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 explains how an option on one asset can be delta-hedged using a different, correlated asset. It defines the hedge sensitivity of the second asset to changes in the first and gives a formula under a bivariate geometric Brownian motion assumption:…
The document frames option selection as a finite search over quoted calls: given a target stock price at expiration, calculate each contract’s profit relative to its premium and select the strike with the highest percentage return. For a discrete set of…
The document asks whether a risk-neutral probability density recovered from option prices using the Breeden–Litzenberger method is the same as option gamma. One answer derives a relationship for models in which the option price is homogeneous of degree one…
The document explains two common ways to define notional for an equity option position. One convention multiplies the number of contracts by the current spot price and the contract multiplier; another uses the strike price in place of spot. Both approaches…
The discussion asks how a trader should decide whether to exercise a physical Bermudan swaption as an exercise date approaches. The accepted answer distinguishes a necessary condition from a sufficient exercise rule. For a receiver swaption, exercising at an…
The post examines a motivation for the SABR model: local volatility models are said to predict implied-volatility smile movements opposite to those observed in markets, whereas the original SABR paper is described as moving the smile in the same direction as…
The document presents a Python implementation of an explicit finite-difference scheme for the Black-Scholes equation, applied to a European call. It lays out a stock-price grid and a time grid, initializes terminal values with the call payoff, and steps…
The document raises a conceptual question about a standard optimal stopping result: for a Markov process, the value function is characterized as the smallest superharmonic function that dominates the payoff. The author questions how this can hold when some…
The document explains the arbitrage implied when a European put's price exceeds its strike, assuming the underlying asset price cannot fall below zero. A put's payoff at maturity is capped at the strike, so a seller's maximum payment is no greater than that…
The question compares a model-free volatility expression based on option prices across strikes with implied volatility obtained by inverting the Black–Scholes pricing formula. The answer clarifies that the two quantities represent different concepts. The…
The note distinguishes option valuation from deciding whether to buy shares. Black–Scholes prices an option under assumptions about the underlying price, volatility, rates, and time to expiry; it does not by itself produce a fundamental value or directional…
The note compares a bond-plus-equity-option approach to valuing convertible bonds with a binomial approach that models both debt and equity features. In the simpler method, the value is treated as the sum of a straight bond and a call option on the…
The note explains how a variance dispersion position can acquire short volatility exposure after a market decline. The example starts with a vega-neutral portfolio that is short an index option and long options on individual stocks, with deltas hedged at…
The document considers how to infer the volatility implied by a convertible note’s issue price when the instrument includes early conversion, issuer redemption, and capped calls. A simple decomposition into bond value and option value may be inadequate once…
The note explains why a long European put can have positive theta under Black–Scholes, despite the usual intuition that long options lose value as expiry approaches. A deep in-the-money put can trade below its intrinsic value when interest rates are…
The document considers how to simulate asset paths under SABR when volatility and vol of volatility are high. It describes a Monte Carlo comparison between a call value obtained from Hagan's implied volatility approximation and an estimate from a discretized…
The note asks whether an American put can be valued by calculating Black-Scholes prices for options with different maturities and selecting the largest. The response explains the key obstacle: an exercise time in the optimal stopping formulation is generally…
The document raises a risk-management question about a reported PIMCO trade involving large short positions in an S&P 500 put and call strangle. It asks how other market participants might hedge long option exposure against a seller of that scale, especially…
The document explains that Black-Scholes-Merton remains common in practice for European options, including settings where markets quote volatility and the model converts that volatility into prices. It also notes that exchanges may use Black-Scholes variants…
The document presents a question about recovering call prices from observed SPX put prices using put-call parity, with zero interest rates assumed. It supplies mixed call and put quotes, option types, and increasing strike values, then points to cases where…
The document explains how option writers may manage exposure before expiration through delta hedging. In the simplified account, they hold a changing mix of the underlying stock and financing assets, adjusting that mix as the option’s sensitivity changes.…
The document explains how an empirically calibrated SVI volatility surface can support option pricing and Greek estimates. SVI maps maturity and strike to an implied volatility; that volatility can then be supplied to a Black–Scholes routine to obtain a…
The document is a curated list of academic papers on historical option returns and strategies. It points readers to work on buy-write portfolios, option returns, volatility and skewness, dispersion trading, expiration effects, and whether options are…
The document explains why implied volatility often rises away from the at-the-money strike, drawing on three complementary interpretations. One links smile curvature to higher moments of the return distribution: convexity can reflect positive excess…