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 reconciles two expressions for estimating the present value of dividends from option prices. It states European put-call parity in terms of the spot price, strike, discount factor, call price, and put price, then rearranges the relationship to…
The problem describes a European call that must be sold if an independent event occurs, at the lower of its current value and its original purchase price. One response reframes the contract as a standard call minus the value of a compound call written on…
The discussion surveys products outside plain-vanilla swaps, swaptions, caps, and floors, while noting that the examples are not all specific to interest rates. It briefly defines Asian, Bermudan, digital, barrier, and window-barrier options, then describes…
The discussion considers why option-pricing models commonly use stock-return processes with finite variance, even when heavier tails are desired. It gives several explanations: the historical influence of Black–Scholes, mathematical and computational…
The discussion asks whether historical stock returns can be used to estimate volatility skew across option strikes. It distinguishes estimating at-the-money volatility from determining how implied volatility varies for out-of-the-money puts and calls, and…
The document asks how to price options quickly during model calibration, comparing the Carr–Madan and COS methods. It highlights Variance Gamma models with stochastic arrivals and related models where cumulants may not be available in closed form. The…
The document investigates why a VIX-like volatility estimate calculated from option prices can fall below the volatility used to generate synthetic options. The example uses Black–Scholes prices with constant volatility, zero interest rate, and zero cost of…
The document compares cash-settled S&P 500 index options with options on S&P 500 futures, focusing on why both contract types may attract users despite similar index exposure. It identifies margin treatment as a key practical difference: futures-based…
The document discusses numerical pricing accuracy in the Heston stochastic volatility model, especially for very far out-of-the-money options near expiry. It contrasts a reported Gaussian quadrature error with option prices smaller than that error, raising…
The document compares the time values of equally deep in-the-money European calls and puts with no dividends, using strikes the same dollar distance from spot. It applies put-call parity to split the difference in spot-based time values into a discounting…
The document defines option open interest as the number of contracts that remain open, rather than the net difference between long and short positions. A contract is counted when one party holds a long position and another a short position; open interest…
The document explains why a Black–Scholes theta estimate can exceed an option’s current value without implying that the option will actually become negative. The closed-form theta is an instantaneous derivative, commonly stated per year; multiplying it by a…
The answer explains which portfolio risk report fields can be scaled by position size and netted across trades. Most sensitivities, including Greeks, typically scale with notional, but aggregation needs care: unlike-for-like positions should be combined, and…
The discussion outlines mortgage valuation as a problem of projecting cash flows under changing interest rates, borrower behavior, and discounting. A proposed simulation approach generates possible rate paths, calculates amortizing payments, and discounts…
The document answers a question about how far an asset price might move over a three-month horizon given its annualized volatility. The response treats the estimate as a static one-standard-deviation range, scaling volatility by the square root of time and…
The document asks how to value a one-year at-the-money European call when volatility is specified as 20% for the first half-year and 90% for the second half-year. The proposed answer combines the two periods’ variances, weighted by their durations, and takes…
The document sets up an arbitrage check for calls on the same underlying with strikes 100, 120, and 130 and prices 8, 5, and 3. It represents a portfolio with signed quantities, requires a negative initial cost, and writes its maturity payoff across regions…
The document corrects a digital-call valuation under the stated risk-neutral dynamics, where the price has drift proportional to its current level and constant diffusion. The original calculation omitted the conditional mean of the terminal price, leading to…
The document asks what information a binomial tree must retain when valuing an up-and-in barrier option. The response assumes a European-style contract and proposes using knock-in/knock-out parity: the value of the corresponding vanilla European option…
The document presents a question about implementing Tilley’s algorithm for pricing American options through simulated asset paths. The shown R approach generates risk-neutral paths, works backward through exercise dates, sorts paths by asset value, estimates…
The document explores whether implied volatility in a diffusion model must be bounded by the model’s instantaneous volatility. It assumes the asset follows a stochastic-volatility diffusion under the pricing measure and considers a setting where…
The discussion explains how negative risk free rates affect equity and index option valuation. It says the Black Scholes framework can accommodate negative rates, and an answer suggests setting very small negative CHF rates to zero as a practical…
The document asks whether a Heston stochastic volatility model and a local volatility model derived from it produce identical option prices. It points to a comparison in Gatheral’s book and asks whether the equivalence depends on spot and variance having…
For puts with the same underlying and expiry, the displayed bid and ask quotes generally increase as strike prices rise. The question arose because some last-traded prices appeared to violate that pattern, leading the answers to distinguish current quotes…