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The document explains how a risk-neutral probability density for the underlying at expiration can be recovered from the second strike derivative of call prices, under a constant interest-rate assumption. Once recovered, the density can be integrated against…
The document works through a derivation associated with Jegadeesh and Titman’s explanation of momentum: the expected cross-sectional covariance between a stock’s return and its prior return. It clarifies that the bar denotes a cross-sectional average, then…
The document concerns fitting a Libor Market Model volatility function to market observations by minimizing a nonlinear least-squares objective over five parameters. The calibration has positivity conditions, including a combined constraint on two…
The document explains why the repo rate on a scarce, specific bond can fall when demand to borrow that bond rises. It corrects the intuition that bond buyers are borrowing more cash and therefore should pay a higher rate: in a repo transaction, one party…
The document asks whether a one-day P/L estimate based on sensitivity to a zero-rate curve includes the effect of time passing. The setup recalculates the curve on the next day using the same bonds, whose remaining maturities have shortened, and applies the…
The note compares a discounting adjustment expressed as the expected discounted integral of the difference between a target rate and a base rate, multiplied by exposure, with a formula attributed to Piterbarg for a fully collateralized position. The apparent…
Bid volume and ask volume refer to trades executed at the bid and ask prices. The note asks whether these quantities can be approximated when only open, high, low, close, and total volume are available. Its response says a model can be constructed, but the…
The note explains why an FX swap can retain a small exposure to the future spot rate even when its near and far legs exchange the same domestic notional. The key is that the notional returned at maturity does not include the interest that would have accrued…
The document compares two ways to connect an Interactive Brokers tick feed to analytics and signal generation: a single integrated application that keeps data in memory and writes it to disk, or separate processes that persist feed data for independent…
The document considers how to select strategy parameters after ranking many combinations on an in-sample period and then checking the strongest performers on an out-of-sample period. Its answer warns that repeatedly selecting the top-performing percentage…
The document asks whether the variance of a price can be derived from the variance of its daily returns, including for use in Bollinger Bands. It explains that treating the previous price as known can yield a conditional relationship for one step, but that…
The document distinguishes the regression meaning of alpha from the broader way traders use the term. In a single-benchmark model, portfolio returns are regressed on benchmark excess returns; beta captures market exposure, while alpha is the intercept…
The document describes a market-making experiment in a video-game item marketplace. It reports that some items appeared to have stable supply and demand, while occasional sales below the best bid created negative spreads. A program repeatedly reposted quotes…
The document explains how to calculate downside beta when the condition is that the benchmark return is below zero. The answer selects only the observations that meet the benchmark condition, then applies the ordinary beta calculation to those paired asset…
The document asks how useful implied volatility remains when the Black–Scholes assumptions do not fit the market. It uses stock-borrowing fees as an example: different borrowing and lending rates can change the pricing equation, so an option price must be…
The document poses a risk-neutral pricing problem for a derivative that pays the square of a non-dividend-paying stock price at maturity. It proposes discounting the expected terminal payoff, but the attempted expression does not satisfy the Black–Scholes…
The document examines whether the single-index model determines the covariance between two stocks from their market betas and the market variance. It explains the textbook formula by expressing each stock’s excess return as a market-linked component plus an…
The document asks why binomial short-rate models often assign equal risk-neutral probabilities to up and down moves, and whether model properties such as no drift or a flat volatility term structure depend on that choice. It highlights a concern: if changing…
The document clarifies the relationship between normally distributed log returns and the expected value of a geometric Brownian motion price. If log price follows a Brownian process with drift μ and volatility σ, exponentiating it yields a lognormal price…
The document describes a dissertation calculation of daily returns for a value-weighted portfolio of large corporations, using prior-day market values as weights to avoid using future information. It outlines normalizing market values across constituents,…
The document considers portfolio shrinkage targets that do not require expected-return estimates, including minimum-variance, risk-parity, equal-risk, equal-weight, and maximum-diversification approaches. Such targets can temper noisy return estimates used…
The document explains how an Ornstein–Uhlenbeck process models reversion toward a long-run level and identifies the inverse reversion speed as its characteristic timescale. It notes that first-passage-time formulas can estimate how long a process takes to…
The document discusses how to interpret a participation threshold in a square-root market-impact model. It presents impact as proportional to volatility and the square root of order size relative to daily volume, and notes that the cited framework assumes…
The document addresses whether different fixed rates in a fixed-for-fixed currency swap create a free return for one counterparty. Its explanation emphasizes that each party pays interest on a principal denominated in its currency and returns that principal…