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 note explains the integral notation used to price a European call. Writing the differential before the integrand is an equivalent way to write the usual integral; the density multiplies the call payoff over terminal prices above the strike. The payoff is…
The document asks how to extend a volatility skew from a set of observed strike and volatility points across a much wider strike range. The initial cubic spline approach produces negative values at distant strikes, raising concerns about overshoot and…
The document explains a practical way to estimate a covariance matrix for a portfolio containing a stock and multiple calls on that stock, with a common maturity and different strikes. Rather than relying on historical return data, simulate the stock forward…
The document raises a modeling question about a two-regime option-pricing setup built around a risk-neutral GARCH process. The author fits crash and normal regimes and observes that options are more expensive in the higher-volatility crash regime. The…
The document clarifies that gamma is the local rate of change of an option’s delta with respect to the underlying price. It is a derivative, so multiplying gamma by a relatively large underlying move gives only a linear approximation; delta and gamma…
The document examines whether a continuously monitored Asian call can be bounded by a vanilla call with the same strike and maturity. Under zero rates and dividends, and assuming continuous asset paths, it uses the integral mean-value theorem to represent…
The document discusses how to estimate the covariance matrix needed to construct a Markowitz efficient frontier. It names maximum-likelihood estimation from historical returns as a standard approach, then describes alternatives: shrinkage of the covariance…
The document explains what happens when an investor sells a European put they previously bought before expiry. Selling the contract to close realizes the change in its market value; it does not leave the investor with an unhedged short put obligation. The…
The document raises an input-selection problem in estimating implied correlation from bivariate option prices. The described approach trains a model on a decade of returns, prices options, and inverts Stulz's formula to infer a correlation coefficient.…
The document explains why Monte Carlo estimates of a European option price can become less precise as volatility rises. The central point is that the estimate averages simulated payoffs, and its sampling error depends on payoff variance and the number of…
The document formulates a contingent claim on an underlying price process that starts between two barriers. The claim pays a specified function of the underlying value at the first time the process exits the interval. This differs from the barrier options…
A cash-settled swaption pays a function of the market swap rate, commonly the intrinsic value multiplied by a cash annuity. The annuity converts the rate payoff into a cash amount. Because the swap rate is observable, this convention can make the payoff…
The document derives the replication error that arises when a trader hedges an option with a Black–Scholes delta while the underlying asset’s realized variance differs from the constant variance assumed by the model. For a one-dimensional diffusion with the…
The document derives the risk-neutral version of the Heston stochastic-volatility model from its historical-measure dynamics. It introduces market prices of risk as shifts to the correlated Brownian motions, then applies Girsanov's theorem to express the…
The document raises a question about how changing volatility affects the distribution of option payoffs across simulated underlying-price paths. Its proposed comparison keeps other inputs and the set of paths fixed, then compares payoff histograms produced…
The document asks whether Monte Carlo option-pricing results from different copula models can be compared when antithetic variates are used only for symmetric copula families. The concern is that the variance-reduction technique may not be applicable to…
The document outlines how to infer caplet implied volatilities from a sequence of cap prices and swap rates, as a step toward calibrating a Hull–White interest rate model for swaption pricing. It assumes caps are at the money and explains that the inputs…
The document considers how to balance hedge quality against bid–ask costs when hedging an exotic instrument with liquid instruments. It proposes measuring the cost of an imperfect hedge as a high percentile of losses over a chosen risk horizon, so the…
The document describes Bloomberg's DLIB suite as a toolkit for pricing and calibrating derivative structures, with templates for some exotic products. It presents BLAN as an option for building bespoke deals from scratch and experimenting with different…
The document introduces Greek surfaces as three-dimensional representations of an option Greek across time to maturity and moneyness, analogous to implied volatility surfaces. Its motivating research question is whether changes in those surfaces, combined…
The document distinguishes contingent claims from forward commitments, two categories of derivative contracts. A contingent claim has a payoff that depends on an underlying asset's value or another uncertain event; an option is the standard example because…
The document corrects an attempted derivation of an option price limit under the Black–Scholes–Merton framework. It explains that an option’s price is an expectation under the risk-neutral measure, where the stock’s drift is the risk-free rate, rather than…
The document asks how to select an Esscher change-of-measure parameter for an exponential Lévy asset so that the discounted asset price is a martingale under the new measure. It writes the Radon–Nikodym density as an exponential tilt and uses conditional…
The document explains how correlation between asset returns and volatility innovations affects the relationship between Black–Scholes volatility and a stochastic-volatility model such as Heston. Negative correlation is reported as typical for equity indices…