คลังความรู้

The document proposes revisiting the portfolio construction rule at every rebalance instead of committing to one allocation scheme throughout an investment horizon. It contrasts optimized approaches, such as minimum variance and maximum Sharpe portfolios,…

The document asks how to handle zero weights from a Markowitz allocation when every asset must be included, and how to determine the risk-free allocation when constructing a portfolio on the capital market line. The response focuses on selecting the risky…

The document explains why investors may hold VXX despite the ongoing value decay commonly associated with contango in VIX futures. It presents two uses: a short-term directional view on volatility, or a hedge for portfolios and trading strategies that are…

The document considers pricing a callable range accrual whose coupon depends on how often a spread between long- and short-tenor constant-maturity swap rates stays above a threshold. The question notes that a LIBOR Market Model can be slow and that a…

The discussion examines whether Black–Scholes–Merton derivations require an unproven self-financing condition. It describes replication arguments in which a portfolio holds stock and a money-market account, and its changes in value come from gains on those…

The document proves convexity for a portfolio risk measure consisting of expected return plus a positive multiple of the expected absolute deviation from expected return. It assumes portfolio return is linear in portfolio weights, so expected return is also…

The discussion distinguishes arbitrage from disequilibrium in models of asset prices. Arbitrage offers a sure profit and creates a strong incentive for market participants to trade it away. By contrast, two stocks with similar volatility but different…

The document examines why the optimal risky portfolio can appear to differ when optimizing with or without access to lending and borrowing at a risk-free rate. Its central correction is that maximizing expected return divided by volatility is not equivalent…

The document addresses fitting a skewed Student t distribution to financial limit order book returns and then simulating from the fitted model. It points to an R distribution-fitting library for maximum likelihood estimation and to a package that implements…

The document explains why multiplying single-period variance by the number of periods can misstate the variance of compounded horizon returns when returns are serially correlated. Under covariance stationarity and a common single-period variance,…

The document discusses a market maker who is assigned an in-the-money American call after the underlying market closes, creating stock delivery and delta exposure by the next open. Because the underlying cannot be traded during that interval, the direct…

The document asks how to interpret theta and gamma for a portfolio holding one call while dynamically hedging its delta with the underlying and financing trades. It gives the local P&L relationship in which theta contributes over time and gamma contributes…

This derivation uses an exponential-affine form for a zero-coupon bond price in the Vasicek short-rate model and substitutes it into the risk-neutral bond pricing PDE. Matching the constant and rate-dependent terms yields differential equations for the time…

Junior tranches of a credit default swap index absorb losses before more senior portions, so a small change in the underlying portfolio can produce a much larger proportional change in the tranche’s value or spread exposure. The document illustrates this…

This discussion addresses a payoff paid at a later date that uses the underlying price at both an earlier and a later date. The standard risk-neutral valuation formula still applies when the payoff is measurable at the payment date: by then, both observed…

This brief spreadsheet example shows how to make a Bloomberg BDP query use a strike stored in another cell instead of a fixed value. The method is to concatenate the ticker’s text components with the cell reference, while formatting the ticker with spaces as…

The question asks how correlations in credit default swap data should be interpreted, particularly when researchers use correlations between equity returns to build relationship graphs. The answer cautions that credit correlation can refer to distinct…

The discussion considers how to infer implied volatility from option bid and ask quotes across strikes, expiries, calls, and puts. It covers estimating forwards through put-call parity, choosing representative option prices, and fitting a consistent…

The question concerns valuing an interest rate swap whose schedule began before the valuation date. It raises two implementation issues: representing a declining notional and including a floating coupon whose rate was fixed earlier. The response gives a…

The question examines whether a security’s price can be recovered by taking expectations of its dividend-to-price return definition and the rearranged price equation. The answer explains that the apparent conflict with the rule that the expectation of a…

The discussion examines steps in a proof about conditional expectations under two probability measures, P and Q. The question focuses on why a conditional expectation under Q can be treated as measurable with respect to a sigma-algebra G, and how that…

The document presents an equity portfolio-selection problem based on monthly returns for a large stock universe and a market index. Its code estimates each stock's average return, beta, residual risk, and Treynor ratio, then applies a single-index cutoff…

The question asks how to obtain the volatility input needed to calculate an option’s Black–Scholes delta. The replies explain that this input is commonly implied volatility: the volatility value consistent with the observed market price of the option and the…

The document discusses pricing a European exchange option that pays the positive difference between two stock prices at expiry. The assets follow correlated geometric Brownian motions. One answer changes to a measure using one asset as numeraire; under that…