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The document explains how a delta-hedging argument leads to the Black–Scholes partial differential equation. A portfolio holds one derivative and shorts an amount of stock equal to the derivative’s sensitivity to the stock price. Applying Itô’s formula…
The document examines a Black–Scholes claim paying the squared terminal stock price times its logarithm. It compares three valuation approaches: risk-neutral expectation using a lognormal moment, changing to the stock as numeraire, and attempting to use the…
The document collects recommendations for learning C++ with quantitative finance applications in mind, particularly for a Python developer who wants to build pricing libraries. One reply suggests choosing a book suited to the learner’s current level: an…
The document explains the currency exposures involved in delta hedging a USDJPY option. In the example, a bank has sold a dollar put and calculates a short dollar hedge from the option’s delta. That hedge is paired with a long yen position at the prevailing…
The document discusses whether quarterly institutional holdings disclosures can inform trading strategies. Replies say that investors and researchers do use Form 13F information, including strategies that follow reported holdings of prominent funds. One…
The document explains how to interpret a Carhart four-factor regression when evaluating a portfolio or trading strategy. Rather than treating a coefficient as a direct comparison between individual small and large companies, it frames the factor coefficients…
The document compares practical considerations for stochastic-volatility and GARCH option models, emphasizing that model choice depends on the option type and the goal. It argues that GARCH can be attractive because its next-step conditional volatility is…
The document surveys practical considerations for options with expiries ranging from hours to minutes. It cautions that very short-horizon returns should not be assumed to follow a lognormal process simply because the horizon is short. Instead, intraday…
The document discusses what can make market prices temporarily differ from fundamental value, in the context of market resiliency. It identifies several microstructure mechanisms: asymmetric information, market makers’ inventory risk, trading costs, and…
The document explains how a forward exchange rate above the level implied by spot and the two currency interest rates creates a covered interest arbitrage. The proposed trade borrows the foreign currency, exchanges it for the home currency at spot, invests…
The document examines a foreign-exchange spot quoted as dollars per euro and compares expectations under dollar and euro risk-neutral measures. The question derives discounted-price expectation identities using each currency's bank-account numeraire, then…
The document asks whether the low-volatility anomaly—the empirical pattern that lower-volatility stocks can earn higher returns than riskier stocks—could be explained by the difference between arithmetic and geometric average returns. It points to the…
The document considers whether a panel model with industry fixed effects is suitable for estimating the Fama-French five-factor alphas of many industry portfolios. The factors can be common across industries and vary over time; constructing separate factor…
The document explains why standard Black-style approximations for cash-settled swaptions imply payer-receiver parity around the forward swap rate. Under that approximation, a zero-wide collar—the difference between payer and receiver swaptions at the same…
The document distinguishes the risk that a borrower or issuer will fail to repay from the risk that a trading counterparty will fail to meet its contractual obligations. A bondholder, for example, faces the issuer’s credit risk; a buyer of credit protection…
Without a risk-free asset, investors cannot draw the usual capital allocation line from that asset or identify an optimal risky portfolio as its tangency with the efficient frontier. The document points to Black’s zero-beta CAPM as a framework for this…
The discussion cautions against assuming that VIX futures will reliably offset losses in a long-only equity portfolio. VIX measures expected volatility, and it does not invariably rise when stocks fall; consequently, an inverse relationship with stock…
The note clarifies that Diebold–Li is a way to estimate a Nelson–Siegel yield curve, rather than a separate curve model with its own distinct estimation procedure. The key modeling choice is to fix the Nelson–Siegel shape parameter; with that parameter held…
The note explains how to interpret the instantaneous forward variance quantity in a model of volatility smiles. It distinguishes that quantity from a forward-starting variance swap, whose fair strike is constructed from the variances of two maturities. The…
This discussion asks how to use volatility from an implied-volatility surface in Reiner-Rubinstein formulas for an up-and-out call, expressed as a vanilla call minus an up-and-in call. The vanilla component may reference the strike while another component…
This example compares delta-gamma and delta-vega hedging for a short call using another call and shares. It gives Black-Scholes Greeks for an in-the-money call and an at-the-money call, then solves for the option quantity that offsets gamma or vega and the…
This discussion examines how to compare a cash basket of S&P 500 constituents with an E-mini futures contract when assessing index arbitrage. It asks whether to aggregate constituent bid and ask quotes separately and discount those totals, or use mid prices…
This discussion frames Funding Valuation Adjustment (FVA) through an uncollateralized FX forward with a client, hedged by a bank under a Credit Support Annex. It asks what an FVA calculated from the trade’s expected exposure profile represents, given that…
The document explains why an unusually high buy limit does not automatically cause a trade at that price. A marketable buy order generally matches against available sell orders at their existing limit prices, so a small order can execute at the best…