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 asks whether a two-factor Hull–White short-rate model, also called G2++, can represent swaption volatility smiles or skews, and whether fitting it to out-of-the-money swaptions would help. The answer points to local- and stochastic-volatility…
The document addresses unstable numerical integration in Heston option pricing, where an oscillatory or slowly decaying integrand can produce negative computed prices, especially for deep out-of-the-money options. A recommended first remedy is the Little…
The question concerns estimating how a multi-day event affects volatility in one-minute equity returns, where extreme observations challenge normal-error ARCH and GARCH specifications. The response recommends matching the conditional error distribution to…
The document explains how a Lévy process such as variance gamma, NIG, or CGMY can be time-changed by the integrated rate of a mean-reverting CIR process. This produces a stochastic volatility Lévy process: the clock remains nonnegative and increasing, but…
In swaption quoting conventions, a volatility surface is often arranged with option expiry increasing down the first column and the maturity of the underlying swap increasing across the top row. Under that layout, the upper-left region represents options…
The document derives the hedging error that arises when a call is priced with Black–Scholes implied volatility but the underlying stock realizes a different volatility. The option price satisfies the Black–Scholes partial differential equation using the…
The document presents a failed calibration of the raw SVI total-variance parametrization for a single options maturity. The example converts market implied volatilities to total variance, computes forward log-moneyness, and minimizes squared fitting error…
The document clarifies how a volatility surface indexed by moneyness can remain consistent with put-call parity. Although the question assumes calls and puts use opposite moneyness ratios, the answer explains that moneyness is generally defined the same way…
The document asks how to scale returns calculated from hourly bars to create volatility-adjusted features at longer horizons, including several-hour and calendar-period returns. It specifically considers whether a multi-hour return should be divided by…
P&L leakage is a small, recurring loss that may be hard to notice in daily results but accumulates into a material drag over time. The note places the concept in hedging and trading, where a reported or expected profit can omit costs and risks that are…
The document presents Gyöngy’s theorem as a way to construct a local volatility process whose one-time probability distributions match those of a given stochastic process. In the stated result, the local drift and variance are conditional expectations of the…
The document addresses how to infer forward rate or caplet volatilities from cap prices or quoted cap volatilities. Its central method is volatility stripping: order caps by maturity, take price differences between consecutive maturities, associate each…
The document asks whether the process driven by an imaginary multiple of real Brownian motion has negative variance. The answer distinguishes the process’s imaginary-valued increments from its variance: at time t, the process is an imaginary multiple of…
The document asks how the return of a function of an asset changes when the underlying asset’s drift rises by a constant, assuming the asset follows geometric Brownian motion. It applies Itô’s lemma to the function, separating the stochastic term from the…
The document examines whether negative forward variance inferred from at-the-money-forward implied volatility across two maturities signals an arbitrage. It defines forward variance from the difference in total implied variance over the two time intervals.…
The answer distinguishes preferences for return skewness according to an investor’s objective. It says that maximizing expected log wealth can favor higher odd moments, such as mean return and skewness, while favoring lower even moments, such as volatility…
The document derives an expression for the expected log return over a time interval under a local volatility stochastic differential equation. Applying Itô’s lemma gives a drift contribution and a negative half-variance contribution; the expected stochastic…
The note considers which call strike to use when a trader expects realized volatility to exceed implied volatility and plans to delta hedge, aiming for gamma gains to outweigh option decay. It explains that the approximate gamma contribution to daily profit…
The document considers whether option pricing should distinguish upward from downward volatility. One response frames asymmetric volatility as a choice of a different underlying model, with risk-neutral pricing still used to value options. It suggests…
The document asks what implied-volatility fitting error is acceptable for an equity-index surface calibrated with stochastic-volatility models such as Heston or SABR. The replies do not establish a universal tolerance. One practical suggestion is to compare…
The document compares distributional choices for modeling interest rates, beginning with a proposed sequence of nonnegative independent observations. Responses caution that this assumption misses persistence and changing volatility: rates tend to cluster…
The document asks how to adapt a closed-form implied volatility estimator from a call option to a put. It presents an equation attributed to Hallerbach (2004) for estimating volatility from a call price, spot price, strike, and time to expiry. This makes the…
The document describes an attempt to infer local volatility from synthetic option prices generated under a Heston stochastic volatility model. Prices are computed with the COS method, while finite differences estimate the time derivative of call prices and…
The document describes using a normal, or Gaussian, model to price caplets and floorlets when rates can be negative. Under the stated setup, the forward rate evolves as a Gaussian martingale with constant absolute volatility, so its value at expiry has a…