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 an asset forward price has the same percentage volatility as its spot price when the interest rate is constant and the cost of carry equals that rate. In this setting, the forward price is the spot price multiplied by a…
The document outlines a model for pricing a call on realized volatility by treating realized volatility as the maturity value of a volatility swap. It assumes the swap price follows a lognormal process driven by the same Brownian motion as the underlying…
The document addresses why the Black–Scholes derivation relates an option’s price dynamics to those of a self-financing stock-and-bond portfolio. Its central explanation is that replication is constructed backward from the option payoff at expiration: a…
The document examines why a set of out-of-the-money FXI puts appears to become much more expensive with added time to expiration than a simple square-root-of-time rule would predict. The quoted example lists ask prices and implied volatilities across several…
The document discusses the common binomial-tree choice of an up factor based on volatility and the square root of the time step, with the down factor set as its reciprocal. This reciprocal relationship makes the recombining tree compact: after each step,…
The document considers how to price European calls and puts on an underlying that can jump, fluctuates with short-term supply and demand, and tends to lose value over longer horizons. It also raises practical questions about physical delivery and indivisible…
The document asks how to interpret the time-dependent drift parameter in the Ho-Lee short-rate model while calibrating a discrete interest rate tree. In the described procedure, each period's parameter is chosen so that model prices under the risk-neutral…
This exchange clarifies the difference between general drift and volatility processes and coefficients written as functions of the current state and time. State-dependent coefficients are a special case of stochastic processes, but specifying the…
This document explains how to calculate the probability that a geometric Brownian motion (GBM) price will fall inside a specified interval at the next time step, given its current value. This conditional probability is used in the stochastic mesh method to…
The document asks whether sharp foreign exchange price moves around short binary option expiries reflect trades or misleading posted bids, and where traders can inspect FX order depth. The answer attributes many such moves to market makers hedging binary…
The document explains why implied volatility calculated from in-the-money (ITM) and out-of-the-money (OTM) equity calls may appear higher for ITM strikes. For calls, ITM contracts have strikes below spot, while OTM contracts have strikes above spot; the…
The document presents a question about extracting a risk-neutral density from SPY weekly options using the R package RND. The author fits a three-lognormal mixture to American option prices and plots the resulting density over strikes around spot, but finds…
The document asks for the maximum price of a European call butterfly with strikes at 103, 106, and 108, assuming no fees and a zero interest rate, without specifying a distribution for the underlying. The accepted reply gives a maximum theoretical value of…
The document summarizes ways to relate bilateral valuation adjustments to unilateral CVA and DVA. One explanation frames bilateral counterparty risk through scenarios in which both parties survive, the counterparty defaults first, or the institution defaults…
The document describes the collateral choice option as an expectation involving the time integral of the maximum among several spread processes, typically modeled as Gaussian. It notes that this expectation is difficult to calculate and that a cited…
The document asks how to compare a numerical European call price from an explicit finite-difference scheme with the analytical Black–Scholes value at an intermediate time step. The model uses spatial grid points corresponding to underlying asset prices, and…
The document considers a four-option position consisting of a long put below the middle strike, two short options at the middle strike, and a long call above it. The questioner calculates one region of the payoff and finds a negative amount, which appears…
The document poses a calibration problem for a one-factor Hull–White short-rate model with constant volatility and mean-reversion speed, using historical zero rates. It proposes inferring mean reversion from the relationship between the historical…
The document explains the probability meanings of the two normal cumulative terms in the Black–Scholes call price. The probability associated with the strike indicator in the discounted payoff is computed under the risk-neutral measure, while the…
This discussion considers whether commercial derivatives-pricing software will reduce the need for pricing quants. The responses describe a changing role: less development of entirely new pricing formulas in some settings, and more adaptation of existing…
This explanation examines the shift parameter in a lognormal model for forward interest rates. It writes a caplet payoff in terms of the shifted rate and strike, assumes the shifted forward follows a lognormal process under the appropriate forward measure,…
This document investigates inconsistent results when fitting a lognormal SABR volatility smile with pySABR. In the failing example, the fitted parameters produce volatility estimates that diverge sharply from the observed quotes when reused in another…
The note describes two ways to estimate implied volatility skew. One approach prices calls and puts across strikes, converts their prices to Black–Scholes implied volatilities, and estimates the smile’s slope. Another uses a call and a digital call at a…
The note explains that the Black model does not depend on the standard spot-to-forward relationship in which the forward price equals spot grown at the risk-free rate. Currency futures are given as an example where borrowing and lending in the two currencies…