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 risk-measurement question for quanto-style options on commodity forwards whose payoff is converted through foreign exchange. It gives a composite-volatility formula that combines the underlying commodity volatility, FX volatility, and…
The document presents a question about how SABR option delta changes when the model is parameterized by alpha versus at-the-money implied volatility. It outlines the chain-rule terms in each delta expression and focuses on the adjustment to alpha required to…
The document derives a risk-neutral value for a derivative that pays a power of the underlying asset price divided by a fixed strike at maturity. It starts with the discounted conditional expectation of the payoff, then assumes the asset follows a…
The document examines why an option chain can show different implied volatilities for a call and put at the same strike, despite the relationship between out-of-the-money puts and corresponding in-the-money calls under put-call parity. For American-style…
The document addresses how to view historical or intraday prices for options on E-mini S&P futures, when readily available futures charts do not provide the desired options data. It notes that some sites show current option prices but may not offer price…
The document describes a difficulty matching implied volatilities for SPY calls and puts across strikes when using Black–Scholes with dividends. Matching the at-the-money vols by calibrating rates and dividends did not remove a discrepancy that widened…
The document poses a risk-neutral pricing problem for a claim paying the positive part of the difference between two stocks at maturity. It rewrites the payoff as the second stock multiplied by a call payoff on the ratio of the first stock to the second,…
The note examines whether an arbitrage-free implied volatility surface is equivalent to a well-defined local volatility surface, and explains why the equivalence fails in edge cases. A discrete martingale model can produce arbitrage-free option prices while…
The document asks where the familiar SABR implied Black–Scholes volatility expression comes from. The replies point to two related derivation routes: a heat-kernel expansion connected with differential geometry, and the derivation in Hagan and coauthors’…
The document addresses whether standard interest rate option models remain useful as markets move away from LIBOR toward SOFR. Its answer is that SABR and LIBOR Market Model approaches are expected to continue in modified forms, rather than being abandoned…
The document explains why option strike availability varies by exchange, underlying, and expiration. Exchanges generally have rules for strike spacing and schedules for adding expirations, while new strikes may be introduced as the underlying price moves.…
The document explains why forward variance swap quotes are expected to have no pricing drift under an equivalent martingale measure. It compares a forward variance position with a futures position: if a self-financing strategy can be entered at zero cost and…
The document considers a rule of thumb for estimating one-day profit and loss on a delta-hedged option position from daily theta and the underlying’s move measured in standard deviations. The question’s proposed expression is compared with actual P&L and the…
The document addresses whether an option must be worth more than its intrinsic value and whether a high-priced option will be difficult to sell. Its answer focuses on the first question: a call can trade below intrinsic value when dividends and other pricing…
The document explains a central distinction in risk-neutral option pricing: expected stock return is not an explicit input to a binomial option model, while volatility is. Under the no-arbitrage replication framework, the underlying price and risk-free rate…
The note explains why a chart might show an implied volatility curve and a flat volatility curve crossing at the at-the-money point. The crossing follows from how the illustration was constructed: the two example curves were chosen to have the same ATM…
The discussion explains a volatility smile as the pattern formed by plotting options’ implied volatilities against strike prices for a common expiry. Unlike the flat volatility assumption in Black–Scholes, market prices often imply higher volatility away…
The document examines the limit of a European call’s Black–Scholes price as time approaches expiration, assuming positive volatility. It writes the standardized variables in terms of time to maturity and considers the three relationships between the…
The document asks whether the single implied volatility shown alongside an option’s bid and ask prices is calculated from either side or from a representative price. The accepted response says that data sources usually calculate it from the bid–ask midpoint,…
The document raises a derivation question for an option that pays the positive difference between two assets at maturity. Starting from the risk-neutral discounted payoff, it splits the payoff into two asset-weighted indicator terms and applies a change of…
The document poses an expiry-day hedging problem for a slightly out-of-the-money option on an illiquid equity. It highlights a practical execution concern: buying the underlying to offset delta could move the market, potentially changing the stock price…
The document asks whether implied volatility has a strict, model-independent definition or is instead obtained by inverting a chosen option-pricing model. It focuses on equity and index options, contrasting European options on a dividend-paying index with…
The document frames a numerical problem from quantitative finance: differentiating a discrete sum with respect to a continuous upper limit to estimate first- and second-order sensitivities. It gives an exponentially decaying summand with a fractional-power…
The document derives the time-zero value of a European chooser option that lets its holder choose at an intermediate date whether the contract will be a call or a put, with a common strike and later expiry. At the choice date, the option is worth the greater…