not simply…

The document compares overnight index swaps (OIS) and Fed Funds futures as market inputs for estimating interest rates expected around an FOMC meeting. It distinguishes a measure based on recent realized Fed Funds trading from futures prices that reflect the…

The document raises a conceptual question about volatility skew: options at different strikes imply different volatilities, even though implied volatility is often treated as a measure of expected future risk. It asks whether those differences represent…

The explanation distinguishes Eurodollar futures from forward rate agreements and interest rate swaps. In this context, Eurodollar refers to futures on a three-month interest rate, though the term can also refer to the Euro–US dollar currency pair. Compared…

The discussion explains how Black-Scholes is used in over-the-counter FX vanilla options. Market quotes commonly express implied volatility in delta space; traders convert those quotes into prices using the relevant market inputs, such as spot, rates, and…

The document asks how to select the optimal bandwidth for a realized-kernel volatility estimate when the desired output uses one-minute bins. It gives the bandwidth formula from the cited reference and notes that its noise-to-signal input uses integrated…

The document asks whether a flat implied-volatility skew implies a normal risk-neutral distribution when option prices are converted into a density using the Breeden–Litzenberger relationship. It notes that the density inferred from its stated formula…

The document seeks to interpret a paper's implied-gamma method for fitting option prices across strikes. In the described approach, gamma is a modeler-selected parameter defining a return uncertainty set. For each in-sample strike, the parameter is…

The document asks how to simulate two positively correlated time series when one is a stock modeled with geometric Brownian motion (GBM). It describes estimating GBM inputs from historical stock data and reports a Pearson correlation of 0.5 between the two…

The document explains how factor models can reduce the complexity of modelling returns and covariance across many assets. It represents each asset’s return as exposures to a smaller set of common factors plus an idiosyncratic component. When the factor count…

The document poses a question about whether a discounted stock price that is a martingale under a risk-neutral measure must fail to be a martingale under the actual or physical measure. It frames the issue in a Black–Scholes setting, with a geometric…

The document asks whether the probability law of a Brownian motion with constant drift is the same as a measure under which that drifted process becomes standard Brownian motion. It correctly distinguishes the process’s original law from the changed measure:…

The document works through the conditional expectation of the squared value of a Brownian motion at a later time, given its filtration at an earlier time. It expands the later value into the known earlier value plus an independent increment, then asks why…

The document examines definitions of a market-making strategy in a supervised-learning context. The cited definitions describe a strategy as a time-dependent vector of quotes selected using a prediction, and a market-making strategy as paired bid and ask…

The document asks how to incorporate company size when constructing the Fama-French winner-minus-loser (WML) equity factor. It notes that winner and loser groups are formed using cumulative returns over the period from month t-12 to t-2, then raises a timing…

The document asks which probability measure should be used to calculate the expected discounted payoff of a replicable claim that lies below a possibly nonreplicable contingent claim. It presents an equality between a supremum over such replicable claims and…

The document contrasts two approaches to pairs trading. One estimates a hedge ratio between nonstationary price levels so their linear combination is stationary, then models that spread as autoregressive or mean reverting. The other combines asset returns,…

The document defines a credit spread from the prices of defaultable and risk-free zero-coupon bonds, then relates default risk to a hazard rate and survival probability. It also introduces a structural company valuation in which equity resembles a call…

The document asks for books that help quantitatively trained readers connect mathematical finance models with real-world practice. The responses offer a range of references across investments, portfolio theory, corporate finance, financial econometrics,…

The document distinguishes basis between curves from the use of OIS discounting and explains how curve differences can affect swap values. It describes basis spreads as rate differences between curves, such as tenors or an interbank curve and an OIS curve.…