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 considers a contract paying the average stock price over an interval ending at maturity. It introduces a running integral of the stock price as an additional state variable and applies Itô’s lemma to a price function of time, spot, and that…
The document discusses whether Malliavin calculus has practical value in finance, especially for computing Greeks on exotic derivatives. Its main use case is a payoff with nonsmooth features, where finite-difference estimates can be difficult or unstable.…
A digital option pays a fixed amount if it finishes in the money and nothing otherwise. The note explains that the payoff diagram’s height is this fixed payment, often denoted by δ. The underlying price determines whether the payment is triggered, while the…
The note explains why the Black–Scholes call pricing formula is dimensionally consistent, despite an apparent problem involving the square root of time in the normal cumulative distribution inputs. The option price, underlying price, and strike are all…
The document explains how to specify correlated Brownian drivers for a two-asset Heston model. Since each asset has a price process and a variance process, the simulation uses four stochastic drivers and therefore requires a joint four-by-four correlation…
The document explains how to derive a local volatility surface from a smooth implied volatility surface and how to handle its derivatives in numerical work. It recommends using finite differences across expiry at fixed moneyness for the time derivative,…
The document asks how to infer negative-moneyness implied volatilities for interest-rate swaptions when broker data provide at-the-money volatility and payer and receiver spreads only at positive offsets from the at-the-money forward. It considers lognormal,…
The document examines the Black–Scholes value of a European call as its expiration moves far into the future, assuming no dividends. In the formula, the call value depends on two probabilities expressed under different measures. As maturity increases, the…
The document raises a pricing question about arithmetic-average Asian options with discrete sampling. The author describes simulating asset paths with the Euler method and estimating expected payoffs by Monte Carlo, then observes that at-the-money Asian…
The document considers early exercise of American options on Treasury futures. One standard argument is that an American call without an underlying cash-flow benefit has no reason to be exercised early, while American puts require a more involved…
The document explains why options with different strikes and expiries can have different implied volatilities even though volatility models describe the underlying asset. Implied volatility is the Black–Scholes volatility that corresponds to an option’s…
The document explains why the correlation between a Brownian-motion process at time t and its time average is not one, even though the average has a distribution that can be written using a scaled Brownian variable. Equality in distribution preserves…
The document examines how to isolate the state price for a particular expiration value when the underlying can finish only at integer prices. It corrects a proposed expression involving calls and puts and gives equivalent portfolios: a second difference of…
The document examines interest-rate sensitivity for an option on a futures contract when the option and futures use futures-style settlement. It uses the Black 76 call-pricing expression, in which the option value is discounted by the risk-free rate while…
The document examines how to interpret variance vega in a stochastic volatility model. It considers changing the variance at the valuation time by a small amount and asks whether that change should be added to the variance path at every later time, or…
The document explains why the meaning of delta depends on whether someone is discussing a model Greek, an individual trade, or portfolio exposure. A single option’s pure delta is a sensitivity to the underlying price and is dimensionless; multiplying it by…
Dupire local volatility calibrates a deterministic function of time and spot to reproduce observed European option prices across strikes and maturities. This makes it convenient for matching the initial vanilla option surface, but the model ties volatility…
The response derives the price of a contract paying the squared log return at expiry. It first expresses the payoff as a function of the terminal asset price and values it as a discounted expectation under the risk-neutral measure. It then applies the…
The discussion explains a possible meaning of the phrase “variance swap volatility.” It interprets the quantity as the volatility level associated with a variance-swap strike that makes the contract have zero value at the relevant time. This is a concise…
The document presents an exercise involving prices of calls across strikes and asks how to derive prices for two other terminal-payoff claims: a digital that pays when the underlying exceeds a strike, and a squared call payoff. It defines the call value as…
The document considers whether a holder of bitcoin can mimic put-option protection by shorting CME bitcoin futures above a chosen price and returning to a neutral position when the price crosses that threshold. The motivation is concern that a rising market…
The document distinguishes forward variance swap volatility from the future implied volatility smile used to value forward-start options. Under idealized assumptions of a pure diffusion and continuously measured variance, the forward variance swap rate can…
The document raises a notation issue in Fourier option pricing with the Heston model: some characteristic functions include a term for the current log-price, while others show only terms involving the model’s variance state. The distinction is whether the…
The document examines two Black–Scholes sensitivities for an at-the-money European option: how Vega changes as time to expiry grows, and how option price changes as implied volatility rises. For the expiry question, it considers the derivative of Vega with…