คลังความรู้

The document explains why a futures quote usually does not predict where the underlying asset’s spot price will be at expiration. For storable commodities and financial products, arbitrage links futures and spot prices: financing, storage, insurance, and…

The document asks how wealth inequality affects an individual’s savings relative to peers, but the response does not provide a calculation of that effect. Instead, it distinguishes three related measures of economic disparity: income, consumption, and wealth…

The document describes a Monte Carlo check of a continuously rebalanced delta hedge for a European call under the Black–Scholes model. The simulated portfolio gains are formed from the option delta and successive underlying-price changes. The initial…

The document explains why stochastic jump processes are commonly modeled as càdlàg: their paths are right-continuous and have left limits. In a Poisson process, the value changes at the jump time, so the jump is visible in the process value at that time.…

The document asks why securitized-product prepayment models often represent borrower status with discrete delinquency states and estimate transition probabilities, rather than directly regressing the dollar amount prepaid in each period. It notes that…

The document gives the product rule for a process multiplied by a deterministic, finite-variation function, then extends it to the product of two stochastic processes. For a finite-variation function and a process, the ordinary two-term product rule applies.…

The document explains how a zero-coupon bond price relates to forward rates in discrete and continuous time. It divides the period to maturity into subintervals and expresses the bond price as the product of discount factors for each interval. With simple…

The document asks when an American put and a European put on the same underlying can have equal value, motivated by checking an American put binomial-tree implementation. It notes that early exercise does not benefit a call on a non-dividend-paying stock, so…

The document poses a simple betting problem about how to profit from information that changes the estimated probabilities of three mutually exclusive outcomes. A bettor can stake money on any outcome, receiving twice the stake if correct and nothing…

The document asks how to value a discretely sampled arithmetic Asian option at an intermediate time using Monte Carlo simulation. It gives a simplified setup with three observation dates, zero interest rate, zero strike, and a single simulated path. At the…

The document explains a change in integration order used when taking the Fourier transform of a call option price expressed as an integral over terminal log prices. The payoff contributes only when the log price exceeds the log strike, so the original nested…

The document asks why the exponential of an accumulated integral of a state-dependent function has differential equal to the current exponential multiplied by the integrand and elapsed time, without the Brownian and second-derivative terms that appear in…

The document asks how to compare options with positive or negative net credits and estimated expected values, including credit trades and a debit trade. Its brief answer defines a simple return measure for a short option position: divide the premium received…

The document considers a stock whose drift follows a correlated Vasicek short rate, and asks how to simulate the stock price when the integrated rate is random. It presents the exact stock-price expression in terms of the integrated short rate and stock…

The document examines an FX option whose payoff is denominated in the foreign currency. It rewrites the payoff of a call on the exchange rate as a scaled put payoff on the reciprocal exchange rate, then compares two proposed Black-Scholes-style expectation…

The note compares the minimum-variance hedge or portfolio choice with minimizing risk capital defined from value at risk and expected return. Under a full-investment constraint and a fixed target return, the efficient frontier is found by minimizing…

To rank days by volatility during a fixed intraday window, use the sequence of closing prices sampled at each minute. Convert those prices to logarithms, calculate consecutive differences, and take the standard deviation of the resulting minute log returns…

The discussion asks how to classify a bid-ask spread as favorable for scalping or ordinary day trading, especially when comparing low-priced, thinly traded OTC shares with more liquid stocks. It notes that absolute spread size can mislead: a wider spread in…

The discussion examines whether randomly selecting a long-only portfolio of S&P 500 stocks should match the index on average, and whether a naïve investor should therefore be expected to underperform. It distinguishes the mean return from the median: with…

The document considers fitting a two-normal mixture to stock returns, with one component intended to represent the distribution's central behavior and another its tail. It highlights the central estimation problem: rare tail events may be absent from a short…

The document cautions that comparing effective bid-ask spreads across assets can be misleading when volatility and trading activity differ. It gives a qualitative relationship for small-tick securities: spreads tend to rise with volatility and fall as the…

The document explains how a constant maturity swap (CMS) rate payoff differs from the payoff of a vanilla swap. A vanilla swap's value depends on both the rate difference and the swap's DV01, which itself varies nonlinearly with the underlying rate. A CMS…

The note outlines a way to express a Eurodollar futures rate when the short rate follows a Vasicek process. It treats the futures settlement rate as the risk-neutral expectation of the three-month LIBOR rate at its fixing date. LIBOR is written as an…

The note diagnoses a failure in an equal risk contribution portfolio optimizer that returns its starting weights after one iteration. The central bug is in the objective function: although the optimizer passes candidate weights into it, the calculations…