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 introduces Emanuel Derman’s volatility approximation for a volatility swap under an assumed linear volatility skew. It describes the approximation as assuming a linear put skew and a flat call skew, with applicability focused on strikes…
The document explores whether an option-implied risk-neutral distribution can be checked by integrating its density twice and comparing the reconstructed call prices with observed option prices. It grounds the idea in the Breeden-Litzenberger relationship,…
The document explains why the relationship between the underlying price and future volatility can affect autocall valuation. Autocalls often include digital payoffs and barrier conditions, so their value depends on the chance that the underlying crosses a…
The document addresses a discrepancy between a calculated FX option delta and a market terminal’s value. The key clarification is that the instrument is a digital vanilla, or binary, option, so ordinary vanilla Black–Scholes delta inputs and intuition do not…
This exchange diagnoses why a Newton–Raphson routine for Black–Scholes implied volatility diverges. The original code divides the option-price error by gamma, but implied volatility is the quantity being solved for, so the derivative needed for the update is…
The document asks whether an option can have a strike that varies randomly over time and what such products are used for. The answers point to several related structures: an Asian option uses an average of the underlying over the contract period in its…
The document explains how binary options differ from vanilla options. A binary contract pays a fixed amount or nothing at expiry, whereas a vanilla call’s payoff grows with the underlying above its strike. The binary’s capped payoff and risk profile can suit…
The document asks whether options on two separate assets, each quoted against a common currency, can create an option on their exchange ratio. Its answer says that a portfolio with payoff equal to the sum of a function of each asset separately cannot…
The document explains why the hedge quantity for an FX option can differ from its standard Black–Scholes spot delta when the option premium is received or paid in the foreign currency. The premium itself already provides exposure to that currency, so the…
The document explains two ways to think about probability distributions for foreign-exchange prices. For a distribution implied by traded options, it describes the risk-neutral density: butterfly spreads across strikes convey information about probabilities…
The document answers a question about research on synthetic options, including their valuation and time decay. It distinguishes simple synthetic positions, such as a synthetic long stock, from more elaborate synthetic structures designed to replicate complex…
The document explains how to simulate a down-and-in barrier call under a local volatility model. At each simulation step, the volatility should be evaluated at the current time and stock level, rather than at the option’s fixed strike and maturity. The…
The discussion examines a four-step binomial valuation of a European put on a stock that moves by a fixed amount each month, with a continuously compounded risk-free rate. The proposed method recursively discounts the risk-neutral expected option payoff at…
The document describes a calibration question for the eSSVI implied volatility surface, where parameters are fitted slice by slice across maturities. It focuses on a calendar-consistency inequality that links the current slice parameters to those of the…
The document explores when early exercise can be optimal for American put options in zero or negative interest rate settings. It contrasts the familiar positive-rate intuition for calls with the corresponding conditions for puts, noting that early exercise…
The document discusses how traders might estimate volatility for option analysis when the appropriate historical lookback is unclear. It describes two choices in historical volatility calculation: the length of the past period being measured and the spacing…
The document distinguishes historical volatility, calculated from past stock returns, from Black–Scholes implied volatility, inferred from traded option prices. Historical volatility summarizes observed movements, while implied volatility reflects the option…
The document derives the expected terminal stock value conditional on the stock finishing above the strike, a term used in the Black–Scholes call price. It starts from geometric Brownian motion, converts the condition that the terminal price exceeds the…
The document raises a backtesting problem for a systematic strategy that repeatedly buys one-year, 90% strike equity index puts and rolls them shortly before expiry. It proposes repricing the option each month with Black–Scholes using the current spot,…
The document asks how much a short straddle could earn in the long run under an efficient market and Black–Scholes assumptions. It assumes the implied volatility used to price options matches future realized volatility, which remains constant over the…
The document asks how to value a perpetual American put in the Black–Scholes–Merton framework when exercise occurs the first time the underlying reaches a chosen threshold below the strike. It sets up the discounted payoff and attempts to transform its…
The document asks how to summarize implied volatility and Greeks for a bull put spread, formed by buying a lower-strike put and selling a higher-strike put. It considers averaging each metric across the legs versus taking the long leg’s value minus the short…
The post investigates a persistent difference between simulated returns on an option account and a delta-hedged account when the Black–Scholes model is run at very high volatility. It asks whether the discrepancy comes from confusing the volatility of log…
The post asks how to manage a short strangle after the underlying rises and the short call develops a large unrealized loss. The trader has been rolling the put to keep the overall position near delta-neutral and wonders whether moving its strike toward or…