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 compares two ways to estimate the distribution of an hour of USDJPY one-minute returns: repeatedly convolving an empirical histogram and Monte Carlo sampling sums of 60 observed moves. The simulated sums produce a wider distribution, while the…
The question examines whether an American option on a deterministic, zero-volatility asset can have time value. Its proposed reasoning treats the option’s value as the best payoff reached along the asset’s deterministic path and argues that early exercise…
The document asks how to estimate portfolio volatility when each of two assets has an exponentially weighted variance estimate. The central guidance is to calculate correlation using the same weighting convention as the variances. Applying common weights to…
The discussion distinguishes independent increments from dependence on the current state and from correlated increments. It notes that Wiener motion has independent increments, while geometric Brownian motion’s price changes scale with the current price; the…
The document addresses the claim that small-cap stocks are more volatile than large-cap stocks and suggests comparing the standard deviations of their returns, a common measure of volatility. It gives an illustrative comparison between a large-cap market ETF…
The note explains how to interpret transition parameters from a three-state Markov-switching GARCH fit in R’s MSGARCH package. Although a three-state transition matrix has nine entries, only six are independently estimated because each state’s remaining…
The document addresses the conditional second moment of a process with linear drift, a deterministic time-varying additive drift term, and two correlated Brownian shocks. The additive term prevents the logarithm of the process from having dynamics…
The document derives the price of a call option that pays the square of the underlying price divided by the strike when the underlying finishes at or above the strike, and otherwise pays nothing. Under a Black–Scholes lognormal model, the direct approach…
The document explains why options on one stock can have different implied volatilities across strikes and maturities. Implied volatility is a way to express an option price using Black–Scholes assumptions; it does not mean the market expects each option’s…
The document addresses whether Dupire’s local volatility formula can be expressed using put prices as well as call prices. Under the stated zero-rate, zero-dividend setup, or when working with undiscounted options on forwards, the formula uses the option’s…
The document asks how traders should select strikes and expiries for option positions such as protective puts and volatility strangles, while expressing concern that choices based only on backtest performance may be overfit. The response offers a practical…
The discussion asks what to do when a historical Value-at-Risk model fails conditional coverage or independence backtests. It recommends treating the failure as a prompt to investigate the model and its inputs, rather than assuming that changing the return…
The document explains two Brownian motion calculations relevant to barrier option pricing: the midpoint distribution conditional on the terminal value, and the probability of crossing a barrier between fixed endpoints. For the midpoint, it uses the joint…
The document explains two ways a long option position can benefit from volatility: a rise in implied volatility can increase the option’s resale value, while large underlying price moves can benefit a long gamma position. A long straddle combines exposure to…
The document asks for the expected product of a Wiener process evaluated at three ordered times. It establishes that the expectation is zero, using properties of Brownian motion rather than a numerical example. One argument exploits the symmetry of the…
The discussion addresses whether maximum-likelihood estimates of a Heston stochastic-volatility model, often used in derivative pricing, can be applied to value at risk. The reply distinguishes risk-neutral calibration for pricing from real-world risk…
The discussion examines a Monte Carlo pricing expression that represents VIX through the risk-neutral expectation of the log of the S&P 500’s future level relative to its current level. The question is why simulated values might make the expression under the…
The document discusses a proposed solution to a modified Black–Scholes equation for a European put when volatility depends on the underlying asset price. The proposed expression uses associated Laguerre polynomials and is evaluated as a series. A numerical…
The document discusses ways to generate returns that reflect skewness and kurtosis, which a normal model does not capture. One approach uses a truncated Gram–Charlier Type A density: its coefficients are linked to cumulants, and the described setup gives a…
The document investigates why a short put’s hedge delta increased even as the underlying price rose slightly. It reports that volatility also declined and two days elapsed, with a long time remaining to expiration. The response identifies charm, or delta…
The document raises practical questions about modeling American options with Black–Scholes–Merton, including early exercise, changing bid–ask spreads, intraday time to maturity, volatility estimation, historical risk-free rates, strike listings, and…
The document explains how put-call parity for year-on-year inflation caps and floors can be used to infer the market rate of the underlying inflation swap. In present-value terms, the cap value minus the floor value equals the value of a swap that pays fixed…
The discussion explains why VIX returns are often negatively associated with S&P 500 returns even though VIX represents implied volatility rather than market direction. In falling markets, investors may seek protection through index puts, raising option…
The question asks why some researchers fit curves to implied volatilities rather than option prices, given that both can vary nonlinearly with strike. The response points out that a standard pricing framework such as Black–Scholes provides a conventional…