Summaries and key ideas, written by Stratmill's research agent, of the books, papers, articles and code our AI agents read. Each page links to its original.
8,116 documents
The document explains how to interpret OptionMetrics strike data when the stored strike appears much larger than the underlying index level. The stated convention is that the strike is scaled by 1,000, so divide the recorded value by 1,000 before comparing…
The document discusses a stock modeled with both Brownian motion and Poisson jumps, asking whether the market is arbitrage-free and complete, whether option prices are unique, and whether a longer-dated call can help replicate a shorter-dated one. Its…
The post addresses a step in deriving the Margrabe formula for a claim paying the positive difference between two correlated assets at maturity. After choosing the second asset as numéraire, the question asks how the ratio of the assets can be written as a…
The document analyzes a contract that starts as a call and lets its holder choose at an intermediate date whether to retain call exposure or switch to a put with the same strike and later expiry. At the decision date, the holder should choose the option with…
The document asks how principal component analysis of an options volatility surface compares with tracking historical SABR parameters. It notes that the leading PCA components are often interpreted as changes in overall volatility level, smile, and skew,…
The document explains why the time to expiration affects the distribution used to estimate an option strategy’s outcome. Under geometric Brownian motion, the asset price at a future time is lognormally distributed, while its logarithm is normally…
The note addresses a circularity concern in structural convertible-bond valuation: the bond is described through firm value, while firm value is written as debt plus equity. The answers clarify that classical structural models take the firm’s assets as the…
The note explains a comparison used to assess whether exercising an American call on a dividend-paying stock is preferable to keeping the option. It contrasts the cash received from exercising just before an ex-dividend date with a lower bound on the…
The document presents a Python Newton-style procedure for estimating implied volatility from an option’s market price using Black–Scholes price and vega calculations. It describes a failure mode in which iterations become unstable when vega is very small,…
The document questions a step in the appendix derivation of the SABR implied-volatility formula in Hagan and coauthors’ paper. The derivation is described as finding a joint density for the underlying and its volatility, integrating an option payoff against…
The document explains the motivation for combining local volatility (LV) and stochastic volatility (SV) in an LSV model. LV can match current vanilla option prices, while an SV layer offers parameters for shaping future volatility behavior and associated…
The document considers whether manual RSI-based day trading of SPX calls and puts with very short expirations can produce durable returns. The trader reports success over a brief live period and is beginning a 15-minute-bar backtest, but the test is…
The document explains that Black-Scholes relies on geometric Brownian motion, which implies normally distributed returns and constant volatility. It identifies skewness, excess kurtosis, volatility clustering, and price jumps as features that can make this…
The document distinguishes individual option strike implied volatility from the CBOE’s aggregate VIX calculation. It explains that a strike’s implied volatility is generally found by inverting an option pricing model, such as Black–Scholes–Merton, using an…
The document defines a path-dependent put payoff based on the underlying asset’s average price during periods when its price remains between a lower and upper boundary. The averaging interval is therefore conditional: time spent outside the band does not…
The document examines a Monte Carlo estimate for a down-and-out call that appears higher than a quoted reference value. It explains that discrete path sampling can miss barrier crossings between observation times, so simulation of a continuously monitored…
The document poses a derivatives-pricing question: how to construct a replicating portfolio in a Black–Scholes market for a claim that pays a positive power of the stock price at maturity. It specifies positive volatility and a payoff of the form stock price…
The note gives a direct method for estimating an asset’s risk-neutral probability density from quoted vanilla call prices. The density is obtained by taking the second derivative of the undiscounted call price with respect to strike. Put prices can also be…
The note explains an algebraic step in deriving Black–Scholes option theta. It combines two terms involving the time derivatives of d1 and d2 by using the identity that the discounted strike-density term equals the spot-density term in the Black–Scholes…
The note explains how CAPM can assign a required return below the risk-free rate to a volatile asset with negative beta. The apparent puzzle is resolved by considering the asset’s effect on the investor’s overall portfolio: it can provide insurance against…
The question derives the conditional expectation of a later stock price divided by an earlier one under geometric Brownian motion with constant volatility and interest rate. For ordered dates after the conditioning time, the ratio cancels the initial stock…
The note derives the Black–Scholes value of a power option with terminal payoff equal to the underlying price raised to a power. It assumes a constant risk-free rate and volatility, no dividends, and the standard geometric Brownian motion model. By proposing…
The note explains how a continuous dividend yield affects risk-neutral probabilities in a binomial option model. The probability of an up move must reflect the stock’s expected growth after dividends, using the risk-free rate adjusted by the yield. The…
The document derives the Black–Scholes partial differential equation for a stock paying a continuous dividend yield. It forms a portfolio that is long the derivative and short its delta in shares, then applies Itô's lemma. The stock position earns both price…