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The document asks why an American put’s value approaches zero near expiration in a specified region above its optimal exercise boundary, even though its terminal payoff is the positive part of strike minus stock price. It introduces time remaining to…
The document raises a diagnostic question about Jarque–Bera test values reported alongside lag-one autocorrelation, a Ljung–Box statistic, skewness, and kurtosis. The author observes that the Jarque–Bera statistic rises sharply when kurtosis is high and asks…
The document asks whether a risky asset modeled as an exponential of a Lévy process is well defined and has finite first and second moments. Its Lévy measure combines negative jumps at integer sizes, weighted by a geometric sequence, with positive jumps…
The document surveys ways to identify candidate pairs for a pairs-trading strategy. It describes a statistical route—searching across instruments for correlated price movements—and names cointegration as a commonly discussed selection approach. It also…
The document examines whether the covariance matrix supplied to R’s quadratic programming solver should be doubled when constructing a mean-variance efficient frontier. The question arises because the solver’s objective includes a one-half factor on its…
The document explains why the drift of log returns under geometric Brownian motion is lower than the drift of percentage returns. Starting from the stochastic price process, it uses a small-change approximation for the logarithm of one plus a return. The…
The document asks how to express a callable bond’s payoff and relate its value to a straight bond and an embedded call option. It presents a proposed payoff formula for a bond callable at a specified date and strike, but does not validate the formula; the…
The document raises a diagnostic question about a pair-trading setup: after using linear regression to estimate a hedge ratio, the residuals appear stationary but visually resemble the price series of the dependent asset. It asks what this appearance means…
The document surveys ways sine, cosine, and related functions appear in financial calculations. Fourier methods use trigonometric components in applications such as option valuation, value-at-risk, and time-series analysis. Trigonometric functions can also…
The document explains why Black–Scholes implied volatility can vary with an option’s strike, forming a smile or skew. It begins from the model’s assumptions of constant volatility and continuous price paths, then frames implied volatility as the…
The document asks whether a trader with limited capital can design a futures strategy that profits regardless of every possible sequence of price moves. Its answers connect the question to arbitrage and the assumption of no arbitrage in market models, and…
The document discusses a QuantLib bootstrapping error caused by combining Japanese yen overnight indexed swap market data from two trade dates in one curve. Instruments from different dates can share maturity pillars, triggering a duplicate-pillar error, but…
The document explains the stochastic-calculus shorthand that the square of a Brownian motion increment behaves like a time increment. It identifies this rule as a lemma associated with Itô calculus, rather than an arbitrary definition, and connects it to…
The document addresses how to calculate expected shortfall when portfolio outcomes have a discrete distribution. It clarifies that the quantile-integral definition remains valid for discrete and continuous outcomes because integration is over the probability…
The document explains the credit triangle linking a credit spread, loss given default, default intensity, and survival probability. Under constant intensity and recovery assumptions, the spread divided by loss given default gives the default intensity, and…
The document explains cash-settled futures convergence through arbitrage and the contract’s settlement mechanism. In a frictionless example, an investor buys the underlying while shorting a futures contract priced above it. At expiry, the futures cash…
The document explains how two expressions for the conditional short-rate distribution in the Hull–White model are related. Starting from the model’s stochastic differential equation, it gives the conditional mean and variance when the drift parameter is…
The document asks whether a currency-conversion identity used in pricing an option with a currency exchange feature holds generally. It describes two valuation routes for a payoff denominated in one currency: convert the payoff at maturity and discount in…
The discussion questions whether CAPM’s risk-return relationship is supported by comparing the high-beta SPHB ETF with SPY over a stated period. The figures show SPHB had higher annualized volatility but lower reported annual average and cumulative returns.…
The document gathers several starting points for learning to price options on weighted combinations of assets. Suggestions include studying a method that approximates basket, spread, and Asian options through sums of Black-Scholes-Merton models, which is…
The document discusses the difficulty of turning an inverse valuation ratio into portfolio weights when its inputs can be negative or near zero. One proposed approach is to maximize EBIT divided by enterprise value instead of minimizing enterprise value…
The document considers how to estimate a portfolio’s sensitivity to movements in one yield curve when positions also depend on correlated prices and rates. It suggests starting with assumed volatilities and correlations: shock the target factor, infer…
The document explains that the signed difference between an order’s average execution price and its arrival mid-price is usually treated as implementation shortfall. It presents an approximate model with a spread-related term and a market-impact term that…
The document discusses whether European sector stock data can support assessments of industry resilience, including univariate forecasts and multivariate models with macroeconomic variables such as inflation. The answer considers the idea workable but…