คลังความรู้

The document compares the cost of obtaining leveraged exposure through a leveraged ETF with borrowing to buy a standard ETF. It raises financing rates and fund expenses as relevant costs, then highlights a distinct issue: daily leveraged products can lose…

The document examines how storage costs enter a commodity forward price and whether monthly costs should accrue interest. It presents the no-arbitrage principle: a forward should reflect the spot asset and the present value of carrying costs, financed to…

The document explains why a historical return histogram may compare simple returns with a normal distribution even though geometric Brownian motion implies normally distributed log returns and lognormally distributed prices. For small price changes, the…

The document outlines a workflow for evaluating a portfolio under non-parallel interest rate shocks. Start with specified scenarios, apply the tenor-specific shifts to the relevant curves, reprice affected instruments, and measure the resulting price…

The document asks how the Merton model’s debt payoff can be written both as debt face value minus a put payoff and as the lesser of face value and asset value. At maturity, debt holders receive the firm’s assets when those assets fall short of the promised…

The note addresses whether negative risk-free rates invalidate the CAPM or the use of a risk-free rate when calculating market excess returns in the CAPM and Fama–French three-factor model. Its answer is that the CAPM does not require the risk-free rate to…

The note considers cross-sectional estimation of factor returns when asset-loading columns are linearly dependent. Ordinary least squares then has no unique solution because the loading matrix’s cross-product is singular. A market factor combined with…

The note distinguishes an up-and-in binary call from a down-and-out binary call and explains their payoffs under a constant-strike barrier. For the up-and-in case, a terminal price above the strike implies that the price must have crossed that strike before…

The note explains why UK RPI zero-coupon inflation swaps can trade at different levels when cleared through LCH versus bilaterally. It identifies two conditions behind such a basis: dealers face an imbalance in client demand across the two markets, and…

The document sets up a portfolio containing an option, the underlying stock, and a second asset whose value depends on both the stock and variance. The stock and variance follow correlated stochastic differential equations. The question is why the…

The note asks whether an American put’s time derivative, or theta, is continuous where the optimal exercise boundary meets the continuation region. It distinguishes this question from the familiar smooth-pasting condition, which says the option’s delta…

The document outlines a proposed workflow for applying an Ornstein–Uhlenbeck process to a pair of potentially cointegrated stocks. The suggested steps are to estimate a linear relationship by OLS, form a residual or auxiliary series, fit an AR(1) model to…

The note explains the intuition behind Hull’s statement that a U-shaped implied volatility smile corresponds to a distribution with more probability in both small and extreme moves than a lognormal distribution. Higher implied volatility away from…

The document examines why adding risky assets can move the mean–variance frontier toward lower volatility without making it approach zero in every case. For equal-variance, uncorrelated assets, it considers an equally weighted portfolio and shows that…

The document asks why an interest rate swap may use separate curves for projecting floating coupons and discounting future cash flows. It identifies the pricing curve with the curve used to infer rates for floating-leg accruals, then asks whether a different…

The document asks how to price derivatives when the underlying follows an arithmetic Brownian motion rather than the geometric Brownian motion used in the Black–Scholes model. It starts with an additive diffusion under the physical measure, then proposes…

The document asks how to estimate a company’s pre-tax cost of debt when its borrowings include bonds with different coupon rates, maturities, and repayment schedules. The proposed method is to calculate the yield on each bond for the relevant fiscal year,…

The document considers how to calculate stock price momentum from weekly observations and whether short lookbacks, such as several weeks, are standard. It presents a simple measure based on the difference between the current closing price and the close a…

The document discusses converting a quoted three-month Treasury bill rate into a weekly risk-free return and subtracting it from an asset’s weekly log return. It endorses scaling the log of one plus the quoted annual rate by the weekly fraction of a 360-day…

The document explains how to price a zero-coupon bond when yields differ across maturities. Its key distinction is between a bond’s maturity-specific zero-coupon spot rate and the sequence of one-year forward rates. Under annual compounding, discounting the…

The document examines negative variance-equation intercept estimates in a GJR-GARCH model with a pandemic indicator as an exogenous input. The response writes a simplified model with a regime indicator and an asymmetric term for negative shocks, then argues…

The document explains how to attribute a portfolio’s volatility to a subset of its assets, such as an equity sector, while accounting for the portfolio covariance structure. It defines an asset’s marginal risk contribution as the change in portfolio…

The document asks how to generate observations from a four-factor t-copula when the dependence matrix changes over time. Its setting is a series of 2,000 correlation matrices from a copula-DCC model, with one matrix intended for each time step. The accepted…

The document explains how to select interpolation schemes when bootstrapping yield curves in QuantLib-Python. It shows a dual-curve setup using deposit, forward rate agreement, and swap helpers, then constructs curves with different interpolation choices,…