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.
10,263 documents
The document explains how to simulate a geometric Brownian motion after changing from the physical measure, where the drift is the expected return, to the risk-neutral measure, where the drift is the risk-free rate. It derives the change in Brownian motion…
The discussion examines why replacing a vanilla option at one strike with a strip of options across nearby strikes can reduce the sharp gamma concentration near expiry. It offers two related explanations: a strip distributes exposure across strikes, and as…
The document uses a one-period, two-outcome stock model to show how stock and call positions can produce a payoff that is the same in either market outcome. With the stock ending above or below the strike, the call’s payoff differs across states. Choosing a…
The note addresses whether Excel’s YIELD function can calculate the yield of an Italian government bond that pays coupons semiannually while its yield convention uses annual compounding. It explains that the bond’s price equation must discount each coupon…
The note explains the inverse relationship between a bond’s yield to maturity and its price, then defines Macaulay duration as the cash-flow-weighted average time until payments are received. Under continuous compounding, bond price is the sum of discounted…
The document challenges the idea that fitting a convex curve in strike-implied-volatility space guarantees an arbitrage-free option surface. Implied volatility need not itself be convex for market prices to be free of arbitrage, and convexity of implied…
The document presents two ways to price a forward-start call whose payoff depends on the asset price at the start date as well as at expiry. One approach conditions on the information at the forward start date and rewrites the value using the price of an…
At expiration, a call with a zero strike pays the stock price when the stock price is nonnegative. The discussion asks whether this means the option should be worth the stock itself before expiration, and whether shareholder rights or counterparty risk…
The question describes an energy option that runs across several months. For each month, its strike resets to that month’s first-of-month index, and daily settlements below that level generate a payoff based on the difference. The question asks how to value…
The document defines an affine term structure model through the relationship between bond yields and model state variables. In the simplest formulation, a zero-coupon bond price is exponential in a maturity-dependent constant and the short rate; taking its…
The document addresses why the Black–Scholes call formula contains different volatility adjustments in d1 and d2 under risk-neutral pricing. It outlines a derivation route: solve the stock price process with its drift set to the risk-free rate, then express…
In a binomial model, the risk-neutral probability is chosen so that the stock’s expected return under that pricing measure equals the risk-free rate. The document explains why a stock’s CAPM expected return is not substituted: option valuation uses…
The document examines the Black–Scholes prices of digital calls and puts as volatility approaches infinity, assuming a zero interest rate. From the model’s formulas, it identifies limiting prices of zero for the call and one for the put, independent of…
The document raises a conceptual question about volatility skew: options at different strikes imply different volatilities, even though implied volatility is often treated as a measure of expected future risk. It asks whether those differences represent…
The explanation distinguishes Eurodollar futures from forward rate agreements and interest rate swaps. In this context, Eurodollar refers to futures on a three-month interest rate, though the term can also refer to the Euro–US dollar currency pair. Compared…
The discussion explains how Black-Scholes is used in over-the-counter FX vanilla options. Market quotes commonly express implied volatility in delta space; traders convert those quotes into prices using the relevant market inputs, such as spot, rates, and…
The document asks whether a flat implied-volatility skew implies a normal risk-neutral distribution when option prices are converted into a density using the Breeden–Litzenberger relationship. It notes that the density inferred from its stated formula…
The document seeks to interpret a paper's implied-gamma method for fitting option prices across strikes. In the described approach, gamma is a modeler-selected parameter defining a return uncertainty set. For each in-sample strike, the parameter is…
The document works through the conditional expectation of the squared value of a Brownian motion at a later time, given its filtration at an earlier time. It expands the later value into the known earlier value plus an independent increment, then asks why…
The document asks which probability measure should be used to calculate the expected discounted payoff of a replicable claim that lies below a possibly nonreplicable contingent claim. It presents an equality between a supremum over such replicable claims and…
The document defines a credit spread from the prices of defaultable and risk-free zero-coupon bonds, then relates default risk to a hazard rate and survival probability. It also introduces a structural company valuation in which equity resembles a call…
The document asks for books that help quantitatively trained readers connect mathematical finance models with real-world practice. The responses offer a range of references across investments, portfolio theory, corporate finance, financial econometrics,…
The document distinguishes basis between curves from the use of OIS discounting and explains how curve differences can affect swap values. It describes basis spreads as rate differences between curves, such as tenors or an interbank curve and an OIS curve.…
The document explains how to determine the floating coupon schedule and rate calculation for a QuantLib zero-coupon swap. It states that the index tenor controls the spacing of floating subperiods, and demonstrates inspecting a floating coupon’s dates,…