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.
5,867 documents
The document derives an approximate single implied volatility for a portfolio of options whose components have different implied volatilities. It begins with the condition that the portfolio’s modeled value at the common volatility should equal the sum of…
The document compares two martingale derivations of the Black–Scholes partial differential equation. With the bank account as numeraire, requiring the discounted option price to have zero drift yields the familiar PDE. The attempted stock-numeraire…
The document offers historical volatility and correlation estimates as starting points for a foreign currency option model with domestic equities, foreign equities, and an exchange rate. Using weekly observations over five years for the DAX, S&P, and EUR…
The document explains why the delta of a binary call becomes sharply concentrated around its strike as expiry approaches. Under Black–Scholes, the option value is expressed using the normal cumulative distribution function, and differentiating gives a delta…
The document derives a European call pricing representation for an asset whose returns combine continuous Brownian movement with independent Poisson jumps. When jump sizes are lognormally distributed, conditioning on the number of jumps makes the terminal…
An implied volatility surface reflects option prices that vary by strike and maturity, unlike the constant volatility assumption in the basic Black–Scholes model. Looking at one maturity at a time, a steep downside wing means out-of-the-money puts are…
The discussion asks whether manipulation of SPX options or equity and volatility futures caused the February 2018 VIX spike, and what data could help investigate. The response points to volatility-linked exchange-traded products as a possible source of…
The document asks why an American put can have a different value from a European put when both are considered under the Black–Scholes framework. It contrasts the pricing inequality and payoff constraint for an American option with the familiar result that,…
The document derives a closed-form price for a European payoff based on the positive part of one minus the strike divided by the terminal stock price, assuming the stock follows geometric Brownian motion under the money-market measure. Its key observation is…
The document explains how to simulate terminal prices for several assets whose returns are correlated, in order to value a multi-asset option by Monte Carlo. In a geometric Brownian motion model, dependence is specified through correlations among Brownian…
The document describes a backward path-integral scheme for pricing an American put on a log-price grid. At each time step, it discounts and integrates the next-step option value against a Gaussian propagator for log prices, then applies the early-exercise…
The document frames a model-selection problem for reinforcement-learning-based dynamic hedging of long-dated swaptions. The proposed application uses 2y2y and 4y2y swaptions, requiring simulated paths that update both a forward swap curve and an implied…
The document shows how to rewrite a European put’s discounted expected payoff as an integral of the underlying asset’s cumulative distribution function. Starting from the payoff integral over nonnegative asset prices, it extends the density’s support to the…
The document frames a research question about abrupt changes in index option prices near major expiration dates. It proposes systematic rebalancing by structured products as a possible source of price pressure and asks what other forces might contribute. The…
The document explains why a European put’s Black–Scholes–Merton value can fall below its immediate exercise payoff. A European option cannot be exercised before expiration, so when the underlying price is far below the strike, the eventual payoff is…
The document considers a weather-linked call whose daily payout depends on maximum temperature mapping to a quantity and a price index average exceeding a strike. The payoff also has daily and contract-wide payout limits, making a direct closed-form…
The document outlines a derivation of the Black–Scholes equation from the Capital Asset Pricing Model rather than from a risk-free portfolio formed by delta hedging. It starts from CAPM’s relation between expected return and covariance-based risk…
The document examines an option whose payoff and premium are expressed in the underlying asset, using an ETH example to compare conversion from a conventional Black–Scholes value with a direct simulation. The key issue is the payoff definition: converting…
The document describes why a digital option’s stock hedge changes sharply as the underlying approaches and passes its strike. A digital option pays a fixed amount when it finishes in the money and nothing otherwise, so its payoff does not rise gradually with…
Energy retailers that promise customers fixed prices while buying power or gas at floating wholesale prices face a mismatch between sales revenue and procurement cost. The risk can grow when demand and prices move together, as during cold weather. The…
A question reports unstable or negative option values when the step count in an FFT-based binomial calculation becomes large. The accepted response suspects numerical precision loss in the terminal stock-price calculation, which raises up and down factors to…
The document asks how to apply antithetic sampling when simulating the Heston stochastic volatility model with a discretized process. The central issue is whether to reverse the random draws only for the stock price or for both the price and variance…
The document presents a QuantLib Python calibration attempt for a time dependent Heston model that fails with a Boost assertion. The code builds a volatility surface, creates a piecewise time dependent model, attaches an analytic pricing engine, and…
The document discusses why an online broker may cap the number of legs in a single options spread order. Its main explanation is that brokers submit orders using structures recognized by options exchanges, and the permitted order formats and applicable rules…