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 compares the value of otherwise comparable European, Bermudan, and American call options through their exercise rights. A European option can be exercised only at expiry, a Bermudan option on specified dates, and an American option throughout…
The document considers pricing American and European options on an unusual underlying with sparse, interpolated observations, very high volatility, and unusually strong recent returns. With no comparable traded options or clearly correlated assets, the…
The document poses a question about whether every discounted price process that is a supermartingale can be replicated or hedged in a Black–Scholes market. It assumes the underlying follows Brownian motion and that the market has the standard Black–Scholes…
The document clarifies the meaning of the cumulative return variable used in Rubinstein’s implied binomial tree method. The questioner has derived terminal path probabilities and recursively aggregated them to earlier nodes, then asks how to obtain the up…
The document discusses upper-boundary conditions in finite-difference solutions of the Black-Scholes equation. Its example considers a European put and explains that imposing zero curvature at a finite upper stock-price boundary can suppress a non-negligible…
The document derives a risk-neutral price for a European claim paying the square of a stock price at maturity. Starting from a geometric Brownian motion with zero physical drift, the answer emphasizes that pricing uses the risk-neutral drift, which is the…
The document examines a minimum-variance portfolio problem using a bank account, a stock, and call options on that stock. Monte Carlo paths under geometric Brownian motion supply estimated returns and covariance. Because the options and stock share a driver,…
The document compares variance swaps with options for expressing a view on volatility. Its central point is that a variance swap's vega is described as independent of the underlying price, while the vega of a single option changes with spot and is greatest…
The document explains why a call and put with the same strike and maturity have deltas that differ by one. Put-call parity relates the call-minus-put position to the underlying asset minus the discounted strike. Differentiating that relationship with respect…
The discussion derives discounted stock dynamics for a Bachelier model with constant drift and volatility, then explains how to choose a Girsanov kernel so the discounted price has zero drift under a new measure. The kernel depends on the current stock…
The document asks why the binomial call pricing formula uses two probabilities: the risk-neutral up-move probability p and an adjusted probability p′ equal to (u/r)p. In the formula, the term weighted by p′ contributes the stock component, while the strike…
The document addresses a common confusion about equity put skew: a farther out-of-the-money put can have a higher implied volatility than a nearer out-of-the-money put, yet still cost less in premium. Implied volatility describes the volatility input that…
The document considers how a shareholder might protect gains after a large stock advance while retaining a longer-term position. The choices raised are selling shares, buying a short-dated out-of-the-money put, or using a collar. A longer-dated put is…
The document distinguishes the pricing formula for a swaption from the volatility model used to represent market prices. It says Black-style pricing is widely used, while arguing that the choice and behavior of the volatility model deserve more attention…
This note describes a question about reconciling a discretely rebalanced Black–Scholes hedge with the payoff of a written call. The questioner tracks a stock-and-bond portfolio, observes that its terminal value matches the option payoff in a simulated path,…
The document examines the continuation value in the Longstaff–Schwartz simulation method for American options. The question asks whether conditional expectations of future cash flows can be separated from discount factors evaluated at the current time. The…
The document explains how settlement after expiry can affect option valuation, distinguishing cash-deferred from physically deferred delivery under deterministic interest rates. In the cash-settled case, the exercise payoff is determined at expiry and then…
This discussion explains why some QuantLib C++ result-access methods were initially unavailable in its Python wrapper and outlines ways to obtain swaption sensitivities. The accepted answer says the generic result and additional-results methods were not…
The document outlines pricing approaches for European options on the difference between two correlated futures prices under a lognormal framework. For a zero strike, it identifies the Margrabe formula, interpreting the pricing problem through one futures…
The document offers a simple way to estimate premiums for near at-the-money calls in a covered-call simulation. It presents an at-the-money Bachelier approximation in which the premium depends on the underlying price, volatility, and square root of time to…
The document derives the first-passage-time density for a price following risk-neutral geometric Brownian motion and clarifies the scaling in a standardized log-distance expression. A change of probability measure removes the drift from the Brownian motion;…
The document raises a practical model validation question for Bermudan swaptions: how to determine whether computed prices and Greeks are correct. It considers comparing numerical methods such as Monte Carlo, PDEs, and trees under the same model, comparing…
The document explains why options on an underlying with a wide bid-ask spread may also have wider spreads. One mechanism concerns options traded without delta hedging: a market maker who takes the other side may need to hedge by trading the underlying and…
The document presents the Black–Scholes formula for vomma, also called volga, and asks whether the expression and its inputs are correct. Vomma is the second derivative of an option price with respect to volatility, so it measures how vega changes as implied…