คลังความรู้

The document explains why a monthly strategy that ranks stocks by price-to-earnings ratios and takes opposite positions in the top and bottom groups can show misleading backtest performance. Historical prices and earnings are not enough to reconstruct the…

The question considers whether a euro holder who wants exposure to EUR/USD could instead trade a futures contract quoted in the reciprocal direction and post margin in euros, avoiding currency conversion. The answer identifies the CME Euro FX futures…

The contract pays the positive part of the difference between the stock price at an earlier date and its price at maturity. At the earlier date, the strike is known, so the claim becomes an ordinary European put with a strike equal to that observed stock…

The document asks how to extend the familiar univariate definition of a median, obtained from the inverse cumulative distribution function at one half, to a bivariate distribution. Its answer is that there is no standard, unique median for multiple…

A binary option pays a fixed amount when the underlying finishes on one side of its strike, so its value is tied to the risk-neutral probability of that event, discounted for interest rates. Under a lognormal Black–Scholes model, increasing volatility can…

The response distinguishes historical par spreads from modern standard CDS market quoting. Par spreads describe the spread associated with a contract that has no upfront fee, as used before the 2009 CDS contract changes. Current trading generally uses a…

The document addresses a kink that can appear when combining call and put implied volatilities to build a fixed-tenor smile from American SPY options. The response recommends deriving one forward price for the option maturity from dividends and the market…

The response discusses how implied volatility differences between expiration dates can affect calendar spread pricing. It suggests that when front-month implied volatility is below back-month volatility, calendars may have broader profit ranges and higher…

An interest rate swap’s current value can be calculated from the yield curve, without specifying a stochastic model for future rate movements. That valuation answers what the swap is worth now under the curve and arbitrage-free pricing conventions; it does…

A replicating portfolio matches an option’s payoff only under idealized assumptions, including continuous trading, no transaction costs, unlimited divisibility, and known constant volatility. Actual markets violate these conditions, so an option combined…

Whether an option has a closed-form price depends on both its payoff and the assumed dynamics of the underlying asset. Some European vanilla options have tractable formulas under certain models, while other combinations of payoff and dynamics do not. The…

The document surveys ways to turn trading signals into mathematical strategy rules. One suggestion is to combine implemented signals in a factor model and estimate coefficients, which can produce a fitted formula and measures of statistical fit. The response…

The document addresses two representations of the solution to a mean-reverting stochastic differential equation driven by correlated Brownian motions. Applying an integrating factor yields a term that integrates the first Brownian motion’s level over time,…

The document raises a modeling question: how to represent prices that may become negative when the usual geometric Brownian motion framework keeps prices positive. It considers whether a different distribution, such as a beta distribution, could be used, and…

The document shows how a payer swaption can be expressed as an option on a coupon bond. It starts from a fixed-for-floating swap with reset and payment dates, writes its value as the difference between a floating leg and a fixed leg, and uses the forward…

The document explains how to hedge the foreign exchange component of P&L from American or global depositary receipts. An ADR position reflects both the local-currency value of its underlying equity and exchange-rate movements. A portfolio can aggregate its…

The document addresses a common sign confusion in put-call parity: when constructing a synthetic forward by buying a call and selling a put at the same strike, the put’s premium is not itself negative. Option premiums are positive prices by definition;…

The document clarifies the probability terms in the Heston call pricing formula. It decomposes the call payoff into two expectations, then uses a change of numéraire to express the price through probabilities under the risk neutral measure and a stock…

The document explains why portfolio variance depends quadratically on asset weights and why covariances enter the calculation. For one asset, scaling its return by a weight scales its variance by the square of that weight; extending this relationship to…

The document collects research leads on how index investing and ETFs may affect equilibrium prices, volatility, correlations, information production, liquidity, and corporate oversight. The motivating model has two risky assets with jointly Gaussian payoffs…

The document considers the issuer’s exposures on an autocallable note linked to the FTSE 100. The note has annual autocall observations, a coupon that accumulates until autocall, a maturity date, and a downside knock-in barrier observed at maturity. The…

The document asks whether a continuously rebalanced, accumulating bond fund can be held long enough to guarantee a positive or specified return, and how that compares with holding individual fixed-rate bonds to maturity. It distinguishes a fund that…

The document asks how a futures backtester should represent the funds required when opening long or short positions. It considers using margin requirements calculated through CME's SPAN system and asks where historical margin data can be obtained, noting…

The document identifies a suspected comparison error in the Optimal Number of Clusters procedure associated with de Prado and Lewis's work on detecting false investment strategies. The algorithm recursively reclusters groups whose silhouette-based…