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 a modeling problem: estimate the probability distribution of the next day’s change in at-the-money Black–Scholes implied volatility using current implied volatility, days to expiration, and the underlying’s percentage move. The author…
The document examines Black–Scholes pricing for a European call on a stock paying known discrete cash dividends. It describes the Musiela construction, which separates the stock’s price from accumulated dividend amounts and models a dividend-adjusted…
The document seeks an intuitive introduction to risk-neutral valuation and arbitrage theory. Its response contrasts pricing derivatives under the historical probability distribution of the underlying asset with pricing under a risk-neutral measure. It…
The document asks for a detailed proof that the value of a call butterfly is nonnegative when its three strikes are equally spaced. Its expression combines a long call at the lower strike, two short calls at the middle strike, and a long call at the upper…
The answer derives the distribution of the ratio of a stock price at an earlier time to its value at a later time under a stock-based numeraire. It starts with the risk-neutral measure associated with the bank account, defines the stock measure using the…
The document poses a derivatives-pricing question: whether a rise in interest rates must always reduce the value of an American put option. The answer says that this conclusion does not hold in general and points readers to a referenced treatment for an…
The discussion asks whether SPX options offer a way to trade the index’s closing value with an offset for forward alignment, similar to a futures index basis trade. One response says SPX combo-on-close trades are available over the counter, with interdealer…
The document describes an attempt to estimate a Treasury bond future’s delivery option by applying principal component analysis (PCA) to yields of deliverable bonds. The author uses the leading components to create yield scenarios, scales them using the…
The document gathers book recommendations for someone with introductory knowledge of stochastic calculus, Black–Scholes, and binomial pricing, plus a background in statistics and time-series modeling. Suggestions span broad derivatives texts, volatility and…
The document describes a curve-building problem encountered while trying to infer implied volatility for a short-maturity Euro floor. The floor price covers the remaining unfixed floorlets, but for some strikes the market quote is below the model price even…
The note explains why VIX futures can converge toward the published spot VIX near expiration even though the spot index itself is not tradable. Futures settle in cash, with the settlement amount determined on delivery day from a calculation closely related…
The note distinguishes the probability that an option expires in the money under different probability measures. In Black–Scholes, N(d2) gives the risk-neutral probability of the underlying finishing above the strike, using the risk-neutral drift; delta,…
The response outlines how interest-rate derivative volatility surfaces are assembled from market-maker quotes and why normal, or Bachelier, volatility became a common convention in low-rate and negative-rate environments. Black volatility describes…
The document gathers proposed explanations for option terminology. It describes call and put through the writer’s obligation: a call may result in shares being called away at the strike, while a put may result in shares being put to the writer. It also gives…
The document explains how to price a call option in a one-period binomial model by constructing a portfolio of the underlying stock and a bond that matches the option’s payoff in both possible states. The stock starts at one, moves up or down by five…
The document explains how a cash-settled Asian call pays at expiration. Its payoff is based on the average of the underlying’s prices over the specified observation dates, compared with the strike. When that average exceeds the strike, the bank pays the…
The document addresses a common confusion between an option's declining time value and its implied volatility as expiry approaches. The answer explains that, near the money, option value scales approximately with annualized volatility multiplied by the…
The document examines how to tell whether an options position is long or short skew, using a risk reversal with an out-of-the-money call and put. It explains that subtracting implied volatilities at nearby strikes estimates local smile slopes, but comparing…
The discussion explains why simply adding a short delta position in the stock to a long option does not, by itself, justify the usual Black–Scholes derivation. Applying Itô’s lemma to the option and stock can cancel the instantaneous stochastic term, but the…
The document clarifies how to interpret the return from a put-call parity position built from a short call, a long put, and shares of the underlying. The example compares the option price difference with the share purchase cost and finds a return close to…
The document considers calibrating a SABR implied volatility surface when an illiquid options market lacks out-of-the-money quotes. It explains that fitting only in-the-money and at-the-money observations leaves part of the implied distribution…
The document explains how to adapt QuantLib’s Heston option-pricing workflow to foreign exchange options. It treats the foreign interest rate like a dividend yield and the domestic rate as the risk-free curve, allowing FX options to be handled similarly to…
The document investigates why a closed-form down-and-out call price differs from a Monte Carlo estimate for a leveraged warrant. The response identifies two problems in the simulation: it uses an estimated real-world return drift instead of the risk-free…
The document describes an option-like contract with two dates. At an initial grant date, the holder chooses a later exercise date; once that choice is made, the option behaves like a European option, with exercise permitted only on the selected date. The…