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The document discusses how to interpret exercise decisions for options with multiple exercise dates. It uses a European call to illustrate the basic rule: at expiry, exercise when doing so gives a positive payoff. For a Bermudan option, it compares immediate…
The document explains why an asset forward price has the same percentage volatility as its spot price when the interest rate is constant and the cost of carry equals that rate. In this setting, the forward price is the spot price multiplied by a…
The document outlines a model for pricing a call on realized volatility by treating realized volatility as the maturity value of a volatility swap. It assumes the swap price follows a lognormal process driven by the same Brownian motion as the underlying…
The document addresses why the Black–Scholes derivation relates an option’s price dynamics to those of a self-financing stock-and-bond portfolio. Its central explanation is that replication is constructed backward from the option payoff at expiration: a…
The document asks how to infer the expected returns that would produce a given fully invested portfolio under mean-variance optimization, given its weights, risk aversion, and return covariance matrix. The accepted answer gives a simple relation: implied…
The document examines why a set of out-of-the-money FXI puts appears to become much more expensive with added time to expiration than a simple square-root-of-time rule would predict. The quoted example lists ask prices and implied volatilities across several…
The document considers which market variables might be studied alongside Twitter activity, including index and stock prices, price differences, trends, and expected returns. A response recommends treating posts as a source of market sentiment rather than…
The document asks why investors are shown historical price and fund-return data if past performance does not guarantee future results. The responses offer several uses: assessing a manager's record relative to a benchmark, examining consistency or hit rates,…
The document explains why a Sharpe ratio is scaled by the square root of the number of periods in a year. It compares daily, weekly, and monthly sampling of a synthetic portfolio and reports broadly similar annualized ratios. The central idea is that…
The document distinguishes factor returns from the risk premia estimated by the second stage of a Fama-MacBeth regression. After estimating assets’ factor loadings in time-series regressions, the second stage regresses cross-sectional returns on those…
The document examines whether a small subset of original variables can represent the first principal component while contributing little to the others. It explains that principal components form a coordinate basis for the original feature space. In a…
The document discusses the common binomial-tree choice of an up factor based on volatility and the square root of the time step, with the down factor set as its reciprocal. This reciprocal relationship makes the recombining tree compact: after each step,…
The document asks how to choose the decay half-life when estimating an x-day value at risk (VaR) from an exponentially weighted moving average of return vectors. It gives the relationship between the half-life and the EWMA decay factor, then asks whether the…
The document considers how to price European calls and puts on an underlying that can jump, fluctuates with short-term supply and demand, and tends to lose value over longer horizons. It also raises practical questions about physical delivery and indivisible…
The document asks how to interpret the time-dependent drift parameter in the Ho-Lee short-rate model while calibrating a discrete interest rate tree. In the described procedure, each period's parameter is chosen so that model prices under the risk-neutral…
The document poses a portfolio mapping problem that arises when factor returns are whitened and then reduced or transformed into secondary factors. It distinguishes the time series of original factor returns from the matrix of asset portfolios associated…
The document examines whether portfolio Value-at-Risk can be derived by treating asset returns as multivariate t distributed. It explains that each individual component of a multivariate t vector has a univariate t marginal with the same degrees of freedom.…
The document explains how the change in a log-price spread relates to the return from a pairs position with a hedge ratio of one. For a spread defined as the log price of asset A minus the log price of asset B, its change equals A’s log return minus B’s log…
The document distinguishes two UK short-rate series that may appear similar but represent different quantities. Treasury bill observations are described as average discount rates from weekly tenders for 91-day bills. The government curve’s three-month spot…
The document surveys ways to estimate covariance matrices when a portfolio universe contains many assets. It identifies two broad approaches: shrinkage, which reduces estimation noise in a conventional sample covariance matrix, and factor models, which…
The document asks how to include irregularly dated dividends when calculating weekly stock returns over a long history. The proposed workflow currently searches for the closest dividend date for each weekly price date, which is slow. The questioner also…
This exchange considers how to update expected asset returns when rebalancing a Markowitz portfolio day by day. The question proposes incorporating each new observation into the historical sample and recalculating the expected returns and portfolio weights.…
This exchange clarifies the difference between general drift and volatility processes and coefficients written as functions of the current state and time. State-dependent coefficients are a special case of stochastic processes, but specifying the…
This document explains how to calculate the probability that a geometric Brownian motion (GBM) price will fall inside a specified interval at the next time step, given its current value. This conditional probability is used in the stochastic mesh method to…