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 examines a Black–Scholes claim paying the squared terminal stock price times its logarithm. It compares three valuation approaches: risk-neutral expectation using a lognormal moment, changing to the stock as numeraire, and attempting to use the…
The document explains the currency exposures involved in delta hedging a USDJPY option. In the example, a bank has sold a dollar put and calculates a short dollar hedge from the option’s delta. That hedge is paired with a long yen position at the prevailing…
The document compares practical considerations for stochastic-volatility and GARCH option models, emphasizing that model choice depends on the option type and the goal. It argues that GARCH can be attractive because its next-step conditional volatility is…
The document surveys practical considerations for options with expiries ranging from hours to minutes. It cautions that very short-horizon returns should not be assumed to follow a lognormal process simply because the horizon is short. Instead, intraday…
The document explains why standard Black-style approximations for cash-settled swaptions imply payer-receiver parity around the forward swap rate. Under that approximation, a zero-wide collar—the difference between payer and receiver swaptions at the same…
This discussion asks how to use volatility from an implied-volatility surface in Reiner-Rubinstein formulas for an up-and-out call, expressed as a vanilla call minus an up-and-in call. The vanilla component may reference the strike while another component…
This example compares delta-gamma and delta-vega hedging for a short call using another call and shares. It gives Black-Scholes Greeks for an in-the-money call and an at-the-money call, then solves for the option quantity that offsets gamma or vega and the…
The document considers whether some equity markets still have nearly flat implied volatility smiles, in contrast with the negative skew often associated with major indices after the 1987 crash. One response reports that options on many Korean equities, and…
The document discusses how to fit a Black-Scholes implied volatility surface when options trade with bid-ask spreads. It explains why separate arbitrage-free bid and ask surfaces are generally not useful for practical hedging, since trades and hedges can…
The document asks how to simulate stock paths under a Merton jump diffusion model, which combines continuous Brownian movement with jumps driven by a Poisson process. The response frames the task as Monte Carlo simulation: choose a time increment, draw a…
The document explains why delta-hedged option profits should not be confused with recovering the premium as trading revenue. Under Black–Scholes assumptions of zero rates, continuous hedging, and realized volatility equal to implied volatility, gamma gains…
The document explains that exchange options, whose payoff exchanges one asset for another, do appear in practice, though listed markets are limited and often less liquid than single-asset options. In commodity futures, a zero-strike spread option can have…
The document asks how to attribute stock-trading profit and loss to individual options in a delta-neutral portfolio when the stock is traded only to hedge those options. It compares allocating the hedge result in proportion to each option’s gamma with…
The document discusses why some customers continue to route large SPX and VIX option orders through open-outcry pits when electronic trading is also available. The accepted response attributes better execution prices to negotiation between floor traders, who…
The document explains how to visualize Black–Scholes Vanna, the sensitivity of an option’s value to changes in both the underlying price and volatility. The central task is to evaluate Vanna over every pair drawn from two input ranges, rather than matching…
The document considers whether an American put calendar spread can have a negative price and examines an attempted no-arbitrage argument based on selling the shorter-maturity put and buying the longer-maturity put at the same strike. The key issue is…
The document discusses how a dealer might quote caps and floors in an emerging market with swaps but no traded interest-rate options. Without option quotes for calibration, the response recommends developing an indicative volatility cube using comparable…
The document distinguishes spot at-the-money options from at-the-money-forward options in foreign exchange. In the described convention, an ATM option has a strike equal to the current spot rate and its delta is hedged in the spot market. An ATMF option uses…
The document discusses difficulties fitting the Heston stochastic volatility model to vanilla option prices. Parameter dependence and flat regions in the calibration objective can make optimization unstable, especially for mean reversion. It considers…
The document explains why historical records for some options may appear to end before expiration. In the QQQ example, puts below a particular strike disappear from one vendor’s data after a date, while a nearby strike remains available. The accepted…
The document examines the profit profile of a call writer who holds stock to offset the call’s delta. Because the stock position has no gamma while the short call has negative gamma, the combined portfolio retains negative gamma. The question is whether this…
The document raises a practical calibration question for a one-factor Hull–White interest-rate model used to price a structured note. The example note pays annual coupons based on the positive part of three-month EUR Libor plus a spread, then returns…
The note explains how to value a put whose premium is deferred until exercise. It rejects treating the arrangement as a standard put with a reduced strike, because that changes the payoff: a conventional option buyer can lose the upfront premium even when…
The document explains how to price a maturity claim that pays the larger of a deterministic money account and a dividend-paying stock. It decomposes the payoff into the money account plus a call option on the stock, with the money account value at maturity…