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.
12,303 documents
The document discusses how often financial institutions recalibrate the SABR model when pricing FX options. Its practical guidance is that calibration is typically performed daily, with additional intraday recalibration sometimes needed on especially…
The document explains how the covariance of two continuous Itô processes follows from the covariance of their Brownian drivers. Drift terms do not contribute to their quadratic covariation; the stochastic components determine it. Applying the covariation…
The document considers a derivative payoff that equals the underlying’s initial price above a threshold and the underlying’s maturity value at or below it. Because the payoff jumps at the threshold, the author asks how to approximate it using a constant, a…
The document describes an attempted QuantLib approach to pricing callable, fixed-rate Danish mortgage bonds with and without amortization. It extends a callable fixed-rate bond class to build coupon cash flows from a schedule of changing notionals,…
The document considers whether a volatility exposure can be approximated by buying at-the-money S&P 500 straddles and delta-hedging them, potentially as a volatility position or a spread against VIX futures. It also describes the William’s VIX Fix, a…
The document explains how to identify candidate pairs for trading relative moves in equity-option implied volatility, even when the individual volatility series are already stationary. Cointegration is not essential: the relevant question is whether a spread…
The document outlines a method for formulating perpetual American option valuation as a free-boundary problem. In the continuation region, where holding the option is more valuable than exercising, the value satisfies the perpetual Black–Scholes differential…
The document addresses which implied volatility to use in a convexity adjustment when valuing a Libor-in-arrears swap cash flow. The relevant volatility comes from an option whose expiry matches the rate reset date and whose underlying rate covers the…
Option pricing models can support trading in several ways. A market maker can use a model to estimate option sensitivities, hedge exposures, and seek returns from the bid–ask spread. A risk taker can compare model values with market prices and trade options…
The discussion distinguishes two quantities often called forward implied volatility. One is calculated from the change in implied variance between two maturities for vanilla options at a fixed strike. It represents a forward measure inferred from today’s…
The document examines an approximation used in deriving a convexity adjustment for a bond forward. It expands the bond price as a function of yield around today’s forward yield, then asks why the expected squared yield change can be approximated by the…
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 discussion evaluates fitting an Ornstein-Uhlenbeck process to daily EUR/USD observations using maximum likelihood. The estimate of mean reversion is close to zero, while the reported stationarity test does not reject a non-stationary series. Since the…
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 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 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 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 combine univariate GARCH volatility models with a copula to estimate expected shortfall for a portfolio of three stocks. It considers whether to obtain conditional volatility by filtering observed returns through each fitted GARCH…
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 explains that left and right tail risk refer to extreme outcomes on opposite sides of a return distribution. The left tail represents unusually poor returns, while the right tail represents unusually high returns. Analysts examine these outcomes…