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 asks how to interpret the ratio of a standard normal upper-tail probability to its density when evaluated at Black–Scholes d₁ for a put option. It identifies this as the Mills ratio and relates its reciprocal to the normal hazard rate, a concept…
The document distinguishes three option price references. The last-traded price records the most recent execution and can remain unchanged when no further trades occur, even as current quotes move. A closing price may reflect the market around the exchange’s…
The document values a perpetual claim that pays one unit when a stock first reaches an upper barrier and pays nothing if it reaches zero. Under geometric Brownian motion, zero interest rates make the claim worth one, assuming the upper barrier is eventually…
The note discusses how a short stock position can affect the carry embedded in an option strategy. In a short sale, the trader receives proceeds that may earn interest, while also paying a stock borrow fee. The net financing effect can reduce the effective…
The document explains calibration in the context of using the SABR stochastic volatility model to price foreign exchange call options. Calibration means choosing model parameters so that model outputs match observed market quotes as closely as possible. The…
The document considers a contingent claim that pays the positive excess of one stock’s terminal price over a strike, provided that a second stock finishes inside a specified price interval. The question proposes pricing it from the two terminal marginal…
The document explains why gamma exposure is expressed using the square of a change in the underlying price. It uses dimensional analysis: option-price changes are measured in dollars, while gamma measures the second derivative of option value with respect to…
Dispersion has a broad statistical meaning: the spread of a distribution around its central tendency. Variance is a common measure of that spread, and standard deviation is its square root. In finance, volatility usually refers to standard deviation as a…
The note asks why American option valuation is expressed as the best conditional expected discounted payoff over stopping times rather than over fixed exercise dates. A fixed date is a stopping time, so optimizing over all stopping times cannot produce a…
The document asks how to price American options when the underlying follows risk-neutral dynamics with volatility that varies with spot. The response identifies this as a special case of a local volatility model, where volatility may depend on time and spot,…
The document asks whether an equity’s implied volatility skew can reveal the probability of an equity move conditional on a change in implied volatility. It defines skew as the difference between implied volatilities at two option deltas and explains why…
The discussion addresses why a Carr–Madan fast Fourier transform calculation can return complex-valued quantities when pricing caplets. The accepted answer says the complex form is expected in the method and that the option price is obtained from the real…
The response points to an intuitive route to understanding Gyöngy’s theorem through European call prices and generalized derivatives. It starts from an asset whose drift follows a risk-neutral specification and whose volatility may depend on the adapted…
The discussion compares QuantLib pricing approaches for geometric and arithmetic Asian options. It reports that the finite-difference Black–Scholes Asian engine supports discrete arithmetic averaging, not continuous geometric averaging. A discrete geometric…
The document considers how to value currency options when a central bank caps an exchange rate, using the Swiss National Bank’s franc ceiling against the euro as context. Its answer argues that standard models calibrated to historical market behavior may be…
The note explains how a finite-difference valuation of an American option should handle a discrete dividend at an ex-dividend date. The stock-price jump is represented by mapping the option value just after the dividend to the value just before it,…
The document offers an intuitive explanation for why standard call and put options have positive gamma in the Black–Scholes setting. As the underlying price rises, a call’s delta rises toward one because the option behaves more like the underlying. A put’s…
The document explains why analyzing option Greeks is not expected to improve the payout of a standard box spread. A box combines calls and puts at two strikes to create a payoff similar to a loan or zero-coupon bond. Its return reflects the risk-free rate…
The post works through a derivation of call-option gamma by differentiating an option-price expectation and using the Dirac delta to evaluate the resulting integral. The key step is to recognize the density of the transformed variable as lognormal, apply the…
The post asks whether linear interpolation of implied volatility can create arbitrage, distinguishing interpolation across maturities from interpolation across strikes. It notes the concern that interpolating volatility over time may violate calendar…
The document challenges a proposed rule linking an option’s time value to its delta. The claim is that when time value makes up at least a specified fraction of an option’s price, delta must be below a corresponding threshold. A Black–Scholes call example…
The document asks how to plot a one-month implied volatility smile when call and put implied volatilities and option deltas are available, but spot and strike prices are missing. It notes that volatility smiles are often displayed against delta, so the…
The document considers a funded equity collar for a shareholder seeking cash against an existing stock position. The proposed structure combines a margin loan secured by pledged shares, a purchased put, and a written call with a nearby strike. The bank would…
The document explains why a slightly in-the-money American option does not generally have a delta of one. Delta reflects how the option’s total value, including remaining time value, changes with the underlying price; it is not determined solely by the slope…