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 raises a forward-pricing problem for options on an ETF that cannot be shorted and pays a single annual dividend. The author is fitting an implied-volatility curve and finds that the nearby future trades below spot, while the options imply a…
The document considers a Black–Scholes stock and a transformed payoff based on inverting the stock price around a fixed level. It introduces a measure associated with a power of the stock and states a pricing identity under that measure. The question is…
The document poses a pricing problem for an American-style cash-or-nothing call: the holder receives a fixed cash amount if the stock first reaches a specified level above its current price. It asks whether the option’s early-exercise feature makes it more…
The document asks whether the Black–Scholes partial differential equation for a continuously sampled arithmetic-average Asian option can be transformed into a heat equation. Its answer argues that the added average-state term makes the problem harder than…
The note works through small- and large-volatility asymptotics for an option price expression involving the normal cumulative distribution function. It uses a Gaussian tail expansion, in which the cumulative probability approaches a step-function limit with…
The document clarifies the meaning of delta hedging a short option in a one-period binomial model. A dealer who sells a derivative is short its value and can offset that exposure by taking a position in the underlying. The delta formula gives the underlying…
The document considers how two parties might price an over-the-counter option when there is no active market to reveal an implied volatility. It distinguishes factors that can be hedged from those that cannot: market prices can inform assumptions for…
The document explains how dividends enter the Black 1976 model for options on futures. For a typical index future, the futures holder does not receive the cash index’s dividends. Instead, the continuously compounded dividend yield and interest rate affect…
The document explains the risks a protection seller takes on in a single-name credit default swap and the limits of hedging that exposure. Selling protection leaves a position short credit spread; there is no generally available instrument that isolates and…
The document examines an attempt to explain option price changes by summing delta and gamma contributions across frequent observations. The proposed estimates use average sensitivities over each interval, but the author finds that the resulting total does…
The document presents a Heston option-pricing calibration implemented as a two-stage optimization: a genetic algorithm searches within parameter bounds, then an interior-point constrained optimizer refines the candidate. The objective compares market call…
The author proposes an alternative foundation for option and return modeling built around uncertain security parameters, rather than assuming those parameters are known. Returns are treated as statistics derived from price and volume observations, with a…
The note works through a derivation linking the discounted risk-neutral expectation of a European-style payoff to the Black–Scholes partial differential equation. It introduces an auxiliary function H that satisfies a drift-diffusion equation with a terminal…
The note considers why a trader might roll a long call to a higher strike after the underlying rises, especially when a quick reversal is expected. It frames the issue through a proposed inequality for Black–Scholes call prices, but the response does not…
The document compares local volatility (LV) and stochastic local volatility (SLV) models, focusing on when the added stochastic factor may matter. LV can fit the current volatility skew, while stochastic volatility adds a representation of how volatility may…
The document explains why a European option’s market price can fall below its immediate exercise value, and why such observations should not automatically be removed from an implied volatility dataset. For European options, positive interest rates can…
The document explains how to estimate value at risk for a portfolio containing options. In a Monte Carlo approach, simulate the underlying equity prices, reprice each option at every simulated state, and calculate VaR as a tail quantile of the resulting…
The document responds to a request for learning resources on calibrating common mathematical finance models, including Black–Scholes, stochastic volatility, jump-diffusion, and Hull–White models. It points readers to Jim Gatheral’s book as an accessible…
The document surveys applications of the Fourier-cosine, or COS, method in finance, focusing on option valuation and recovery of terminal price distributions. The method approximates a probability density with a cosine expansion whose coefficients come from…
The answer explains delta neutrality as an aggregate exposure condition: sum the delta contributions of all positions, accounting for each leg’s size, and offset the total with an appropriate trade. It frames delta as the sensitivity of portfolio value to…
Least Squares Monte Carlo (LSM) estimates the continuation value of an American option by simulating risk-neutral price paths and fitting regressions as it works backward through possible exercise dates. At each date, the estimated value of waiting is…
The document explains why mean reversion in a stock’s historical price process does not, by itself, change its no-arbitrage option value. Under the risk-neutral measure, the stock’s drift becomes the short rate while its instantaneous volatility remains the…
The answer contrasts two types of OTC options by their practical use. Asian options are common in foreign exchange hedging, where companies with recurring foreign-currency revenues may care about the average exchange rate over a period rather than the rate…
The discussion clarifies why foreign exchange volatility smiles are commonly quoted by delta rather than strike. Delta offers a maturity-aware way to describe an option’s position in the smile: the same strike can represent quite different delta levels at…