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.
12,303 documents
The document explains why options on the same underlying can imply different volatilities across strikes even though the Black-Scholes model assumes constant volatility. It presents several interpretations: the smile can reflect model misspecification,…
The question attempts to find the distribution of the stochastic term in a Heston-style asset process by substituting an integral expression for the variance process, taking a square root, and applying Brownian differential identities. The response…
The document explains why foreign exchange option data may arrive as volatility quotes and delta-based smile points rather than as a ready-made table of implied volatilities by strike and maturity. In much of the over-the-counter market, options are quoted…
The document discusses the arbitrage-free valuation of a claim paying a power of the terminal stock price in the standard Black–Scholes model. The questioner applies Itô’s lemma first to the power process and then to its logarithm, but obtains an exponent…
The document explains how to interpret a VaR limit specified over a fraction of a year and convert the volatility implied over that horizon into annualized volatility. For a horizon represented as one of m equal yearly intervals, first identify the VaR…
The document asks whether a European call is worth more when volatility is random with a given mean than when volatility is fixed at that mean. It introduces Jensen’s inequality as the key tool: if the option price is a convex function of volatility, then…
The document investigates the integrated value of a call option across maturities, defined as the time integral of its price, and asks why its leverage ratio becomes hump-shaped near the at-the-money spot level under low initial variance in the Heston model.…
The document asks for the mean and variance of a mean-reverting process whose noise is proportional to the state. The answers first present an explicit-solution expression and moment formulas, but a later response flags that expression as incorrect and gives…
This exchange considers how the Avellaneda–Stoikov reservation price responds to a stock's price scale and an inventory measured in shares. The question compares a stock before and after a large share consolidation, where the share price and volatility per…
This discussion explains how to estimate annualized volatility from a sequence of weekly stock prices. The method is to calculate consecutive log returns, find their sample standard deviation to estimate weekly volatility, then scale it by the square root of…
The document asks how to implement time-shifted power-law kernels used in a path-dependent volatility model. The model forms volatility from a baseline plus weighted past returns and weighted squared returns. Separate kernels assign the lag-dependent weights…
The discussion explains how a local-volatility model relates an implied-volatility surface to an instantaneous volatility function of spot and time. Its central clarification is that local volatility is calibrated from the current implied surface; if that…
The document frames a one-month portfolio problem involving cash, a stock, and a six-month European put. Returns are evaluated under the physical measure, but the option’s future market value depends on risk-neutral pricing. The author explains why valuing…
The document concerns Greeks for the Bachelier model, which assumes normally distributed price changes. The author says delta, vega, and gamma are straightforward to derive, then focuses on deriving rho and theta when the interest rate is nonzero. They…
The document explains how the Stochastic Volatility Inspired (SVI) parameterization can represent implied total variance across option strikes. Its formula describes a smile using five parameters: a level, slope scale, skew, horizontal shift, and curvature…
The document discusses adding an explicit high- or low-volatility regime to a Fama–French regression for a stock portfolio. The proposed research question is whether a manager’s factor exposures differ by regime, such as favoring quality during high…
The document addresses how option vega relates to gamma and explains a formula attributed to Nassim Taleb’s Dynamic Hedging. Under Black–Scholes, gamma is expressed using the asset price, volatility, time to expiration, and the normal density at d1; vega…
The document derives the conditional variance of a price process that changes through Poisson-timed jumps with independent, lognormally distributed multipliers. It first expresses the price at the horizon as its current value multiplied by all jump factors…
The document examines an interpolation rule for a Wiener process whose volatility is piecewise constant between known endpoint values. The proposed rule scales the endpoint increment by the square root of a ratio involving accumulated variance terms. The…
The document asks why an FX 25-delta butterfly is commonly expressed using the average of 25-delta call and put volatilities adjusted by the at-the-money straddle volatility. It also asks what the 25-delta label means in this construction, placing the focus…
The document asks how to build a function that returns implied FX volatility for an arbitrary strike and maturity from quoted market points. The available inputs are 10-delta and 25-delta put and call volatilities, along with zero-delta straddle or…
The document explains how to construct a forward swap rate from discount factors or forward Libor rates, then connect it to swaption valuation. It derives the swap rate as the ratio of the swap’s floating-leg value to its fixed-leg annuity. Given Libor…
The document outlines pricing approaches for derivatives with general payoff functions under European, American, and perpetual exercise rules. For European claims, it gives the risk-neutral discounted expected payoff and explains that a specified underlying…
The document describes a Monte Carlo setup for studying the profit and loss of a long out-of-the-money call that is delta hedged over time. The option is purchased using the volatility assumed at inception, while simulated underlying price paths reflect the…