คลังความรู้

The discussion compares possible ways to track changes in option volatility skew relative to at-the-money volatility. One answer proposes measuring the implied-volatility difference between the strikes where Black–Scholes d2 equals zero and d1 equals zero.…

The answer explains asset liability management through the example of a pension plan, where invested assets must support future benefit payments. Liability assumptions include the size and timing of payments, mortality, inflation, and potentially political…

The question sets up a discretized forward-rate model in which each forward is lognormal under its associated zero-coupon bond measure. It derives the resulting normal distribution for the log forward at its fixing date and uses that distribution to motivate…

The answer models an interest rate swap as an asset leg and a liability leg, then expresses generalized duration as the asset value times its duration minus the liability value times its duration. The sign reflects the liability’s negative contribution to…

The note contrasts pricing in complete and incomplete financial models. In a complete model, it describes a unique martingale measure and a derivative price given by the discounted expectation under that measure. In an incomplete model, multiple martingale…

The note concerns a measurability step in proving that the first time an adapted càdlàg process enters an open set is a stopping time. The stated argument expresses the event that the hitting time is strictly before a fixed time as a union over rational…

The note asks how to differentiate a Carr–Madan Fourier integral for option prices with respect to a model parameter. Since the integration limits do not depend on that parameter, the proposed application of differentiation under the integral sign leaves the…

The note explains why option prices generated by a stochastic volatility model can imply a volatility smile when those prices are converted using the Black–Scholes formula. Under Black–Scholes pricing with one fixed volatility, the prices of calls across…

The document derives the futures price for an asset whose price process and short rate are driven partly by the same Brownian motion. Since the futures price is the expected future spot price, the solution first expresses the integrated short rate as a…

The document raises a replication question about estimating a monthly factor regression for European equity portfolios, with particular interest in the intercept as abnormal risk-adjusted performance. It describes calculating monthly portfolio returns from…

The discussion asks whether a floating-rate note’s discount margin (DM) change can approximate its credit-related return, in the way spread changes and duration are used for fixed-rate bonds. One proposed method estimates the bond’s price sensitivity to a…

The document describes a comparison between intraday VIX-style implied volatility and realized variance computed from one-minute prices aggregated into hourly intervals. The implied measure is calculated from option prices for two maturities around 30 days,…

The document distinguishes a bid or ask tax from a transaction tax: the charge applies when a limit order is submitted, even if the order never trades. It describes two laboratory double-auction studies in which participants traded against experimenter-set…

The discussion considers GPU selection for parallel quantitative finance workloads, with the original question asking whether hardware should favor multiplication over addition. One answer suggests that for MATLAB or Python on a single machine, the…

The document considers the probability that one of two correlated geometric Brownian motion prices finishes above the other, while the lower price also exceeds a threshold. It rewrites the event using shifted Brownian variables as two simultaneous…

The document derives the variance of a weighted sum of two correlated Brownian increments. For coefficients applied to each increment, the variance includes both individual variance terms and a covariance term determined by their correlation. Since each…

The document shows how to estimate geometric Brownian motion parameters from a series of daily prices. It first maps prices to log returns, which follow a normal process with drift equal to the price process’s drift minus half the variance rate. When the…

The document explains how derivatives can change a borrower’s practical liquidity and recovery profile even when derivative claims rank equally with senior debt in bankruptcy. Under collateral agreements, a firm may have to post cash or securities as market…

The document frames a feature-selection problem using order book snapshots with five bid and ask price levels, corresponding displayed quantities, and a midprice defined from the best bid and ask. The proposed target is the change in midprice across a few…

The document asks how to extend a CAPM certainty-equivalent present-value formula to several priced risk factors. The proposed expression subtracts each factor’s price of risk multiplied by the cash flow’s covariance with that factor, then discounts the…

This note applies Itô’s lemma to the logarithm of a price following geometric Brownian motion. Starting from a stochastic differential equation with drift proportional to the price and volatility proportional to the price, it differentiates the logarithm…

The discussion distinguishes policy rates, interbank rates, and other borrowing rates, then explains how changes in interest rates can relate to currency values. One answer describes the chain from central bank rates to bank and business borrowing costs,…

The document asks how to decide whether a portfolio consideration belongs inside an existing objective function or should be optimized as a separate objective. It starts with a conventional formulation that seeks higher expected return while controlling…

The document examines a transaction described in a book about Enron. It says Enron held futures on its own stock whose gains could not be recognized as profit under accounting restrictions, while it had recorded a mark-to-market gain on RhythmsNet shares.…