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 explains when the second strike derivative of call prices can be interpreted as a risk-neutral density. The stated conditions include no-arbitrage bounds, a lower bound on the strike derivative, a vanishing derivative at very high strikes, and a…
The document considers a basket call whose components are denominated in different currencies while the payoff is paid in US dollars. It asks how to price and hedge the payoff formed from the sum of the two asset values less a strike, floored at zero. The…
The document asks how log moneyness differs from log strike, particularly when describing coordinates used in volatility-surface work. It also asks why practitioners use the term “moneyness” and whether it refers to an option being in or out of the money.…
The document asks whether combining a volatile underlying asset with options can improve portfolio diversification, measured by a higher Sharpe ratio than holding the asset alone. It motivates the question with a position consisting of one share and one…
The document asks how to transform the expected stock price equation in a one-step binomial model. Under the risk-neutral measure, the stock moves up by factor u with probability p and down by factor d with probability one minus p. The expected terminal…
The document questions whether calibrating a pricing model to observed derivative prices can produce reliable beliefs about future asset paths, given that market prices reflect human judgments and incomplete knowledge. The answer frames calibration as…
This question examines an intuitive explanation of the Black–Scholes formula: treating option value as an expected payoff based on the chance of exercise. It tests that idea with an at-the-money call and a simplified outcome in which the stock either reaches…
The document examines a proposed derivation of the Black–Scholes call formula. It changes variables to log spot, writes the call price as a combination of an asset-or-nothing term and a discounted cash-or-nothing term, and substitutes that form into the…
The document asks for the probability that a call buyer earns more than a specified profit at expiry when the stock follows geometric Brownian motion. It gives the initial stock price, drift and volatility parameters, strike, expiry, and risk-free rate, then…
The document defines an index-linked certificate of deposit (CD) return multiplier as the CD’s return per invested dollar relative to the underlying index’s return per invested dollar. It asks whether the multiplier is below one for an at-the-money index…
The document derives the time-zero price of a cash-or-nothing binary put that pays one unit when the stock finishes at or below the strike. Under risk-neutral Black–Scholes dynamics, the payoff value is the discounted probability that the terminal stock…
The document clarifies why Heston option vega may include a factor of two times spot volatility. In a stochastic volatility model, an option value is naturally expressed as a function of variance. If a trader instead wants sensitivity to volatility, the…
The document explains two ways to attribute an option’s price change to movements in its underlying price, implied volatility, and time. Risk theoretical P&L uses sensitivities measured at the starting point, multiplying them by observed input changes;…
The document derives the expectation of an exponential applied to a standard normal variable, a step used in the risk-neutral log-normal model for an asset price. It sets the volatility-time scale as a parameter, writes the expectation as an integral against…
The document explains why an ETF that maintains exposure by rolling futures can have a forward value close to its current net asset value, despite differences between spot and futures prices. Because the fund sells expiring contracts and reinvests the…
This note discusses pricing an accreting swaption in the Libor Market Model when a closed-form solution is unavailable. One response says that when the fixed and floating legs accrue nominal in the same way, the Rebonato approximation for equivalent Black…
This note explains pin risk in options whose underlying can itself expire or be exercised, using an over-the-counter Bermudan swaption-style example. Pin risk arises when the underlying is near the option strike and the holder’s exercise decision is…
The note explains why put-call parity can appear to differ between formulas for options on spot and options on futures. For spot options, the call-minus-put relationship uses a discount factor applied to the forward-minus-strike amount. For options on…
The note corrects a proposed Monte Carlo payoff for a barrier knock-out call. Multiplying asset prices observed at maturity and at an intermediate time does not represent the stated contract. The payoff should be the positive part of the maturity asset price…
The note clarifies what a GARCH model estimates when implied volatility (IV), or its growth rate, is used as the dependent variable. A typical GARCH setup has a conditional-mean equation for that variable and a conditional-variance equation for the mean…
The document derives a dynamic hedge for a quanto option whose underlying stock is priced in a foreign currency while the payoff is fixed in domestic currency. It models the foreign stock and exchange rate as correlated geometric Brownian motions and…
The document presents a derivation for a European option that expires at one date and gives the holder the right to enter a forward contract maturing later. It uses a zero-coupon bond as numeraire and assumes the forward price is lognormally distributed…
The document explains how to handle the Black–Scholes delta in a European option replication strategy as time approaches expiration. Although the usual formula contains a term that becomes zero at maturity, its limiting behavior gives delta of one for an…
This question describes a local-volatility model for a stock and asks how to price an up-and-out call by solving the Black–Scholes partial differential equation. The example specifies a barrier, strike, spot price, rates, maturity, and a volatility function…