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 discusses how to choose boundary conditions when numerically solving the Black–Scholes partial differential equation on a finite underlying-price grid. The appropriate conditions depend on the payoff and on how the finite grid corresponds to the…
The document explains why desks may use SABR alongside Black-76 or Bachelier pricing for vanilla swaptions and other interest rate options. In this account, the vanilla pricing formula supplies the option value, while SABR models how implied volatility…
The discussion addresses how to calculate Bartlett-adjusted vega when a QuantLib SABR swaption volatility cube derives its forward rates internally from a spot curve. Since the cube does not expose forward rates as direct inputs, the proposed workaround…
The discussion explains how to calibrate a Hull–White interest-rate tree for valuing callable bonds. It recommends using swaption volatility quotes when those instruments resemble the embedded options in the bonds, and suggests constructing an equivalent…
The document presents an attempt to fit Heston stochastic-volatility parameters to a grid of market call prices using SciPy optimization. It includes a characteristic-function-based pricing routine and loops across strikes and maturities to produce model…
The discussion shows how to turn sampled, potentially non-normal log returns into asset price paths. For each asset, take its starting price and multiply it by the exponential of the cumulative sampled returns. A copula can supply joint return draws across…
The discussion distinguishes expected, implied, and realized volatility in option risk calculations. It clarifies that the recommendation to use expected volatility does not mean using realized volatility: realized volatility describes past price variation,…
The question concerns a one-year call on a euro-denominated stock with a euro strike, subject to a knock-in condition based on relative performance. The option pays only when the euro stock finishes above its strike and its growth over the year exceeds the…
The discussion distinguishes trading for exposure from quantitative work that develops valuation models. A practitioner may use outputs from stochastic-calculus-based models, such as Black–Scholes, without applying stochastic calculus directly to routine…
The note clarifies the meaning of the function Φ in the Cox–Ross–Rubinstein binomial option model. It denotes a complementary binomial distribution function: for a random variable X, this gives the probability that X exceeds a specified value, or one minus…
The document asks how the VIX or its short-term counterpart compares with implied volatility backed out from a near-the-money S&P 500 call. The response highlights that a Black–Scholes implied-volatility estimate is conditional on the model used to translate…
The document asks how to adapt a bisection-style implied-volatility calculation for an American option on a dividend-paying stock. The key principle in the accepted answer is to keep the root-finding procedure and replace the Black–Scholes pricing function…
The document frames a modeling problem: estimate before trading how much price concession may be needed to execute a multi-leg option spread, such as a vertical or butterfly. It proposes candidate inputs, including each leg’s bid–ask width across the…
The document asks why short-term repo implied by Euro Stoxx 50 options can be materially higher than repo implied by options on constituent shares, and whether index financing should match an average of single-stock financing. The response emphasizes that…
The document discusses real-time risk estimates for US equity option portfolios when implied volatility is unreliable at deep in-the-money or out-of-the-money strikes. The questioner describes deriving volatility from current market prices, then revaluing…
Put–call parity can imply an arbitrage when the prices of a call, a put, the underlying, and the present value of the strike do not align. The document asks what could make such a trade fail beyond insufficient liquidity to assemble the positions. Its answer…
The answer distinguishes the real-world stock-price process, whose drift is the expected return, from the risk-neutral process used to price options, whose drift is the risk-free rate. It presents the geometric Brownian motion solution under each measure and…
The document surveys research using information from option prices and implied volatility to forecast returns. It groups examples into several themes: predictors derived from implied volatility, expected-return estimates recovered from option prices, and…
The document explains how to price a European put option on a zero-coupon bond under the Cox-Ingersoll-Ross (CIR) interest-rate model. It derives the put price from the corresponding bond call price using put-call parity: the put equals the call less the…
The document presents the setup for comparing European call prices under Black–Scholes and a GARCH-in-mean model following Duan (1995). It gives the physical-measure return and conditional variance equations, then describes the risk-neutral dynamics used to…
The document explains a variance-swap discussion of a short call that is hedged daily, where the trader loses money overall despite realized volatility being below the option’s implied volatility. It clarifies that the charted trading profit and loss is the…
The document derives the time sensitivity of a European call under the Black–Scholes–Merton model with constant interest and dividend rates. It rewrites time to maturity as a separate variable, applies the chain rule, and differentiates the discounted stock…
The note explains the usual sign of option gamma: under Black–Scholes assumptions, a long call or put has positive gamma, while a short position has negative gamma. Gamma describes how an option’s delta changes as the underlying price moves. For a long put,…
The document proposes valuing a simplified, single-underlying phoenix autocall as a zero-coupon snowball component plus the discounted expected value of coupons paid at observation dates. It models coupon eligibility using the underlying’s level relative to…