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 discusses implementing Carr–Madan Fourier pricing for European calls under Black–Scholes. It outlines a characteristic function, a damped transform, and a fast Fourier transform approximation using Simpson weights. The author reports that an…
The document addresses how to interpret the yield of a forward bond price when the available input is a zero curve quoted with continuous compounding. The example concerns a European put on a coupon bond and a relation that uses duration to translate forward…
The document examines how to combine the delta, vega, and theta of a long at-the-money SPX put and a short at-the-money VIX call. It gives example option Greeks and a historical VIX-to-SPX beta, then applies that beta to estimate a combined delta. The author…
The document responds to a student seeking manageable machine-learning project ideas with two research directions from quantitative finance. The first is deep hedging for risk management, along with data-driven estimates of option Greeks such as delta.…
The document raises a question about the relationship between the familiar gamma contribution to the P&L of a delta-hedged position and a proposed calculation based on the change in delta during an underlying price move. It presents the gamma-and-theta…
The document asks whether familiar models for an underlying asset can be derived from assumptions about the discounted profit and loss of a delta-hedged European option. It writes the hedged portfolio P&L in terms of the option payoff, initial premium,…
The document frames European call pricing on a finite binomial stock tree as a superhedging problem. With zero interest, the goal is to find the least initial wealth that can fund a bond-and-stock portfolio, adjusted at the tree’s nodes without later cash…
This discussion identifies markets where Asian options may trade actively and where historical prices might be available. It points to options on federal funds futures: because the futures settle against the average effective federal funds rate over a…
The document gives a brief example of trading volatility with options: buy or sell an out-of-the-money strangle, consisting of a call and a put, and delta hedge the combined position using the underlying asset. The hedge is intended to reduce directional…
The document discusses an approximation that relates implied volatility for a spot and strike pair to the average of local volatility across the price interval between them. The motivating explanation is that paths ending in the money may spend much of their…
The document derives a local delta-gamma hedge for a target option using a second option and the underlying asset. It writes the hedge as a combination of the underlying and the second option, then matches the target option's gamma by setting the second…
The discussion distinguishes a linearly interpolated swaption volatility cube from a cube generated using SABR. Linear interpolation of market quotes can describe a static set of Black-model volatilities across strikes and maturities. SABR instead models…
The discussion asks why some state-contingent prices in an example from Ross's Recovery Theorem exceed one. A response offers an intuition: a guaranteed payment in a particular future state can be highly valuable when that state involves catastrophe or when…
The discussion surveys possible indicators of market tail risk and financial stress. It points to the St. Louis Fed Financial Stress Index as a public measure, mentions sell-side stress indicators, and describes research proposing VVIX—the volatility of VIX…
The document presents several ways to estimate theoretical option prices when a new underlying has little or no options-market history. As a basic starting point, it suggests using historical return volatility as a constant volatility for near-the-money…
The document asks how to interpret the payoff of a European floating-strike lookback call. It describes the strike as the maximum stock price reached over the option’s life and questions whether the payoff must therefore be zero or negative at maturity. This…
The document derives the probability that a geometric Brownian motion stays below a barrier over a time interval, conditional on its terminal price. A direct Brownian-bridge formula seems tempting after taking logarithms, but the resulting Brownian-motion…
The document discusses how to infer which participants may be market makers in an options order book. Visible traits such as quoting at the best prices, balanced bid and offer size, repricing with the underlying, or using non-round ticks can offer clues, but…
The note explains how to choose volatility-surface inputs when calibrating a model for autocallable products. Its central recommendation is to calibrate one model to the autocalls held in the portfolio, reflecting that hedging is generally managed at…
The document derives a Fourier representation for pricing a European option under risk-neutral valuation. It identifies the payoff as a function of terminal log price and explains why a call payoff does not have a conventional Fourier transform: it grows…
The document raises a currency-convention question about the delta of a USD-denominated USDJPY option when the holder measures value in another currency. It distinguishes the option’s sensitivity to USDJPY spot, expressed as the derivative of its dollar…
The document considers a fund guarantee that pays the customer the greater of the initial investment or the fee-adjusted fund value at maturity. The answer rewrites the payoff as the initial amount plus a call option on the fund, with a strike increased to…
The document explores how a Black–Scholes-style option equation changes when the hedged portfolio is assumed to have its own stochastic diffusion. Starting with geometric Brownian motion for the underlying, it applies Itô’s lemma to the option value and…
The document explains how to interpret opposite moves in implied volatility for options expiring on adjacent dates. It emphasizes that implied total variance, rather than implied volatility alone, is the quantity to compare across expirations. Total variance…