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The document derives an approximate single implied volatility for a portfolio of options whose components have different implied volatilities. It begins with the condition that the portfolio’s modeled value at the common volatility should equal the sum of…
The document explains how to estimate a bond’s value by discounting each scheduled coupon and principal payment. It adds the bond’s z-spread to the relevant spot rate, uses the resulting rates to calculate discount factors, and sums the discounted cash…
The document compares two martingale derivations of the Black–Scholes partial differential equation. With the bank account as numeraire, requiring the discounted option price to have zero drift yields the familiar PDE. The attempted stock-numeraire…
The document offers historical volatility and correlation estimates as starting points for a foreign currency option model with domestic equities, foreign equities, and an exchange rate. Using weekly observations over five years for the DAX, S&P, and EUR…
The document explains why the delta of a binary call becomes sharply concentrated around its strike as expiry approaches. Under Black–Scholes, the option value is expressed using the normal cumulative distribution function, and differentiating gives a delta…
The document discusses how to calculate p-values for estimated GARCH coefficients and whether the degrees of freedom should account for the model’s parameters. One response recommends using the sample size minus the total number of estimated parameters,…
The document examines how Actual/Actual ISMA determines coupon amounts for a fixed-rate bond with a short or long stub period. Its example has a first coupon running from the issue date to a February payment date, followed by monthly coupons. The initial…
The document examines whether a rising risk-free rate necessarily raises the earnings yield, defined in the discussion as earnings divided by price. It begins from an earnings-based relation between earnings yield, the equity risk premium, and the risk-free…
The document derives an expression for the expected value of a process described by a stochastic differential equation with drift and diffusion terms. Rewriting the equation in integral form separates accumulated drift from the stochastic integral. Under the…
The document outlines a basic workflow for studying whether investors favor value or growth stocks during crises. It suggests obtaining constituent stock prices from market data sources, using an established equity research classification to separate value…
The document explains how deterministic constants and trends can be specified in Johansen cointegration models. It lists five model forms, ranging from unrestricted constants and trends to no constant or trend, and says nested specifications can be compared…
The discussion distinguishes uncertainty in portfolio allocations from uncertainty in the inputs used to construct them. Mean-variance optimization can produce a precise allocation from estimated returns and covariances even when those parameters are poorly…
The document derives a European call pricing representation for an asset whose returns combine continuous Brownian movement with independent Poisson jumps. When jump sizes are lognormally distributed, conditioning on the number of jumps makes the terminal…
An implied volatility surface reflects option prices that vary by strike and maturity, unlike the constant volatility assumption in the basic Black–Scholes model. Looking at one maturity at a time, a steep downside wing means out-of-the-money puts are…
The discussion asks whether manipulation of SPX options or equity and volatility futures caused the February 2018 VIX spike, and what data could help investigate. The response points to volatility-linked exchange-traded products as a possible source of…
The document asks why an American put can have a different value from a European put when both are considered under the Black–Scholes framework. It contrasts the pricing inequality and payoff constraint for an American option with the familiar result that,…
The document describes an attempt to estimate value at risk (VaR) and expected shortfall (ES) with a peaks-over-threshold method using a generalized Pareto distribution (GPD). In a rolling sample of Petrobras returns, the author encounters a software error…
The document concerns parametric expected shortfall (ES) when returns are modeled with a four-parameter Paretian stable distribution. It describes a question about implementing a closed-form ES method attributed to Stoyanov, with VaR defined for returns as a…
The document surveys constraints that practitioners may impose when optimizing a portfolio. It notes that the formulation matters: a fully invested portfolio typically has weights summing to one, while an active portfolio expressed as deviations from a…
The document derives the conditions under which the unconstrained minimum-variance portfolio of two assets has no short positions. Starting from the formula for the weight on the first asset, it requires that weight to be nonnegative and no greater than one.…
The document asks how a short-term VIX futures index’s daily roll weights translate into the holdings and cash flows of an exchange-traded product. It uses a dated example with two adjacent futures prices to question how a roll handles a price difference,…
The document asks whether the conditional expectation of an exponential Brownian increment, given information available at an earlier time, can equal its unconditional expectation. It assumes the earlier time is no later than the endpoint and questions…
The document derives a closed-form price for a European payoff based on the positive part of one minus the strike divided by the terminal stock price, assuming the stock follows geometric Brownian motion under the money-market measure. Its key observation is…
The document discusses practical uses of equity return factors such as the Fama–French factors, momentum, and liquidity or tail-risk measures. It describes factor investing as a portfolio construction approach and notes that predicting factor returns, often…