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 how options volume, dollar volume, and open interest across strikes and expirations might inform implied sentiment, volatility surfaces, option premiums, or Greeks. It contrasts modeling each contract on its own with measuring activity…
The document explains why the drift of log returns under geometric Brownian motion is lower than the drift of percentage returns. Starting from the stochastic price process, it uses a small-change approximation for the logarithm of one plus a return. The…
The document explains why Black–Scholes implied volatility can vary with an option’s strike, forming a smile or skew. It begins from the model’s assumptions of constant volatility and continuous price paths, then frames implied volatility as the…
The document explains that the signed difference between an order’s average execution price and its arrival mid-price is usually treated as implementation shortfall. It presents an approximate model with a spread-related term and a market-impact term that…
The document explains why a continuously delta-hedged option has zero net P&L in the idealized Black–Scholes setting. Applying Itô’s formula splits the option’s change into theta, delta, and gamma contributions. Once the delta hedge removes the underlying…
The document asks whether the Brownian motions in the SABR stochastic differential equations are specified under the physical measure or a risk-neutral measure. Its answer explains that a forward rate is modeled as a martingale under the measure associated…
The document explains how to estimate an option price at an unquoted strike or expiration by first converting observed option prices into implied volatilities, fitting or interpolating a volatility surface, and then pricing the target contract with an option…
The response addresses early exercise of American calls and puts on a stock that pays a continuous dividend yield. It presents an early exercise premium decomposition: each American option’s value is expressed as its European value plus terms involving…
The document explores extending the Heston stochastic volatility model to represent mean reversion in the asset price or returns as well as in variance. The answer proposes a price-level adjustment in which the drift of log price pulls it toward a long-run…
The document examines how volatility affects the risk-neutral probability that an option finishes in the money under Black-Scholes. For a call, that probability is represented by the cumulative normal value at d2, so its response to volatility depends on…
This note explains why a Bloomberg VCUB option matrix may show different values for strike and straddle views. The distinction is the displayed measure: the strike view presents volatility, while the straddle view presents a premium. A premium-based straddle…
The note proposes a set of indicators for monitoring foreign exchange risk at different horizons. For short-term market stress, it recommends tracking implied-volatility indices across equity, bond, and FX markets, alongside funding and credit signals such…
The document explains that implied volatility is the Black–Scholes volatility that matches an observed option price; it is a way to express market prices through that model, not proof of a uniquely correct price. It also cautions that the strike pattern…
The document compares pricing an at-the-money European call on a two-stock basket with a multivariate local volatility model using constant correlation and a Gaussian copula using the same correlation. It reports a result for models calibrated to each…
The document describes an observed inverse relationship between strike and Black–Scholes implied volatility for Ibex 35 options. The author says the option prices and strikes came from daily exchange data, plots spot alongside the volatility curve, and…
The document explains how to create an implied-volatility curve from option prices generated by a Heston model. For each strike, first evaluate the Heston call price using the chosen model parameters and maturity. Then find the Black-Scholes implied…
The document asks how a continuous-time price and variance process relates to a discrete-time stochastic volatility model. Applying Itô’s lemma to log price and log variance gives a mean-reverting log-variance process and a log-return process whose…
The document explains why an RQuantLib valuation of an American option can return a price while showing missing values for delta, gamma, vega, theta, rho, and dividend rho. The accepted answer attributes this to the selected pricing engine: the simplified…
The document explores an analogy between the short side of a zero-investment momentum portfolio and selling a call on the broad equity market. It references research describing how momentum can suffer sharp losses when markets rebound rapidly after declines:…
The document describes problems backing out implied volatility from 30-minute options bars, especially for in-the-money contracts near expiration. The question uses a numerical root-finding approach with a short-term Treasury yield as the risk-free rate…
The document asks how to derive a 99% expected shortfall for a long position using a two-day forecast from an ARMA(1,1)-GARCH(1,1) return model. It outlines the model’s conditional mean and volatility equations, assumes independent standard normal…
The question concerns pricing an American put when the underlying follows a square-root diffusion, and asks whether QuantLib-Python can use its SquareRootProcess class. The author proposes parameter values for mapping the stated process to that class, but…
The document explains the sign of convexity for payer and receiver swaptions by starting with the underlying interest rate swap. A payer swap pays fixed and receives floating, so it benefits when rates rise. However, the gains from higher rates are…
The document explains how traders summarize the shape of an implied volatility smile or skew. It defines the risk reversal as the difference between 25 delta call and put volatilities, and the butterfly as their average relative to at-the-money volatility.…