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 describes diagnosing weak convergence in Monte Carlo simulations of an option-pricing model with state-dependent volatility. The original comparison estimates expectations at two time-step sizes, but uses independent simulated paths. Sampling…
The document presents Gyöngy’s theorem as a way to construct a local volatility process whose one-time probability distributions match those of a given stochastic process. In the stated result, the local drift and variance are conditional expectations of the…
The document addresses how to infer forward rate or caplet volatilities from cap prices or quoted cap volatilities. Its central method is volatility stripping: order caps by maturity, take price differences between consecutive maturities, associate each…
The document examines claims that artificial neural networks outperform Black–Scholes in option pricing. The cited study generates Black–Scholes estimates using simplified inputs, including a closing spot price, fixed volatility and interest rate, no…
The document examines whether negative forward variance inferred from at-the-money-forward implied volatility across two maturities signals an arbitrage. It defines forward variance from the difference in total implied variance over the two time intervals.…
The document compares shorting borrowed shares with constructing a synthetic short using a long put and short call at the same strike and expiry. Its example uses a hard-to-borrow stock and reports that the option combination produces a credit and an…
The document gives an intuitive argument for uniqueness of the risk-neutral probability measure in the Black–Scholes setting. It invokes Arrow–Debreu claims, each paying one unit in a particular state. If the same claim must have the same arbitrage-free…
The document examines a one-period market with three assets and states, asking whether a European call can be priced by replicating its payoff when the market admits arbitrage. The stated replication produces a cost of 0.5, but solving for state prices…
The document asks how to calculate the variance of a discounted European call or put payoff under the Black–Scholes assumptions. Its answer outlines a second-moment approach: variance equals the expected squared payoff minus the square of its expected value.…
The note considers which call strike to use when a trader expects realized volatility to exceed implied volatility and plans to delta hedge, aiming for gamma gains to outweigh option decay. It explains that the approximate gamma contribution to daily profit…
The document considers whether option pricing should distinguish upward from downward volatility. One response frames asymmetric volatility as a choice of a different underlying model, with risk-neutral pricing still used to value options. It suggests…
The document asks what implied-volatility fitting error is acceptable for an equity-index surface calibrated with stochastic-volatility models such as Heston or SABR. The replies do not establish a universal tolerance. One practical suggestion is to compare…
The document asks how to adapt a closed-form implied volatility estimator from a call option to a put. It presents an equation attributed to Hallerbach (2004) for estimating volatility from a call price, spot price, strike, and time to expiry. This makes the…
The document explains how Greeks for swaptions, caps, floors, and other interest rate options are defined and calculated. They measure sensitivity of a derivative’s value to pricing inputs. Because interest rate pricing depends on an entire curve, the…
The document asks how to identify the most likely value associated with a call option when the underlying price is lognormally distributed. It starts from the discounted expected call payoff and distinguishes the mode of the underlying price distribution…
The document examines whether adding a swap payment period changes the value of American and Bermudan swaptions in a predictable order. It defines co-terminal and co-initial swap structures, their exercise schedules, and the proposed comparisons between…
The document addresses two points about an example of VaR manipulation: a likely sign error in an inequality and how a put strategy changes reported risk and profit. The answer says the displayed ordering is consistent if the intended condition is that both…
The document considers a bond with a call and a put exercisable on the same date, using different strikes to illustrate how the embedded options affect value. Under its stated assumptions—a forward bond price of 100 and no discounting—the example values the…
The document describes an attempt to infer local volatility from synthetic option prices generated under a Heston stochastic volatility model. Prices are computed with the COS method, while finite differences estimate the time derivative of call prices and…
The document describes using a normal, or Gaussian, model to price caplets and floorlets when rates can be negative. Under the stated setup, the forward rate evolves as a Gaussian martingale with constant absolute volatility, so its value at expiry has a…
The document explains why options on the same underlying can imply different volatilities across strikes even though the Black-Scholes model assumes constant volatility. It presents several interpretations: the smile can reflect model misspecification,…
The discussion considers whether high-dimensional Sobol sequences are practical for large market simulations and Monte Carlo pricing, where the required dimension may exceed the limits of common implementations. One answer points to established Sobol…
The document explains why foreign exchange option data may arrive as volatility quotes and delta-based smile points rather than as a ready-made table of implied volatilities by strike and maturity. In much of the over-the-counter market, options are quoted…
The document derives a static option replication for a payoff equal to the square of a non-dividend-paying stock’s terminal price. It rewrites the square as twice the integral, over strikes from zero to infinity, of the call payoff at each strike. This…