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 describes a proposed Monte Carlo approach to valuing a leveraged down-and-out call under geometric Brownian motion. The contract sketch has the buyer contribute part of the asset price while financing the remainder, pay interest on that…
The document addresses how an at-the-money call and a short at-the-money put with the same underlying and expiry relate to a futures position. Its answer states that one call paired with one put creates one synthetic long futures position. This establishes…
The document compares an estimate of historical volatility from three years of index returns with the volatility implied by a one-week, out-of-the-money put on an Indian market index. The question describes historical standard deviation as roughly twice the…
The document explains an arbitrage test for call options sharing an expiry. Call prices should be convex as a function of strike, so the price at a midpoint strike cannot exceed the average of the prices at two equally spaced outer strikes. The example…
The document asks how to move from FX vanilla and barrier option quotes to a local volatility model. The author describes fitting smiles at quoted maturities with Vanna–Volga, then interpolating total variance across maturities using a flat-forward approach.…
The document explains the role of gamma when replicating a protective put by adjusting positions in the underlying asset. Under idealized continuous hedging with no transaction costs, delta hedging can replicate the option payoff, so gamma does not create a…
The document explains implied volatility as the volatility input that makes a Black–Scholes option price match an observed market price. Under Black–Scholes assumptions, the underlying follows geometric Brownian motion with constant volatility, so the model…
The document proposes choosing a European call strike by comparing its premium with the payoff at expiration, then asks whether historical price data can help select the strike that maximizes profit. It attempts to replace the uncertain call payoff with a…
The document discusses ways to obtain a stock’s options chain, including multiple expiries, and make the data usable in Excel without manually copying each entry. One response points to a commercial source of historical options data supplied in CSV format,…
The post examines a formula from a paper on European option pricing that expresses instantaneous mean drift using a call’s vega, delta, and gamma. The author compares the formula with current option prices and finds negative estimates, despite reading the…
The document considers a proposed options relative-value trade: buy gamma on one underlying and sell gamma on a correlated underlying whose same-moneyness call has higher implied volatility. It describes the setup using a return beta, an implied-volatility…
This argument derives an asymptotic upper bound on Black–Scholes implied variance for calls with increasingly large strikes. It assumes a fixed maturity and that the underlying's terminal price has a finite second moment under the risk-neutral measure. A…
The document shows that a European call payoff can be represented as a long asset-or-nothing digital call combined with a short cash-or-nothing digital call, with both components expiring at the same time. When the underlying finishes above the strike, the…
The document introduces an American derivative exercisable at discrete times and states three properties of its price process: its value is at least its immediate exercise payoff (and zero), its discounted value is a supermartingale, and any other process…
The document considers whether the Central Limit Theorem makes normal-return assumptions unnecessary in Black–Scholes–Merton option pricing. It shows how asset prices compound through simple returns and how log prices sum log returns. Under suitable…
This short exchange considers books for an engineer who has strong mathematics and statistics skills and wants rigorous option-pricing material without extensive pure-mathematics terminology. The responses recommend several texts with different levels of…
The document examines three quoted European call prices with the same underlying and strike but different maturities. The longest-dated call is quoted below the nine-month call, prompting a request for an explicit arbitrage argument. The accepted response…
The document considers the time-t price of a claim paying the integral of a stock price over a fixed horizon under the Black–Scholes model. The questioner expands the risk-neutral expectation using the stock’s geometric Brownian motion and asks whether the…
The document considers Monte Carlo valuation of a forward-start call under the Heston stochastic volatility model. Its payoff depends on the ratio of the asset price at the later date to its price at the reset date, so the strike is effectively set by the…
The document distinguishes two related pricing equations. The Black–Scholes or local volatility PDE values one option with a fixed strike and maturity as time and the underlying price vary. The Dupire forward PDE instead describes vanilla option prices…
The document explains why option implied volatility, especially for stock indexes, often exceeds the volatility later observed in returns. It frames this gap as a volatility risk premium and considers whether selling at-the-money straddles can capture it.…
The document considers how to value an option whose underlying is a listed company’s earnings. One proposed framework simulates share price with geometric Brownian motion and the price-to-earnings ratio with a mean-reverting process, then derives earnings…
The document compares variance swaps with volatility swaps, focusing on replication, convexity, and market practice. Variance is described as the more directly replicable quantity: a portfolio of options across strikes can replicate variance exposure, while…
The document raises a question about approximating option sensitivities, using rho under the Black–Scholes partial differential equation as a proof of concept. The author proposes differentiating the PDE with respect to the interest rate to obtain an…