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 whether traders can forecast future option prices directly, then outlines volatility-based approaches. It describes gamma trading as selling options when implied volatility is expected to exceed realized volatility, with delta hedging…
The document explains the origin and limits of a simple approximation that relates implied variance at a strike and maturity to local variance near the initial spot and the strike at expiry. It places this shortcut alongside short-maturity results: the…
The document considers how to obtain long exposure to the VDC ETF with options when its dividend payouts are irregular. It raises a tradeoff between possible option positions and dividend-related price changes around an ex-dividend date. The author is…
The document explains that volatility skew and statistical skewness are related terms with distinct meanings. Implied volatility skew describes how option-implied volatility varies across strikes for a fixed expiry; it is observable from option prices after…
The document discusses why Yahoo Finance may report zero implied volatility for some in-the-money options, especially shortly after the market opens. It offers possible explanations rather than a confirmed account of Yahoo’s calculation: missing or unstable…
The document resolves a question about replicating a European call in a zero-interest-rate market where the stock’s diffusion coefficient depends on its price. The key result is that the number of shares in the replicating portfolio is the option’s stock…
The document raises the problem of converting probabilities inferred from stock option prices under a risk-neutral measure into real-world, or physical, probabilities. Its only proposed method is the Recovery Theorem, named in a brief answer. No derivation,…
The document explains how to estimate the probability that an underlying asset reaches an option strike before expiry. It distinguishes this first-passage event from the probability of finishing in the money, noting that option delta is not generally the…
The discussion examines whether theta-to-gamma ratios can guide exits from short at-the-money options paired with long-tail options. It relates the ratio of cash theta to cash gamma to implied variance, and extends this relationship to a portfolio by…
The document explains the role of the underlying asset position in a variance swap replication strategy. A short variance swap can be hedged with a strip of options across strikes, often represented in practice by an at-the-money straddle together with…
The document distinguishes the fair strike of a plain variance swap from the price of a variance swap whose realized variance is capped. A strip of vanilla options can be used to replicate the uncapped variance exposure under standard assumptions, so…
The document examines a stop-loss style strategy for replicating a call option and asks how its hedging costs relate to the time the underlying spends near the strike. The answer explains the call payoff using the Tanaka-Ito-Meyer decomposition: intrinsic…
A Bermudan swaption allows exercise on multiple specified dates, giving it more exercise flexibility than a European swaption. The document notes that these products are actively traded rates volatility instruments and are used in callable bond structures: a…
The document explains how to revalue a swaption one week after observing its implied volatility in a rates volatility cube. The option retains its underlying forward swap and strike, while its remaining expiry shortens. Its value at the later date is…
The document addresses pricing European options on the ASX 200 index when dividends paid by constituent companies affect the index level. It reports that using a continuous dividend yield in Black–Scholes–Merton produced a price consistent with the observed…
The discussion explains how a contract combining one touch barrier with one no-touch barrier can pay, using upper and lower barriers as an example. A one-touch feature generally triggers a payout when its barrier is reached, while a no-touch feature can…
The document explains how prices of European receiver swaptions relate to the risk-neutral distribution of the underlying swap rate. It corrects the idea that an out-of-the-money premium, compared directly with an at-the-money premium, gives the probability…
The document outlines maximum-likelihood estimation for a GARCH(1,1)-M return model under the physical measure, then discusses an implementation problem in an R optimizer. The model specifies conditional returns using a known risk-free rate, an estimated…
The document distinguishes implied volatility from historical volatility. Implied volatility is the volatility parameter that makes an option pricing model reproduce the market price of an option; it is inferred from current prices rather than calculated…
The explanation links negative correlation between an asset’s spot price and its instantaneous volatility to a downward implied-volatility skew. It contrasts the result with the standard Black–Scholes assumption of constant volatility, under which…
The discussion explains why an index does not itself have a single gamma position and how market commentary may instead estimate the aggregate gamma exposure of option dealers. A simplified calculation assigns dealers the opposite side of customer put and…
The document discusses why the differential equation for an American put may be written as an inequality in the continuation region above the optimal exercise boundary. One response links the inequality to the option’s delta at the boundary and the condition…
The document explains that short-rate models describe the evolution of an instantaneous interest rate and use it to derive discount factors as expected accumulated discounting over time. Because the instantaneous rate is not directly observable, the response…
The document recommends a practical scenario approach for pricing down-and-in puts when vanilla option prices and implied volatilities are available. Apply a spot shift and test a small set of implied volatility shifts, using the existing volatility surface…