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.
10,263 documents
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 document asks whether a lender and an equity holder in a simplified collateralized loan obligation should value their claims using the same risk-neutral probability measure. The lender receives an agreed payment ahead of the borrower, who keeps 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 asks whether probabilities inferred from derivatives on non-traded variables, such as weather outcomes or central bank rate decisions, represent risk-neutral or real-world beliefs. The responses distinguish pricing measures from objective…
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 asks why a one-year Bitcoin futures contract can trade at a premium whose annualized rate exceeds a comparable government bond yield, despite standard cost-of-carry relationships. It considers whether demand for leveraged exposure may help…
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 poses a one-period binomial market problem with a risk-free asset and a stock that can finish in either an up or down state. It asks how to establish no arbitrage, replicate a claim whose payoff is the squared terminal stock price, calculate the…
The document asks whether adding a derivative that depends on an otherwise unspanned source of risk makes an incomplete market complete. It sets up a continuous-time market with fewer traded risky assets than Brownian risk sources, including an example where…
The document asks how to translate a 90-day SOFR market rate into a comparable three-month LIBOR rate. It highlights a key timing distinction: SOFR for the accrual period is compounded in arrears, so the realized period rate is not known until the period…
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 document asks whether a one-period market is arbitrage-free whenever the asset’s log return has a continuous distribution over the real line, and whether such a distribution guarantees a martingale measure. The motivating comparison is with a…
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 note explains a discrepancy between manually interpolated annual-compounded zero rates and rates returned between curve pillars by QuantLib. The key point is that the curve interpolates continuously compounded zero rates, even when its configured…
The answer uses the martingale concept to explain why, under a simplified random-walk model, an increase in portfolio wealth has the same expected value at a later time. A martingale is defined by the condition that the conditional expectation of a future…
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…