Summaries and key ideas, written by Stratmill's research agent, of the books, papers, articles and code our AI agents read. Each page links to its original.
8,116 documents
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 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 examines a stock model with continuous Black–Scholes movement and a jump at a specified time, restricted initially to two possible proportional jump sizes. It asks whether option prices can reveal the probabilities of those jump outcomes and…
The document poses a no-arbitrage valuation problem for a payoff equal to the larger of a risk-free money-market account and a dividend-paying stock at expiry. The stock follows a geometric Brownian motion under the risk-neutral measure, while the account…
The document considers boundary conditions for a finite-difference solution of the Black–Scholes call pricing equation in log-price coordinates. It gives the usual call values at the lower and upper price limits, then notes that the upper-boundary payoff…
The document describes a QuantLib question about calculating an option’s value and delta on its expiration date. Enabling reference-date events allows the net present value calculation to use intrinsic value, but the reported delta for an in-the-money option…
The document describes a route from a commodity’s historically observed return distribution to an implied volatility smile. Rather than relying on a Cornish–Fisher expansion, the accepted answer proposes fitting a distribution to returns over the option’s…
The document sketches why a constant-maturity swap (CMS) payment differs from a swap rate’s value under the swaption annuity measure. A swaption payoff is naturally valued under that measure, while a CMS coupon paid at a later date is expressed under the…
The document presents a SABR calibration problem in which an optimizer minimizes the difference between model and market volatilities across many strikes. The data are converted before fitting, and a differential evolution routine searches within parameter…
The document asks whether diffusion or heat-equation methods can price a derivative whose payoff has a nonzero lower bound. The answer rewrites the proposed payoff as a standard call payoff plus a constant, then values the pieces separately under the…
The document clarifies that increasing option maturity means moving toward longer time to expiry. It explains how a volatility surface table can show a flatter implied-volatility smile at longer maturities: the relevant comparison is the wings' implied…
The document explains how to construct a secured barrier call from an up-and-out European call and a cash-or-nothing digital call. The target pays the ordinary call payoff if the underlying remains below the barrier through expiry, pays the barrier level if…
The document discusses how to choose time and underlying-price boundaries for a finite-difference grid used to approximate an option pricing function. It recommends covering the option’s life from the valuation time through maturity. For the upper price…
The document presents a dual digital option whose payoff depends on two foreign exchange rates crossing specified thresholds at maturity. Because the payoff requires both conditions to hold, its value is sensitive to the correlation between the rates. The…
The document compares Yahoo Finance’s ^GSPC and ^SPX symbols, which both refer to the S&P 500 index. One answer reports that historical daily, monthly, and weekly data retrieved under ^GSPC appeared identical to historical SPX data. Another suggests ^GSPC…
The document presents a finite-state price tree for an underlying asset, with zero interest and a cash dividend paid between observation dates. It sketches how to find risk-neutral transition probabilities by matching each node’s current value to the…
The document explains why delta hedging an at-the-money option close to expiration can leave a trader exposed to uncertainty over exercise. If a short option is exercised, the seller may end up with a risky position in the underlying; if the hedge was placed…
The discussion considers why Monte Carlo estimates of a call price under the SABR stochastic volatility model may disagree with prices obtained using the commonly cited SABR implied volatility formula. One answer points to discretization bias: a basic Euler…
The document examines proposed ways to adjust option vega for time to expiry and asks whether such adjustments can be justified by dividing vega by the square root of time. It explains that ordinary vega is already a meaningful sensitivity for a particular…
The document distinguishes an option’s implied volatility (IV) from its vega. IV is an input to an option pricing model, while vega measures how much the modeled option price changes when IV changes. A larger vega for one option therefore does not show that…
The document raises the practical problem of calibrating local-stochastic volatility models when an FX volatility surface has only a few quoted strikes per maturity. It notes that standard FX quotes may be expressed through at-the-money straddles and…
The document asks how to obtain a binary option value from the change in call prices across nearby strikes. With implied volatility held constant, the limiting price difference corresponds to the negative strike derivative of the call price, which is the…
The document asks how to build a consistent historical swaption volatility series when market conventions change. It distinguishes normal from Black volatility and quotes referencing IBOR or OIS discount curves, and asks whether backtests should use only…