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 why changing interest-rate or dividend-yield inputs can make an option implied-volatility smile appear unusual, especially for in-the-money options. Under Black–Scholes intuition, a higher interest rate raises a call’s modeled value; if…
A deliverable FX forward exchanges the two contracted currencies at maturity. A non-deliverable forward instead calculates a net payment in a settlement currency from observed exchange rates on a fixing date, usually before maturity. NDFs are used where…
The question asks whether Black–Scholes can use an expiration measured in hours after converting that duration to years, and why an option calculator returns a different value when the input changes from a day to an hour. The answer interprets the change as…
The document explains how derivative volatility relates to the volatility of its underlying asset through elasticity, also called leverage or lambda. Elasticity measures the derivative’s percentage price response to a one-percent change in the underlying,…
This document asks how to translate an earnings event volatility estimate into an implied one-day expected return using a formula from a volatility trading text. The formula combines the exponential of half the event variance with the normal cumulative…
The document frames discrete delta hedging as a tradeoff: more frequent rebalancing can reduce hedge error but raises transaction costs, while less frequent trading lowers costs and leaves greater residual risk. It argues that there is no single practice…
The document asks how to apply multidimensional Itô’s lemma when a function maps a vector of stochastic processes to another vector. It illustrates the question with a vector of assets divided by a stochastic numeraire, a transformation that can arise when…
The document addresses how to tell whether listed call options are European or American when searching for real market data. Its key distinction is that US options on individual securities are generally American, while options on indexes are generally…
The document derives the value of a floating-rate note before its next coupon reset in a continuous-time zero-coupon bond framework. Each future floating coupon is represented as the difference between discount bond prices at its reset and payment dates.…
The document asks whether a sharply inverted credit default swap curve can create arbitrage. Its example compares protection at two maturities: buying longer-dated protection and selling shorter-dated protection may appear to leave a positive premium balance…
The document discusses how the move from LIBOR to risk-free reference rates affects swap discounting and curve construction. It describes the market shift to SOFR discounting for US swap quotations and cleared swaps, and explains that an existing projection…
The discussion explains why an arbitrage-free call price curve should be convex in strike. A butterfly spread combines calls at neighboring strikes so that its payoff is nonnegative across terminal prices and positive over an intermediate range. If the call…
The discussion considers what mathematical background can help a reader understand and implement models in Brigo and Mercurio, including HJM, LMM, and local volatility models. The questioner has studied introductory stochastic calculus and is working through…
The document explains how to value a payoff equal to a forward LIBOR rate fixed and paid at the start of its accrual period by translating it into a payoff at the period's end. Under the assumed risk-neutral framework, the rate known at the start can be…
The document asks whether every twice-differentiable call-price surface across maturities and strikes can be generated by a local volatility function of time and the underlying asset price. It points to the assumption used in Dupire’s approach and seeks a…
The document examines a change-of-measure density for an asset with constant dividend yield, whose price follows a geometric Brownian motion under the physical measure. It solves the price process and substitutes that expression into a proposed density,…
The document explains dual delta as the sensitivity of an option’s value to its strike, distinguishing it from ordinary delta, which measures sensitivity to the underlying price. In the Black–Scholes–Merton framework, call dual delta is negative and relates…
The document derives Black–Scholes prices for two contingent claims under geometric Brownian motion. The first pays the positive part of the difference between the stock price at maturity and its price at an earlier observation date. Conditioning on the…
The document explains the cash-flow interpretation of a vanilla interest rate swap when its fixed rate is negative. In the example, the fixed-rate payer pays a negative fixed amount and receives the floating Euribor leg. A negative fixed payment means that…
The document asks whether a forward Libor rate remains a martingale at intermediate times under the forward measure associated with its payment date. The accepted response applies the tower property of conditional expectation: if the rate is a martingale…
The document presents a Python workflow for fitting a SABR volatility smile with QuantLib. The example supplies option strikes and implied volatilities, constructs a NoArbSabrInterpolatedSmileSection, sets initial values for alpha, beta, nu, and rho, and…
The document derives the stochastic dynamics of a forward rate from the risk-neutral dynamics of zero-coupon bond prices. It first defines the instantaneous forward rate as the negative maturity derivative of the log bond price, then uses the…
The document describes a local volatility pricing implementation that derives volatility from an SSVI implied volatility surface using Dupire's formula and finite differences. The author checks the implementation by repricing the vanilla options used to…
The document concerns numerical evaluation of the integral used to price a call option under the Heston stochastic volatility model. The question asks whether Simpson’s rule or Gauss–Legendre quadrature is suitable and seeks MATLAB or Scilab implementations.…