คลังความรู้

The note addresses whether a rates Bermudan callable or swaption remains convex as its strike changes, even though possible exercise dates correspond to swaps with different remaining maturities. It represents the exercise payoff at a given date as the…

The note explains why an exponential moving average can be matched to an N-period simple moving average by equating their average data ages. EMA observations k periods old receive weights αβ^k, where β = 1 − α. The simple moving average has an average age of…

The document distinguishes financial bars from statistical resampling methods. Time bars divide observations by clock intervals, while tick, volume, and dollar bars aggregate irregular transaction data until a chosen trade count, traded volume, or traded…

The document asks whether residuals from regressing equity returns on principal component factors can support a mean-reversion strategy, given the common regression assumptions of normally distributed, independent errors. The response clarifies that fitting…

The document discusses how to express the Heston stochastic volatility pricing equation using a forward price, log-moneyness, time to expiry, and an undiscounted option value. It gives the spot and variance processes, their correlated Brownian shocks, and a…

The document considers how to model hedge fund returns for simulation, noting that return moments alone do not specify a distribution. One proposed approach represents returns as exposures to a broad set of tradable ETF factors. The cited research uses…

The document explains why the Black–Scholes delta of an in-the-money call can first decline and later increase as volatility rises. Interpreting delta as the risk-neutral probability of exercise, it separates two effects: greater volatility makes the strike…

The document discusses how to convert daily US Treasury volatility to an hourly measure and cautions against assuming that trading activity is evenly distributed across the full day. Treasury trading is described as over the counter, without official…

The document examines how a Black–Scholes call price approaches its terminal payoff as time to maturity shrinks to zero. It rewrites the price into a term involving the discounted intrinsic-value difference and a second term expressed as an integral of the…

This worked example uses two risky stocks, their volatilities and correlation, and a risk-free asset under CAPM. It distinguishes the market portfolio from the minimum-variance portfolio: market capitalization determines the market weights, which are then…

The Snell envelope is explained through a sequence of choices where each decision may be taken only when its opportunity arises. In the highway example, each exit represents a stopping time, and the value at a given point is the best outcome still achievable…

The document compares the continuous-time Ornstein–Uhlenbeck (OU) process with the discrete-time AR(1) model for stationary, mean-reverting data such as a spread. It presents OU as a continuous-time model with an analytic solution and explains that a simple…

The document explains how to find the probability mass under a normal distribution between a lower and upper bound. It uses the cumulative distribution function (CDF), which gives the accumulated probability from negative infinity up to a chosen value.…

The document defines bullish RSI divergence as price making a lower low while RSI makes a higher low, and bearish divergence as price making a higher high while RSI makes a lower high. These patterns indicate that RSI is not confirming the latest price…

The document introduces Fischer Black’s idea that the nominal short rate can be viewed as an option: it is the greater of zero and the sum of a shadow real interest rate and inflation. The shadow rate is described as tied to the investment opportunity set,…

The note explains how delta-hedging error arises when an option is valued using an assumed volatility that differs from the underlying asset's true volatility. The option's Black–Scholes pricing equation links its theta to its price, delta, gamma, interest…

The note asks why options on leveraged exchange-traded funds can appear cheaper than options on an unleveraged fund tracking the same index and expiry, despite an expectation that leverage should multiply implied volatility. It explains that a daily…

The document considers how to convert independent country signals into equity portfolio weights while preserving a neutral allocation and bounding each country's deviation. Its proposed construction treats the signal vector as a point in a multidimensional…

The note addresses why a call spread built from two outer strikes does not create arbitrage simply because its price exceeds the price of an intermediate-strike call. The intermediate call is cheaper than the weighted combination, so buying it and selling…

The document compares two numerical approaches for pricing European calls under the Heston–Nandi GARCH model using an inversion integral. One implementation evaluates the integrand through adaptive quadrature, while another precomputes…

The document describes a way to extend the standard top-of-book microprice to several price levels. At each level, it weights the ask price by bid-side size and the bid price by ask-side size, then divides the combined weighted prices by total depth across…

The document outlines practical approaches to estimating the value of a corporate bond in default. Distressed-debt analysis can consider possible bankruptcy or restructuring recoveries, including cash, replacement debt or equity, and other securities. Those…

The document explains how interest enters the overnight profit of a delta-hedged option position. It represents the portfolio as a short option combined with a position in the underlying sized by the option’s delta. The net portfolio value determines whether…

The document considers how to display an S&P 500 futures moving average as a comparable reference level on a Nasdaq futures chart. A direct price-ratio adjustment is proposed by the questioner but is reported not to work as intended. The answer instead…