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The document clarifies that implied volatility is a Black–Scholes-equivalent volatility inferred from a European vanilla option price for a given strike and maturity. A smile implied by a different pricing model is constructed in two steps: fix that model’s…
The document explains why an interest rate swap’s sensitivity to its forecast curve can change when its discount curve moves. Changing discounting alters the present value assigned to projected floating-leg cash flows; as those values change, the swap’s…
The document describes a two-asset risk-parity objective: compute each asset’s share of portfolio variance from its weight and the covariance matrix, then minimize the distance between those shares and equal contributions. The initial attempt uses a…
The document corrects the claim that a European option’s binomial price does not depend on probability. It distinguishes physical probabilities, which describe actual beliefs about market moves, from risk-neutral probabilities, which are implied by the…
The document examines whether an implicit finite difference scheme for the Black–Scholes partial differential equation guarantees nonnegative derivative values and numerical stability. It sets out a uniform price grid, the tridiagonal system for stepping…
The document clarifies the relationship between a bond’s price and its yield to maturity. Yield to maturity is defined as the rate that discounts the bond’s cash flows to its observed price, so a quoted price can be used to calculate the corresponding yield.…
The document asks how a bond carry approximation based on the difference between yield to maturity and the repo rate is derived. It considers whether yield to maturity can stand in for running yield, defined as coupon income relative to price, and suggests…
The document considers how to infer whether a trade was initiated by a buyer or seller when streaming data include the bid, ask, and last trade price. The proposed rule compares the trade price with the contemporaneous quote midpoint, classifying prices at…
The document poses a one-year CAPM calculation for three assets. It supplies each asset’s annual volatility and market capitalization, assumes the returns of distinct assets have equal pairwise correlation, and gives the risk-free rate and expected market…
The document investigates a mismatch between a computed Federal Reserve dollar index value and its published value. The index formula compounds each constituent exchange-rate ratio raised to that currency's weight, then multiplies the resulting product by…
The response proposes representing chart patterns as sequences of numerical movements and scanning a rolling window of volume-weighted average price candles for matches. It illustrates the representation with examples such as a double top, head and…
The discussion explains how to turn a stream of intraday trade profits and losses into periodic returns for a Sharpe ratio calculation. Choose a starting account balance, aggregate the trade results to the end of each chosen interval, and compare that…
The document considers hedging a one-year exposure in an underlying asset with Brent crude futures. It raises whether to match futures maturity to the exposure or use shorter-dated contracts when their historical price changes show a stronger relationship,…
The document compares constrained minimum-variance portfolio optimization with an unconstrained objective that penalizes deviations from a target return and linear exposure constraints. It explains that choosing penalty weights by hand can make the original…
The document works through a limit involving the upper tail probability and density of a normal distribution, arising in a shortfall-to-quantile ratio. The key step is to evaluate a related tail expression as the threshold tends to infinity. The response…
The document considers whether investors with long horizons can trade liquidity and drawdown advantages for greater long-term returns. Its answer identifies potential benefits of less frequent trading: lower brokerage costs, possible tax advantages for…
The document explains how combining two uncorrelated return streams can raise a portfolio’s Sharpe ratio. It presents a regression argument: when the streams are uncorrelated, the first strategy’s beta relative to the second is zero, so its excess return is…
The document discusses how Federal Reserve facilities and administered rates affect overnight money markets and the yield curve. It distinguishes the reverse repo facility, described as supporting a floor, from interest on reserve balances, which can act as…
The document asks how to distinguish expected from realized average transaction cost for a market sell order using a limit order book. The questioner walks through consuming visible buy-side depth to fill an order and computes a volume-weighted average price…
The document explains how to derive the cumulative distribution function of the maximum of several random variables when they are independent. To find the probability that the maximum is at most a threshold, it first considers the complementary event that…
The document asks how to decompose the one-day mark-to-market PnL of a position receiving a forward six-month interest rate into changes in rates and carry. It gives initial and next-day quotes for a six-month rate and a six-month rate starting six months…
The document discusses why Python is widely used for quantitative finance, while arguing that no single language dominates every part of financial technology. Its practical advantages include familiar syntax, a large developer pool, extensive libraries for…
The document examines two normally distributed commodity price changes with zero means, unequal standard deviations, and perfect positive correlation. It asks whether their relative value changes and whether one commodity’s move is simply a fixed multiple of…
The document compares three ways to estimate an annual Sharpe ratio from monthly index data: overlapping rolling annual returns, non-overlapping calendar-year returns, and annualized monthly excess returns. Its worked example uses monthly stock and risk-free…