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 asks what implied volatility represents in vanilla option pricing and whether fitting it to market prices can reveal mispricing. It contrasts treating volatility as the adjustable input to Black–Scholes with an alternative parameterization that…
An inverted volatility surface can reflect market conditions, but sparse or unreliable option quotes may make its shape misleading. In the ONDK example, strikes are widely spaced and bid-ask spreads are broad, leaving too few useful observations to estimate…
The document describes how to estimate cross-gamma for an instrument whose value depends on multiple underlying spot prices. Cross-gamma is the mixed second derivative of value with respect to two different spot levels, and it appears off the diagonal of a…
The document asks whether a forward market model (FMM), extending the classic LIBOR market model to handle backward-looking risk-free rates (RFRs) and term rates together, has been combined with stochastic volatility such as SABR. The motivation is pricing…
The exchange discusses estimating an option’s future price after a specified move in its underlying when future implied volatility is unknown. One approach is to make explicit assumptions about both the underlying price change and the change in implied…
The document raises a derivation question about two binary call options under a risk-neutral geometric Brownian motion model. It states the familiar form for an asset-or-nothing call, whose payoff is the underlying asset when its terminal price exceeds the…
The document distinguishes the drift used to price derivatives from the drift used to simulate future underlying prices. Under the risk-neutral measure, derivative valuation uses the relevant risk-free rate; for a real-world simulation of potential future…
The document considers a portfolio that is short EUR/USD and long GBP/USD, and asks how options could limit losses while preserving the possibility of gains. The key observation in the response is that the two positions create an implicit short EUR/GBP…
The document asks where to obtain historical overnight index swap rates across a range of maturities, motivated by pricing foreign exchange options. It distinguishes a single overnight benchmark such as the Effective Federal Funds Rate from a full OIS curve,…
The document raises a statistical question about uncertainty in option Greeks estimated with Monte Carlo simulation. It starts from a simulated option value with a standard error, then applies central finite differences to estimate Delta from prices at…
The document explains why option pricing becomes less determinate when an underlying follows a jump-diffusion process. In the Black–Scholes framework, market completeness yields a unique risk-neutral probability measure and therefore a unique arbitrage-free…
The document discusses estimating sensitivities for a European arithmetic basket call, whose payoff depends on the sum or average of several underlying assets. Because the basket has multiple underlyings, it has separate deltas, vegas, and other…
The document examines a Monte Carlo simulation of a European call under the Black–Scholes–Merton model. It contrasts compounding normally distributed simple returns with exponentiating normally distributed log returns, then gives the drift adjustment needed…
The procedure estimates an at-the-money implied volatility for a target maturity by using two listed option expirations that bracket it. It first selects nearby call and put strikes, estimates a forward price for each expiration from put–call parity, and…
The explanation clarifies that the Black–Scholes stock process is written for the ordinary price level, with drift and volatility proportional to the current price. The logarithm enters when Itô’s lemma is applied to that process: the log price then has a…
The document raises a pricing question about a down-and-out call when the underlying follows a jump process, with the Kou double-exponential and normal inverse Gaussian models suggested to represent asymmetric jumps. It considers raw Monte Carlo simulation…
The document poses a question about constructing a risk-neutral density from out-of-the-money options. Since the density is related to the second derivative of option price with respect to strike, it asks whether convexity of option prices is needed to…
The document explains why an FX option’s delta convention must match the convention used to build a volatility smile or surface. It lists spot and forward delta, each with premium-adjusted and unadjusted forms, and notes that converting between conventions…
The note addresses whether Black–Scholes prices a call and a put equally when their strikes are equally far from the current underlying price in opposite directions. It distinguishes terminal payoffs from option values before expiry. At expiry, if the…
The note explains why callable bonds should not automatically be analyzed as if they run to maturity. When an issuer is likely to call, calculating spread and interest-rate risk to maturity can give misleading results. A common practical approach is to find…
The document explains how a Monte Carlo simulation can align with Black 76 without first converting futures prices into spot prices. It starts from a lognormal spot-price process with drift and volatility, then substitutes the spot-forward relationship so…
The document explains that a numeraire used for pricing does not always need to be a traded asset. It presents the risk neutral pricing relation under a measure associated with the chosen numeraire, then describes expressing derivative and underlying prices…
The document asks whether the publication of the Black–Scholes option-pricing paper in 1973 was directly connected to the founding and launch of the Chicago Board Options Exchange that year. The cited historical account says the formula was not developed to…
The document explains how to price a claim paying the cosine of a terminal stock price under risk-neutral pricing when the stock follows geometric Brownian motion. It writes the discounted expected payoff as an integral over the risk-neutral lognormal…