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.
21,023 documents
The document explains a Value at Risk calculation for a mixture of a standard normal variable and an independent, rare discrete shock. In the example, the shock shifts a small-probability component far into the loss tail. Because that component accounts for…
The document compares two ways to obtain future distributions and price claims from an Itô process. A Fokker–Planck equation evolves the probability density through a deterministic partial differential equation, while Monte Carlo estimates outcomes from…
The document considers how to compare two optimized portfolios whose returns are positively skewed and have positive kurtosis. The author asks whether to estimate each portfolio's alpha against value-weighted and equally weighted benchmarks, or regress one…
The document explains two special cases of the SABR model by conditioning on a realized volatility path. With beta equal to one, the asset follows a multiplicative diffusion, so its logarithm has a normal distribution and the asset level is lognormally…
The document shows how to extend the covariance matrix of underlying assets to include portfolios formed from those assets. It represents the asset returns or random variables as a vector X, the portfolio weights as a matrix W, and the portfolio variables as…
The document explains why a Double Heston model uses two stochastic variance processes. A second factor adds flexibility to the relationship between asset returns and volatility, allowing that correlation to change as the factors’ relative contributions…
The document asks how to interpret cross-asset moves after an FOMC announcement and whether deleveraging can be measured through changing covariance, signal responses, or trend formation. The response cautions against explaining market moves with a single…
The document considers how to estimate a house’s current value from property characteristics and an earlier sale date. It contrasts including date directly in a gradient-boosting model with first adjusting historical prices for market inflation, then…
The document explains why a bond’s implied spot rate can differ from a rate curve built with a SciPy univariate spline. The bond has two remaining cash flows, and its price and yield-to-maturity equation imply a final-cash-flow spot rate of about 2.8%, while…
The document outlines a way to analyze a portfolio’s style exposures by linking securities’ factor characteristics to subsequent returns. It proposes a cross-sectional regression across the investment universe, with next-period stock returns as the outcome…
The document poses a numerical modeling question about discretizing a mean reverting Ornstein–Uhlenbeck process that combines continuous Brownian noise with random jumps. The jump component is modeled as a normally distributed jump size multiplied by a…
The document explains why historical value at risk can be unreliable when an asset has only a short return history: the observed sample may not include the full range of plausible losses. It considers using sector returns as a proxy, but notes that averaging…
The document discusses how to interpret and empirically assess Geometric Brownian Motion (GBM). Under GBM, log returns follow a normal distribution whose mean combines the drift and a volatility adjustment. The drift is therefore a model parameter: its…
The document presents practical ways to estimate the parameters of the Vasicek short-rate model without relying only on an unconstrained fit to bond prices. One approach discretizes the model into an approximate AR(1) process and matches theoretical and…
The document explains why daily forex observations cannot be transformed into tick bars or volume bars. Daily prices are aggregates of underlying market activity, so they do not retain the sequence of individual quotes, trades, or trade sizes needed to…
The document discusses how to choose regression basis functions in the Longstaff–Schwartz least squares Monte Carlo method for valuing American-style options, including real options with multiple risk factors. The question is whether the few Laguerre…
The document questions the assumption behind conventional strategy optimization: that the process generating historical market data remains stable. It notes that out-of-sample checks and periodic reruns can help address overfitting, but may not resolve the…
The document introduces a two-good exchange economy with two individuals and asks how a Walrasian auctioneer's price adjustments relate to equilibrium stability. The proposed tâtonnement rule raises a good's price when it has excess demand and lowers it when…
The note considers an Itô integral of a deterministic function against Brownian motion and explains why it has zero mean, a covariance determined by the overlap of the integration intervals, and Gaussian distributions. The mean follows from the martingale…
The note works through a 97.5% expected shortfall calculation for two independent bond portfolios. Each portfolio has two possible loss amounts, and independence lets the combined outcomes be enumerated: both portfolios incur the smaller loss, one incurs…
The note explains how to reverse the order of integration in a time integral of Brownian motion, expressing the result as a stochastic integral with the differentials in the opposite order. The original integration region has time variable s between zero and…
The document derives a Black–Scholes–Merton value for a call-like payoff that is positive when the terminal asset price exceeds the strike and scales with the reciprocal of that price. It rewrites the payoff using an indicator and inverse asset, relating the…
The document examines whether arbitrage-free pricing forces zero-coupon bond prices or spot rates to move monotonically with maturity. The question derives a presumed declining spot-rate curve from an assumed monotonic increase in bond prices, then notes…
The document explains why the choice of discount rates changes a bond duration measure. Traditional Macaulay duration discounts each cash flow using the bond’s single yield to maturity, then weights its payment time by the present value of that cash flow…