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20,364 dokumenter
The document explains the Carr–Madan formula as a way to represent a sufficiently smooth payoff as a linear position in the underlying, a cash amount, and integrals of European put and call payoffs across strikes. The decomposition expands the payoff around…
The document shows how put–call parity can identify an arbitrage across European options with two different strikes, even when the current stock price is unknown. Applying parity separately at each strike and subtracting the equations removes the stock-price…
The document discusses extracting implied volatility from American option prices using a Cox–Ross–Rubinstein binomial tree. The tree prices the option by working backward from terminal payoffs, comparing continuation value with early exercise value at each…
The document explains how to simulate stock paths when periodic log returns are modeled with a Normal Inverse Gaussian (NIG) distribution, with the aim of pricing options by Monte Carlo. It warns that an NIG random variable cannot simply replace the normal…
The document questions a boundary condition for a travel-time option whose state includes a mean-reverting process with additive Brownian noise and its time integral. In the familiar geometric Brownian motion example, zero is absorbing: once the underlying…
The document discusses using Empirical Mode Decomposition (EMD) of EUR/USD prices as input to a machine learning model. It addresses a discrepancy between promising test-period results and poor predictions on the latest observation. The proposed explanation…
The document explains why a geometric Brownian motion simulation can appear to rise consistently when its drift is strongly positive. Applying Itô’s lemma to the GBM model shows that the log return over a fixed horizon is normally distributed, so the price…
The document explains how to compare spreads on fixed-coupon bonds and loans or floating-rate notes. Its answer says that the fixed bond’s additional interest-rate exposure need not justify a separate spread adjustment, since that exposure can be hedged at…
The document raises a practical problem in estimating covariance for portfolio construction when asset price histories have different gaps. Missing observations arise from holidays, isolated data omissions, and longer stretches without quotes, leaving…
The document presents two parametric ways to describe equity implied volatility across strikes and maturities. One uses ATM forward volatility plus linear skew and quadratic curvature on a log-moneyness scale normalized by volatility and the square root of…
The document discusses using the PerformanceAnalytics Return.portfolio function to apply a time series of portfolio weights to daily asset returns. The author observes that specifying different rebalance_on frequencies appears to produce identical portfolio…
The document clarifies why repo and reverse repo can refer to opposite sides of the same collateralized transaction. A party that delivers a bond and receives cash is commonly said to repo out the bond; economically, that party is borrowing cash against the…
The document considers how to select and prepare market and macroeconomic time series for a portfolio return regression. The portfolio example is denominated in Swiss francs and includes global equities, fixed income, gold, commodities, currency pairs, and…
The document addresses whether a martingale-corrected Andersen discretization for the Heston model needs adjustment when the model has a nonzero constant interest rate. The answer endorses adding the risk-free accrual over each time step to the exponent…
The document shows how to solve a stochastic differential equation whose process has zero drift and a time-varying adapted volatility coefficient. It applies Itô's lemma to the logarithm of the process, turning the multiplicative equation into an additive…
The document explains why a foreign exchange forward is not generally the conditional expectation of the future spot rate under the domestic risk-neutral measure when interest rates are stochastic. The forward price instead comes from discounting the payoff…
The document asks whether vanilla options on two assets, together with the assets themselves, can create insurance that costs money when correlation is high and pays when correlation falls. The motivation is to hedge a strategy that benefits from the assets…
The document outlines requirements for historical global equity research data: programmatic access, coverage of delisted companies and corporate outcomes, long histories of annual and quarterly fundamentals, daily closing prices, and broad international…
The document frames the rate sensitivity of a pay-fixed, receive-floating interest rate swap as three contributions: discounting floating coupons, discounting fixed coupons, and projecting floating cash flows. It asks how the magnitudes of those components…
The document asks how to move from evaluating a machine-learning model on historical test data to producing forecasts from newly received broker data. The author describes training several model types on price and fundamental inputs, currently using a random…
The document compares three ideas used to explain why the Black–Scholes–Merton option-pricing formula can hold: profitable trading opportunities, market efficiency, and market equilibrium. It asks why equilibrium might be considered the most important of…
The document asks for methods to manage asset allocation across discrete transition states, with attention to liquidity, trading costs, and delayed cause-and-effect relationships. It contrasts moving from cash directly into an asset with transitions between…
The document raises the problem of checking whether prices produced by models such as Black–Scholes or Heston are accurate. It distinguishes a priori validation, which includes choosing a model and fitting it to market data, from a posteriori validation…
The document discusses how to choose the spot-price bump when approximating call-option Greeks numerically and comparing them with Black–Scholes analytical values. It favors central differences, which reprice at spots on both sides of the current level, over…