คลังความรู้

This exercise asks whether a market with two assets and three states admits arbitrage, and how to bound the price of a risk-free asset. State prices assign a price to receiving a unit payoff in each state; multiplying the payoff matrix by the state-price…

This note reconciles three explanations for the inverse relationship between bond prices and yields: the return implied by paying a price for a fixed payoff, the opportunity cost of holding a bond when other interest rates rise, and the discounting of future…

This discussion examines why sharp equity declines can unfold quickly, including when observed trading volume appears modest. One proposed mechanism is leverage: falling stock prices can trigger margin calls, forcing investors to sell positions and sometimes…

This note explains why conversion factors affect hedge ratios during the trading life of a deliverable Treasury futures contract but do not change the physical delivery quantity. Because the futures price does not move one-for-one with a particular bond, the…

The document derives the probability density of the minimum of a standard Wiener process over a fixed time interval using the reflection principle. It first observes that the minimum has the same distribution as the negative of the process maximum, since…

The document raises a mathematical question about expressing the self-financing condition for a continuously traded portfolio when gains are defined through Lebesgue–Stieltjes integration. It contrasts this setting with discrete trading, where portfolio…

The document explains how the bond price dynamics in the Vasicek short-rate model follow from the model’s closed-form bond price. That price is written as a time-dependent factor multiplied by an exponential involving the current short rate and the function…

The document discusses how to choose a risk-free rate when valuing bonds issued by Ecuador, which uses the US dollar and may not have a readily available local government yield curve in common financial databases. One answer recommends using the US dollar…

The document describes a calibration problem for a one-factor Hull–White interest rate model with constant mean reversion and volatility. The author reports that Monte Carlo prices match market prices for three swaptions in a set of four, while the first,…

The document considers a daily classification task: using index features to predict whether the next day's open-to-close move is up or down. The proposed workflow forms a direction label, selects and scales features, trains an SVM on a rolling window, and…

The discussion explores how discrete binomial models can help teach Itô's lemma and its correction term. One proposed illustration transforms a symmetric random walk with a convex function, such as squaring it, then compares the expected transformed value…

The document examines whether one stock divided by another remains a martingale after changing from the money-market account to a stock numeraire. It sets out two uncorrelated stock processes under the risk-neutral measure and applies a Radon–Nikodym density…

The document addresses how to identify and plot peaks in a stock price series. It notes that peak definitions vary and that a threshold parameter in a peak-finding function may be unclear. As an alternative, it describes a streak-based method: classify…

In the classical mean-variance setup, expected returns and covariances are treated as known inputs, so optimization produces deterministic portfolio weights. The weights become random when the inputs are estimated from return data: changing samples can…

The document examines a second-order approximation for the certainty equivalent of a risky payoff. Expanding utility around the payoff’s mean approximates expected utility using the utility curvature and payoff variance; a first-order expansion of utility at…

The document presents a question about applying the Capital Asset Pricing Model to a two-stock portfolio. The investor supplies portfolio weights and estimated betas, then asks how to obtain the expected return and whether a correlation inferred from the…

The document asks why the characteristic function used in Heston stochastic-volatility option pricing satisfies the same partial differential equation as the probability terms in a call-price representation. It presents the governing PDE for those terms,…

The document describes an inconsistency in Yahoo Finance data access for Chinese equities: a legacy quote endpoint returned values for a Shenzhen-listed symbol but missing values for Shanghai-listed symbols, even though a separate historical-data endpoint…

The document explains the direction of a USD FRA-OIS swap position. The answer states that a trader expecting the Libor-OIS spread to widen would pay on the swap, paying OIS plus the spread and receiving Libor. This clarifies the cash-flow direction behind…

The document asks how to extend Black–Scholes pricing when the stock follows both a diffusion and a resettable Poisson jump process. One response points to the jump-diffusion version of Itô's formula and the generator-based Dynkin formula as the tools for…

The question considers forecasting a future multi-day return from a history of rolling returns. Because adjacent rolling windows share most of their daily observations, the inputs are strongly autocorrelated. The response says that autocorrelation alone does…

The position buys two deep out-of-the-money calls and puts, while selling one nearer out-of-the-money call and put. One response describes it as a ratioed strangle switch; another views it as an iron condor combined with a long strangle. These are…

The document compares a normal model for a forward swap rate, where volatility measures absolute rate changes, with a lognormal model, where volatility measures proportional changes or log returns. It derives the relationship by considering the lognormal…

The document outlines prudential valuation adjustments (AVAs), which bridge fair value and a more conservative estimate of realizable value under orderly exit conditions. It places these adjustments in a regulatory and accounting context, noting that…