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The document considers a three-day, 95% value-at-risk estimate for a long European call when the underlying follows geometric Brownian motion. It describes valuing the option with Black–Scholes, deriving a lower-tail stock-price quantile from the lognormal…
The document discusses how to choose a long-run growth or discount assumption for a perpetuity in firm valuation. One response suggests linking the rate to the valuation model and using a risk-free rate reduced by three percentage points for earnings-based…
The example explains why restoring target weights in a leveraged long-short portfolio can require buying an appreciated asset and increasing a short position in an underperformer. After a strong gain in the long asset, the portfolio’s total value has also…
The document discusses how to build mathematical foundations for quantitative finance and statistics. The questioner seeks rigorous, proof-based study of relevant tools such as linear algebra, probability, set theory, stochastic calculus, and measure theory,…
The discussion asks how to infer a stock price probability distribution from option prices and track changes in that distribution over time. One response identifies implied probability densities, also called state price densities, as the relevant framework.…
The document raises a portfolio selection problem: maximizing the Omega ratio for non-normal returns can favor assets with unusually high returns and produce a portfolio whose holdings have similar, correlated return patterns. The investor describes…
The discussion surveys possible ways to obtain historical daily quotes across VIX option strikes and expirations. One response says reliable public data may be unavailable and cautions that analysis may also require the corresponding VIX futures history,…
The answer explains how to move from a Bachelier underlying with an unspecified physical drift to a risk-neutral process for pricing a European call. It starts with the discounted asset price and uses the martingale condition to identify the required drift…
The document reviews a Heston option-pricing implementation that produces negative probabilities or prices under extreme inputs, despite giving plausible at-the-money results for other parameter choices. The implementation applies a characteristic-function…
The document examines whether ETF market makers can secure the bid–ask spread by immediately hedging trades with correlated instruments such as futures. The questioner models the spread earned in the ETF against the spread paid in the hedge and introduces a…
The document considers whether ESG can be added as a factor to the Fama–French five-factor model and how such an extension should be interpreted. It distinguishes a factor that captures exposure to priced cash-flow risk from a preference for firms with…
The document explains why borrowing at the risk-free rate to scale a risky portfolio increases its expected excess return and volatility by the same factor. The Sharpe ratio uses excess return, so the leveraged portfolio’s numerator is the leveraged…
The note resolves how to obtain a short-maturity yield expansion from a Taylor expansion of a zero-coupon bond price. Starting with the price written as one plus successive powers of time to maturity, it applies the logarithm series and then negates and…
The note explains why the profit and loss of a portfolio hedged with stock must include dividends received or owed on the stock position. It writes the portfolio value with a money market account, stock, and short derivative, then tracks those holdings over…
The document asks how to estimate term structures of default probabilities when market data is unavailable, including for non-listed firms. It notes that migration-based Markov chains can turn historical rating transitions into rating-specific default…
The note rewrites the payoff of a power option, defined as the underlying raised to a power times a call payoff, as a difference of two asset-or-nothing binary call payoffs. It first expands the payoff into two terms, each gated by the condition that the…
The document discusses a long-short portfolio formed by taking equal-weight average monthly returns for two groups of stocks, subtracting the mature group’s returns from the technology group’s returns, and regressing the spread on Fama-French market, size,…
The document addresses a choice of risk-free rate when calculating CAPM quantities or evaluating a portfolio strategy. The question contrasts a three-month Treasury rate with a ten-year rate for a strategy whose lookback window is one year and asks whether…
The document describes a way to account for recurring time-of-day patterns when estimating volatility with GARCH on hourly returns. It recommends estimating a deterministic intraday periodicity component and filtering returns by that component before fitting…
The document frames a model-design problem for estimating the concentrated positions component of an additional valuation adjustment for derivatives. It proposes relating price or volatility impact to the volatility of valuation inputs, bid–ask spread…
The document proves that for thresholds a ≥ b, the expected value of a random variable conditional on being at least a is no smaller than its expected value conditional on being at least b, assuming the higher-threshold event has positive probability. This…
The document proposes measuring possible conflicts of interest between credit rating agencies and countries through the share of agency revenue associated with each country. The idea is to look beyond direct payments from governments and consider fees paid…
The note asks whether the expected product of increasing functions of two correlated stochastic processes rises as their instantaneous correlation rises. The response sketches an Itô-calculus argument in a simplified setting: it assumes zero drift and unit,…
The note considers whether a Treasury issue that trades special in overnight repo is cheap relative to a fitted Treasury spline. Its central lesson is that the value of repo specialness depends on how long the bond is expected to remain special. A current…