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 raises a conceptual question about what delta neutrality protects against. A position with zero delta at the current underlying price has no first-order price sensitivity at that point, but that condition alone does not describe how the payoff…
The document clarifies a common confusion between a quantity being random over time and being known at a particular time. A call’s delta is the derivative of its price with respect to the underlying price; it is not calculated by dividing by a future, random…
The document discusses implementing a finite difference pricing model for equity index options using local volatility derived from a fitted SVI volatility surface. The question is how to align the surface’s log-forward-moneyness coordinate with the pricing…
The document explains how Fourier pricing formulas for European options can be adapted from spot assets to futures. It focuses on the Variance Gamma model and notes that its raw process characteristic function needs a drift adjustment so the exponentiated…
Underlying stock return correlation does not directly determine the correlation between two option positions. The document illustrates this with perfectly correlated stocks and options whose exposures point in opposite directions: a deep in-the-money call on…
The document asks what information the filtration represents when defining a risk-neutral measure for the Heston stochastic volatility model. A filtration describes the information available up to each time, and the response gives a market-oriented example:…
The document explains a no-arbitrage trade when a European call costs less than its intrinsic value, defined here as the underlying price minus the strike. The described position buys the call and shorts the underlying. At expiration, if the underlying is…
The document introduces variance swaps as instruments for taking a view on an underlying asset’s volatility, described in the question as historical volatility. It asks whether a variance swap can practically hedge vega or volga exposure, and in what market…
The document explains how a near at-the-money payer and receiver swaption pair behaves as the underlying forward swap rate moves. Its initial delta is approximately zero, though the exact value depends on the pricing setup; spot-premium treatment can also…
The document examines a backtest of a collar formed with shares, a long put, and a short call, with both options struck at the current stock price. Its payoff expression simplifies the option payoff terms at expiration, leaving the initial call-put premium…
The document describes estimating an event-related move from options by separating variance expected across ordinary time from the extra variance attributed to the event. It uses at-the-money implied volatilities for the first two expirations after the event…
The document explains why implied volatility can still be useful even though it is calculated from an option’s market price. It presents Black–Scholes as a way to transform observed prices into volatility values, which can then be used to estimate prices for…
The document explains how to price a barrier option with a finite-difference engine in QuantLib when volatility is represented by a market implied-volatility surface. It points to the finite-difference Black–Scholes barrier engine and notes that its…
The discussion considers whether an American call can be exercised early when a continuous dividend yield is high relative to the risk-free rate. Its key point is that the sign of the adjusted rate alone does not determine the exercise decision. For…
The document derives a European call value when a forward price follows a two-factor diffusion with volatility components that vary over time. It applies the exponential solution for the forward process and combines the factor variances into an integrated…
The question describes problems retrieving an implied volatility smile from a SABR volatility surface through QuantLibXL. A positive time input below one year produces a negative-time error, while another error appears to contradict the function's default…
The document asks whether a European put’s price can be calculated as the expected payoff when the underlying’s return over the option’s life follows a specified density. Assuming zero interest rates, it writes the put payoff as the strike minus the terminal…
The document explains why an in-the-money option does not necessarily become more valuable simply because expiration is closer. It separates an option’s value into intrinsic value, determined by the current underlying price and strike, and time value, which…
The document outlines the Black–Scholes derivation: apply Itô’s formula to an option whose value depends on the underlying, combine it with the underlying in a portfolio, hedge away the source of randomness, and use no-arbitrage to obtain the pricing PDE. It…
The document explains why an up-and-out call can have negative delta when the underlying approaches its upper knock-out barrier. Near the barrier, a rise in the underlying has two opposing effects: it increases the ordinary call payoff conditional on…
The document describes a pricing problem involving a multi-asset collar on European natural gas and US oil products, with an additional currency conversion because the contracts are denominated in different currencies. The questioner is considering Monte…
The document asks about an approximation discussed in a paper on delta hedging with a volatility smile: treating the sensitivity of implied volatility to the underlying price as equal to its sensitivity to strike. Under a downward-sloping smile, the cited…
The document explains how aggregate dealer gamma may affect hedging flows near options expiry. When dealers are long gamma, movements in the underlying can lead them to trade against the move, buying after declines and selling after rallies. When dealers are…
The document considers whether a Euro Stoxx futures contract and matching European call and put options offer an arbitrage when only one of each contract may be traded. It introduces put-call parity and rearranges it to compare the call-minus-put price with…