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 defines a convex family of semimartingale wealth processes that start at zero, never fall below minus one, allow certain combinations of strategies, and are closed in the Emery topology. It asks for the financial motivation behind the…
The note derives the expected squared value of a geometric Brownian motion (GBM), starting from its stochastic differential equation. Applying Itô’s lemma to the square of the process shows that the squared process is itself a GBM, with adjusted drift and…
The document asks how to derive average value at risk, also called conditional value at risk, from a piecewise value-at-risk curve for a triangular fuzzy number in a credibilistic setting. It gives separate formulas for VaR below and above the midpoint…
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 discussion considers how to estimate the market return input to CAPM when an index has flat or declining historical performance and its average return falls below the risk-free rate. One response says a negative market risk premium is mathematically…
This recreational logic puzzle assigns different cat counts to three speakers. Each tells the truth when addressing someone with fewer cats and lies when addressing someone with more cats. The accepted solution first derives the ordering of their counts from…
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 discussion considers how to estimate a smooth price distribution when most observed intraday futures prices occur only once. It suggests starting with a histogram, where the chosen price-bin width changes the visible clusters, and describes kernel…
The discussion addresses how to estimate or manage the time a mean-reversion trade takes to return toward its mean. It notes the Ornstein–Uhlenbeck process as one way to model half-life, but shifts attention to whether a candidate spread has a credible…
The discussion considers how to obtain a nine-month LIBOR fixing when that tenor is unavailable. It emphasizes that interpolation depends on the intended use. In a loan whose contract required a nine-month rate, the parties agreed to create a synthetic…
The document works through pricing a zero-coupon bond under a Vasicek-style short-rate model with zero mean reversion level. It starts from the rate’s stochastic-integral solution, inserts that solution into the discounted payoff, and changes the order of…
The document derives the duration of a single fixed cash flow under annual effective compounding. Starting from the negative sensitivity of present value to the interest rate, differentiation gives a duration equal to the payment time divided by one plus the…
The document addresses whether a positive-weight, minimum-variance portfolio optimization maps each expected-return vector to a distinct set of weights. Its answer is no: distinct return inputs can lead to the same portfolio, so the mapping from expected…
The document explores how to shock selected points on an on-the-run Treasury yield curve, propagate those shocks to other maturities, and reprice an off-the-run bond. The proposed method applies principal component analysis to historical yield changes, then…
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 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 examines a claim from the Longstaff–Schwartz interest-rate model that the short rate, expressed as a weighted sum of two factors, is positively correlated with the variance factor. The answer reduces the problem to the sign of the covariance,…
The document applies the Kelly betting formula to historical results from selling option spreads and asks whether a calculated allocation of about 20% of capital is too high. The response explains that Kelly is aggressive and assumes a well-specified bet…
The document explains why the first principal component of a stock return covariance or correlation matrix is sometimes used as a proxy for a market factor. PCA finds the direction that accounts for the greatest cross-sectional variation. When the first…
The document asks whether returns that are highly skewed or fat-tailed over short intervals become more normal when compounded over a longer period. It describes an example of accumulating weekly yields across a year and observes that, when yields are…
The document presents a proposed mean-variance portfolio optimization using residual covariance from a diagonal model. Its constraints set a target expected return, require the portfolio weights to sum to one, impose a beta-related condition, and specify a…
The document explains why an inverted yield curve is often viewed as a warning about economic weakness. Under expectations-based accounts, long-term rates reflect expected future short rates; an inversion can therefore indicate that markets anticipate rate…
The document raises a question about what option open interest represents across different stocks. For small-cap options, the author sees open interest mainly as a sign of liquidity. For highly traded large-cap names, where liquidity may be less limiting,…
The document derives a consumption process restriction from a supermartingale condition used in an optimal contracting problem. The key step applies Itô’s lemma to discounted consumption raised to a power, relating its differential to a martingale component…