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 examines why an at-the-money European call and put on a non-dividend-paying stock can have equal value when interest rates are zero, despite their different-looking payoff diagrams. Its central explanation is that option prices reflect both…
The document addresses how to obtain intraday prices for an ETF when standard daily price retrieval is insufficient. It identifies Interactive Brokers as a possible data source through its API and says the API itself does not require payment, while noting…
The document explains how market-implied probabilities for Federal Reserve rate hikes can be estimated from fed funds futures. Under a simplified assumption that only two outcomes are plausible, the expected rate increase embedded in futures prices can be…
The document investigates how to reproduce CBOE’s discontinued single-name 30-day normalized skew field using historical options data. It describes the target as the 25-delta put implied volatility minus the 25-delta call implied volatility, divided by…
The document asks how to define an arbitrage-free price process for an American-style derivative when the buyer may make payments over its lifetime and choose a random exercise time. It proposes selecting a process so the supremum over exercise times of…
The example shows how put-call parity and relationships across option strikes can be used to infer some option prices from quoted market data. Ignoring discounting, the response applies parity at the 70 strike to the given futures level and straddle price,…
The document discusses background reading for someone struggling with a mathematical finance text. It frames Shreve’s work as a calculus-based treatment of finance and suggests building mathematical preparation before tackling it. For derivatives-focused…
The document asks how to judge whether a derivative-pricing model improves on Black–Scholes when traders already use implied volatilities and interpolation to fit option prices. It distinguishes pricing from prediction: a pricing model can interpolate or…
The problem asks for Black–Scholes prices of two terminal payoffs: the square of the stock price and a unit payment when the stock finishes below a strike. It applies risk-neutral valuation by discounting each payoff’s expected value. The squared payoff uses…
The document explains how options systems represent time remaining on expiration day. Rather than using whole-day buckets, pricing and risk software generally represents expiry tenors fractionally and may measure time internally in years, seconds, or…
The document distinguishes a caplet written on a forward Libor rate from one whose payoff uses the spot Libor rate observed at the option’s reset date. The forward-rate caplet can be expressed under the relevant forward measure as an option on a martingale…
The document asks how quickly the value of a Bermudan option approaches the value of an American option as exercise dates become more frequent. It describes the practical setting in which numerical methods use discrete exercise opportunities, and proposes…
The answer explains an intuitive link between a probability and the price of a binary claim. If an event pays one unit when it occurs and nothing otherwise, its expected payoff under the relevant pricing assumptions is the event probability. On that basis,…
The document poses a pricing problem for a path-dependent option on a geometric Brownian motion asset. Its European payoff depends on the running maximum and a time-averaged quantity formed from the integral of the squared asset price. The question asks how…
The document considers choosing a call option when a stock’s terminal price is forecast to be uniformly distributed over a specified range. Assuming the available calls are priced with the Black–Scholes model, the answer recommends the lowest available…
The document asks whether a claim’s price, computed as an expectation under the risk-neutral measure, can be used directly to construct a replicating strategy by holding its derivative with respect to the stock price. The claim pays the larger of a fixed…
The question considers an option that pays the difference between two stock prices at maturity only if stock A has remained above stock B throughout the contract. The accepted response reframes the condition by using stock B as the numeraire. In units of B,…
The document explains how to value a European put that knocks in at a lower barrier and knocks out at an upper barrier. It expresses the payoff using the terminal put payoff and indicators for whether the underlying has crossed the lower barrier and remained…
The document asks whether the effective parameters in a time-decay SABR model can be expressed by one closed-form formula across all values of the decay start time. It presents two results from Sander Willems’s backward-looking SABR work: one for decay…
The document explains that there is no standard rule for how many past days to use when estimating historical volatility. It distinguishes historical, or realized, volatility from implied volatility: option prices reflect the market’s view of future…
The note explains that options on volatility control indexes exist, even though they may not appear as exchange-listed contracts. It says these options are generally embedded in structured notes and are mainly traded over the counter. This provides a reason…
The note asks whether portfolio gamma and vega, like delta, can be calculated by summing the Greeks of each position. The answer says that in linear valuation models, prices and their Greeks add across positions. It specifically identifies Black–Scholes as a…
The note discusses numerical instability when using the Bjerksund–Stensland 2006 option pricing model at very low volatility, especially when solving for implied volatility on out-of-the-money options. It offers a practical parameter choice: use a minimum…
The document describes calculating Black–Scholes delta and vega in R for a sample of executive compensation contracts containing multiple options. The author derives the Black–Scholes call sensitivities and reports that computed vega values are zero,…