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.
8,116 documents
The document explains why an option skew measure based on volatility differences divided by log-strike distance, with a square-root-of-maturity adjustment, can remain similar across listed expiries. The key comparison uses strikes set at a fixed number of…
The document considers whether options can reveal volatility over a specific upcoming interval, such as a trading day containing a scheduled central-bank announcement. It notes that implied volatility is only directly observable for horizons with liquid…
The document reviews a call ratio spread designed to start delta-neutral and gamma-neutral, with a short stock position offsetting option delta. Although the structure may collect theta, the answer emphasizes its payoff asymmetry: a credit and frequent small…
Co-terminal, or diagonal, swaptions share a common final maturity while their expiries step forward and their underlying swap tenors shorten. The document illustrates the structure with a callable bond that can be redeemed at regular intervals before…
The document compares a portfolio of call options on individual assets with a call option written on a weighted portfolio of those assets. For nonnegative weights summing to one, the payoff of a call is a convex function of its underlying value. Convexity…
The question derives daily log returns under geometric Brownian motion and estimates their sample variance from a month of stock prices. Since the variance of a return over a daily interval is the annual variance scaled by the length of that interval, the…
The document clarifies how carrying costs and dividends enter equivalent forms of put-call parity. Its central explanation is that parity equates a stock-minus-bond position with a call-minus-put position. A stockholder bears financing or other carry costs,…
The document poses a derivatives-pricing problem: valuing a European payoff based on the square of an underlying stock price when the stock follows a geometric Brownian motion. It applies Itô's lemma to the squared price and observes that the resulting…
The document gives a brief rule of thumb for choosing an option-pricing model by underlying: Black–76 is suggested for options on index futures, while Black–Scholes is suggested for options written directly on an index, such as SPX options. It points readers…
The discussion compares two ways to express the profit and loss of a long delta-hedged option when realized volatility exceeds implied volatility: marking the option using realized volatility, or retaining the implied-volatility mark and allowing…
The document asks how to obtain Black-Scholes implied volatility from a model’s characteristic function using an integral relation attributed to Gatheral’s treatment of Lewis inversion. It focuses on the contour used to value a covered call, why the payoff…
The document asks whether a basic Black–Scholes spreadsheet for European options can be adapted to price American options. It suggests Whaley’s approximation as a possible spreadsheet method when limited accuracy is acceptable. The approach requires…
The document explains why binomial option pricing models often set the up and down factors as reciprocals. That condition makes the price tree recombine: an up move followed by a down move reaches the same node as a down move followed by an up move, keeping…
The document addresses how to compare equity option implied volatilities across the transition from LIBOR to SOFR discount curves. Its answer points to contractual benchmark fallback conventions: a replacement risk-free rate can be paired with a spread…
This note compares sensitivity methods for discrete Asian options when the underlying follows geometric Brownian motion. A geometric-average Asian option has an analytic price under Black–Scholes assumptions, allowing Greeks to be obtained from the…
This discussion asks how to estimate the exercise probability of the embedded American call in a convertible bond using QuantLib, and distinguishes that probability from delta. It describes approximating the option’s sensitivity to strike, also called dual…
The document concerns technical and practical pricing of insurance-linked securities, especially catastrophe bonds. Its answer characterizes many underlying contracts as excess-of-loss insurance covers, suggesting that actuarial literature on those contracts…
The document compares linear interpolation of quoted rates with constant-forward-rate interpolation when constructing a rate curve for pricing American and European options. It highlights that linear interpolation can create discontinuities in instantaneous…
The document considers how to estimate an implied volatility surface for an asset that has spot data but no options market. It outlines several practical approaches: estimate realized volatility with a model such as GARCH and add a chosen risk premium; scale…
The document addresses why a bank would sell an option when a favorable move in the underlying could make the payout larger than the premium collected. The example assumes a call premium of five dollars and a later stock price of one hundred ten dollars, but…
The question seeks an integral-based route to barrier option prices, motivated by the reflection principle and a guessed reflected probability density. The answer points to a derivation for Black–Scholes barrier options that uses reflection together with…
The document clarifies that negative variance values can arise from numerical discretization even though the continuous-time Heston variance process does not become negative. Flooring values, taking absolute values, or otherwise altering simulated variance…
The document compares local-volatility and stochastic-volatility valuations for a double no-touch option that pays a fixed amount if spot remains between two barriers until maturity. It assumes both models are calibrated to the same implied volatility…
The document explains how to relate an option on a delivery-date futures contract to a Black-76 valuation using a different futures contract associated with the option's expiry. Under deterministic rates and basis, it expresses the delivery-date futures…