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The document works through a covariance involving the stochastic integral of cubed Brownian motion and the time integral of squared Brownian motion. It rewrites the latter using an Itô-integral representation plus a deterministic term. The deterministic…
The document asks whether the time integral of Brownian motion is a martingale. Although integration by parts rewrites it using a stochastic integral, the resulting integrand depends on the terminal time. The accepted answer explains that the familiar…
The document answers which exchange rate to enter as the spot parameter when pricing an FX option with the Garman–Kohlhagen model. Its central point is to use the current spot rate; the forward rate for the option’s expiry is already implied by the model’s…
The document concerns constructing a portfolio of individual bonds selected from a benchmark, with exposure to a chosen investment factor. The questioner seeks research on factor investing using actual bonds rather than exchange-traded funds or sovereign…
The document explores an empirical approach to estimating fair implied volatilities for SPX options. It proposes using a proxy for future volatility, such as VIX, to select similar historical observations, fitting a distribution to those observations, and…
The note asks how to define risk factors for scenario testing of a portfolio containing several currency pairs. It outlines a basic stress-testing workflow: identify portfolio-relevant drivers, specify scenarios, shock those drivers, and interpret the…
The note corrects a short-sale profit calculation that incorrectly credits interest to both the sale proceeds and the haircut collateral. The proceeds are reduced by the opening commission, while the haircut amount is restricted collateral whose interest…
The document explains why applying Itô’s lemma to a function of a diffusion requires a second-order term. For a process with drift and Brownian volatility, a Taylor expansion includes the squared process increment. Ordinary time increments have negligible…
The document questions a claim about compounding a constant overnight rate over a three-month period when calculating a spot swap’s roll-down. The questioner interprets the quoted rate as an annualized overnight rate and compounds it daily, finding that the…
The document explains how to identify the standardized random increment in a risk-neutral trinomial asset model and connect it to a weak convergence argument for Black–Scholes. Taking logarithms of each of the three possible one-step multipliers gives three…
The document describes a discrepancy between a closed-form price for a geometric Asian call and a Monte Carlo estimate. The author uses a formula parameterized by spot, strike, maturity, volatility, interest rate, and a monitoring count, and reports that the…
The document explains how to calculate a portfolio’s logarithmic return from its value at two consecutive times. With fixed holdings represented by weights, first calculate the weighted sum of component prices at each time, then take the logarithm of the…
The document discusses why Fama–French work testing the CAPM may not cite Ross’s arbitrage pricing theory (APT) as its foundation. The answer distinguishes the frameworks: it characterizes Fama–French factors as added variables used to test whether the CAPM…
The document raises a practical concern about combinatorial purged cross-validation (CPCV) in financial backtesting. By combining separated time blocks into a training sample, the method can make observations on either side of a gap appear adjacent. That can…
The document reviews a Monte Carlo implementation for a payoff capped at an upper barrier and floored at a lower barrier at expiry. The accepted answer identifies errors in the simulation: each price update should multiply the current spot by the geometric…
The document responds to a beginner’s difficulty identifying widely traded ETF symbols by outlining how funds can be grouped into asset classes and investment styles. One answer points to a commercial fund-classification service and gives an example of how a…
The document explains how a European call behaves in the Black–Scholes setting when volatility is zero. With no dividends, the stock follows a deterministic path growing at the risk-free rate. The call’s payoff is therefore known at expiry; discounting that…
The document explains how to turn a Dupire local volatility model into a European option price. Its example starts with an implied volatility surface, constructs a Black–Scholes–Merton process that uses the surface, and sets up a vanilla call. It then…
The document explains why option pricing can use an equal-probability illustration for upside and downside outcomes even when historical returns are asymmetric. It distinguishes the risk-neutral measure used to price derivatives from historical return…
The document works through the time-zero value of a claim in a one-period, two-state market. A portfolio holding the risky asset and a bond is chosen so that its terminal value matches the claim in both the up and down states. The hedge ratios are obtained…
The document considers how to derive a pricing PDE for a derivative under the Heston–Hull–White model, where the stock, stochastic variance, and short rate are correlated. It identifies a problem with applying the usual discounting transformation directly:…
The document examines how to calculate vega sensitivity under the Basel Fundamental Review of the Trading Book standardized approach. The question contrasts multiplying Black–Scholes vega by implied volatility expressed as a decimal with an example using a…
The document discusses two interest-rate products that pay based on absolute changes in future rate volatility. One references successive six-month LIBOR changes; the other references changes in a long-dated forward-start swap rate and includes capital…
The document poses a static arbitrage problem using a contract that pays a negative logarithm of the terminal underlying price, a forward contract, and a discount bond. Given prices for the three traded assets, it infers the underlying’s spot value from the…