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
Dupire local volatility calibrates a deterministic function of time and spot to reproduce observed European option prices across strikes and maturities. This makes it convenient for matching the initial vanilla option surface, but the model ties volatility…
The response derives the price of a contract paying the squared log return at expiry. It first expresses the payoff as a function of the terminal asset price and values it as a discounted expectation under the risk-neutral measure. It then applies the…
The discussion explains a possible meaning of the phrase “variance swap volatility.” It interprets the quantity as the volatility level associated with a variance-swap strike that makes the contract have zero value at the relevant time. This is a concise…
The discussion addresses whether log-periodic power law (LPPL) models used in crash prediction should be treated as continuous- or discrete-time processes. Its main answer is that the model is defined continuously in time, while discrete price observations…
The document reviews a Python implementation of the Parkinson volatility estimator, which uses high and low prices rather than close-to-close returns. Its central calculation squares the log ratio of each period’s high to low, sums those values over a…
The document distinguishes forward variance swap volatility from the future implied volatility smile used to value forward-start options. Under idealized assumptions of a pure diffusion and continuously measured variance, the forward variance swap rate can…
The document examines two Black–Scholes sensitivities for an at-the-money European option: how Vega changes as time to expiry grows, and how option price changes as implied volatility rises. For the expiry question, it considers the derivative of Vega with…
The explanation distinguishes the volatility parameter in a geometric Brownian motion model from the standard deviation of the asset price itself. For an Ito process with proportional diffusion, the price is lognormally distributed, so its variance is shaped…
This discussion addresses whether SABR should be calibrated directly to cap quotes or to the individual caplets that compose those caps. Its proposed workflow is to first strip market cap quotes into a caplet volatility surface, then calibrate SABR…
The document defines rolling returns from stock prices using a chosen observation window and shift, then notes that the resulting return distribution may have multiple peaks. It asks how volatility should represent risk when returns are multimodal. The…
The document explains an approximation from options literature that links the volatility of hedging error to the uncertainty in an estimated volatility. For a lognormal stock process sampled at discrete intervals, the stated uncertainty is proportional to…
The discussion considers whether Monte Carlo simulation can calibrate the Heston stochastic volatility model and which discretization to use. One answer recommends calibrating vanilla European options with the model’s known characteristic function and…
A geometric Brownian motion process has a continuous-time solution that remains positive when its starting value is positive. The note addresses why a numerical simulation of the stochastic differential equation may nevertheless produce negative values. Its…
The document separates three questions: whether next-day variance is forecastable, whether the full return distribution can be forecast, and how to evaluate those forecasts. It describes ARCH/GARCH models and explains why early comparisons using squared…
The document gives an intuitive comparison of three stochastic-process concepts used in quantitative finance. A random walk evolves in discrete steps, illustrated by repeated coin tosses that move a path up or down. Brownian motion is continuous in time and…
The document discusses how someone without access to an OTC derivatives desk can study prices and payoff risks for customized options. It explains that dealers generally quote only to established desk customers, who may need product-specific legal agreements…
The document corrects a proposed variance formula for a stock modeled with geometric Brownian motion. Under Black–Scholes dynamics, the terminal price is lognormally distributed. Its conditional mean is the current price grown at the risk free rate less the…
For European interest rate options, the document distinguishes pricing from hedging. Vanilla prices across strikes can be represented by an implied volatility surface or interpolated total variance, so stochastic volatility is not required to price each…
The document asks why an up-and-out call is said to have a lower delta than a standard call, while a down-and-out call is said to have a higher delta. The answers reason from how the underlying’s movement affects both the call payoff and the chance that the…
The answer introduces Lévy processes as a way to model jumps in asset prices and connects them to risk-neutral pricing. It contrasts two constructions of price dynamics in the Black–Scholes setting, then describes generalizing them by replacing Brownian…
The document explains how to account for scheduled earnings announcements when modeling an asset’s conditional volatility with GARCH. It proposes adding announcement indicators as exogenous terms in the variance equation, allowing past announcements to have…
The discussion addresses how to set an upper-domain boundary when numerically pricing options with finite differences. It recommends using linear behavior in an extreme model state as a generic boundary approximation for many products. In mathematical terms,…
The discussion distinguishes estimating implied volatility from observed option prices from using machine learning to approximate prices for complex derivatives. Inverting a standard option pricing model to find implied volatility is described as a simple,…
The document solves a linear stochastic differential equation with mean-reverting drift toward a constant level and noise proportional to the current state. One answer constructs an exponential integrating factor and applies Itô’s product rule; the…