คลังความรู้

The answer describes how to derive a forward rate from a simulated Hull–White short-rate path. At each simulated future time, use the current short rate in the model’s zero-coupon bond pricing expression, with the model’s time-dependent A and B terms, to…

The exchange clarifies that CME’s term SOFR figures are not simply another observation of the overnight rate. They are forward-looking term rates implied by trading in SOFR futures. This explains why the benchmark page can show rates for horizons such as six…

The discussion shows that the number of stocks worldwide depends on how the universe is defined and which source is used. It cites MSCI global index factsheets as a way to estimate investable coverage: one broad index included large, mid, small, and micro…

The discussion explains why researchers normalize financial data: transformations can make measurements from different assets or return series more comparable and can simplify some mathematical or computational tasks. It emphasizes choosing a method to fit…

The document raises a question about constructing a variance-swap replication portfolio from options. It describes an apparent jump in implied volatility when the option wing switches from puts below a strike to calls above it. The quoted explanation is that…

The document investigates whether an FX option’s time to expiry in an at-the-money delta-neutral strike calculation includes hours and minutes or counts only whole days. It applies the standard forward-based strike formula to a one-day example, using the…

The document reconciles several formulations of Durrleman’s condition for preventing butterfly arbitrage in an implied volatility surface. It explains how expressions differ when volatility is represented as raw volatility, total volatility, or total…

The document compares ways to choose the drift parameter in a recombining binomial tree. It describes the Cox–Ross–Rubinstein construction, which uses symmetric log-price steps, and an alternative that adds a fixed drift to the up and down multipliers. It…

The document asks how to price a one-year call when a stock can move to either of two values, given a risk-free rate and equal real-world probabilities. It uses a one-step binomial model: calculate the call payoff in each state, form a replicating hedge, or…

The document examines whether returns should be normalized after converting price observations into returns. The questioner notes that returns are already relative changes and asks what further normalization by a measure of prior performance accomplishes,…

The document asks why volatility models often treat out-of-the-money calls and puts as a group even though their implied volatilities can differ. It identifies liquidity as a reason for focusing on OTM contracts and notes that OTM puts commonly carry higher…

The document explores whether financial asset fungibility is an all-or-nothing property or can depend on selected attributes. It uses hypothetical digital currency tokens to illustrate how issuer, currency symbol, decimal precision, and encumbrances might…

The document explains how Vasicek’s single-factor model uses a shared economic variable and firm-specific residual risk to represent default. A firm defaults when its latent asset variable falls below a threshold. The unconditional probability of default…

The document clarifies how gross basis relates a bond’s cash price to a bond futures contract. It gives the relationship as bond price minus futures price multiplied by the bond’s conversion factor. This frames gross basis as the difference between the bond…

The document asks why an American call on a stock paying continuous dividends has a finite stock price above which early exercise is optimal. It contrasts the option’s value, which cannot fall below intrinsic value, with the possibility that the two values…

The document asks how to interpret the Hansen-Jagannathan bound, which relates the volatility of a stochastic discount factor to the Sharpe ratio of excess returns. The question focuses on constructing a lower bound on discount-factor volatility when its…

The document raises a question about why short-term Treasury bill rates are used in research to forecast the equity premium, while acknowledging that their predictive significance is disputed. It asks whether the rate might reflect firms’ borrowing costs and…

The document asks why correlations, regression coefficients, and R-squared change when monthly observations are expanded into daily observations using linear interpolation. Its central point is that the computed correlation depends on the observations…

The document compares a two-outcome Kelly formula, expressed using win and loss payoffs, with the approximation that relates the optimal fraction to expected return divided by return variance. It asks why the variance term does not appear to match the payoff…

The document asks how to infer zero-coupon discount factors from clean and dirty prices of coupon bonds with annual coupons and staggered maturities. It proposes expressing a bond's dirty price as the sum of its coupon and principal cash flows, each…

The document discusses maintaining interest rate curves built from swap and other market data, with particular attention to calibration failures and kinks. It recommends keeping user-adjustable configuration manageable, supported by instrument symbology and…

The document explains how to convert a perturbative approximation for a vanilla call's equivalent Black volatility into an expansion centered at the midpoint between the current forward and strike. The original approximation expands the local volatility…

The document describes a daily-rolled backtest of EURUSD options using one-month and two-month straddles, 25-delta risk reversals, and 25-delta butterflies. Each position is held for one trading day, with the next day's valuation using the full volatility…

The document works through the present value of an annuity due over twelve years when payments are made in four installments per year and the annual interest rate is 2%. The replies clarify the assumptions needed to interpret the problem: each annual payment…