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.
12,303 documents
The document explains why the difference between two correlated Brownian shocks can be represented by a single Brownian motion when pricing an exchange option. The payoff can be rewritten using the ratio of the two stock prices, and Itô’s lemma gives that…
The document explains why a farther-dated option can show lower implied volatility than a near-term option while still having a higher premium. It describes an inverted volatility term structure: near-term implied volatility is elevated, while the market…
The document presents asymptotic boundary conditions used when numerically solving generalized Black–Scholes models on a truncated price domain. At the low-price boundary, the call value tends to zero. At the high-price boundary, it approaches the discounted…
The answer explains the tendency for equity volatility to move inversely with stock prices by comparing equities with credit. For credit, falling prices can reflect growing concerns about whether borrowers can meet obligations; uncertainty can increase…
The document examines how to extend Cornish–Fisher Value-at-Risk from a single return period to a longer holding horizon. One proposed method assumes returns are independent and identically distributed and infinitely divisible, so their cumulants add…
The document considers how to assess a proposed interest-rate cap volatility surface for arbitrage. A cap consists of caplets, and the response recommends stripping caplet values from cap prices across maturities. In a simplified setting with a flat yield…
The document identifies a modeling connection between interest-rate options and volatility options: both markets have term structures. It points to a variance-curve model in which variance across maturities evolves as a martingale driven by Brownian shocks,…
The note asks how to take a directional view on volatility while limiting exposure to the underlying’s direction. It points out that straddles, strangles, and iron condors respond to volatility but also carry exposure to other option risk factors, including…
The note considers how to forecast currency exchange-rate volatility and compares broad approaches, including simple purchasing-power-parity ideas, factor models, univariate GARCH, and multivariate dynamic conditional correlation models. The author is…
The note explains how a stochastic volatility model distinguishes physical-measure dynamics from risk-neutral dynamics used for option pricing. It describes a volatility risk premium that shifts the variance process’s drift, while the stock drift is adjusted…
The explanation links sticky-delta implied volatility to scale invariance in asset-price models. If scaling the initial spot and strike by the same factor scales a European option’s value by that factor, the option price is homogeneous of degree one.…
The discussion examines whether an option’s quoted price captures the cost of hedging when implied volatility varies across the life of the trade. In an illustrative case, volatility is expected to be concentrated in one part of the option’s life. The answer…
The document asks whether a stochastic volatility model and its Gyöngy local volatility counterpart imply equal option prices and hedging costs. It describes the relationship as matching the terminal distribution of the underlying under the two processes,…
The document compares Heston stochastic volatility with Vanna–Volga (VV), an interpolation approach used to construct an FX implied volatility smile. It describes VV as requiring a small set of market volatility quotes, commonly at-the-money and 25-delta…
The note concerns constructing an FX volatility surface when market quotes are expressed by moneyness or delta rather than by fixed absolute strikes. It describes limitations encountered with Python bindings for QuantLib and an Open Source Risk Engine…
The note explains why option models are often calibrated directly under a risk-neutral measure using observed option prices, without first specifying a corresponding physical-measure process. The calibrated model is used to extrapolate prices beyond the…
The document asks whether a discretely sampled arithmetic Asian option depends on forward implied volatility. It compares the question with forward-starting and barrier options, whose values can be sensitive to the volatility smile at a future reset date,…
The document explains why an option’s time value is generally greatest near the money and declines as the option moves deep in or out of the money. It uses delta hedging for intuition: a long option hedged with the underlying can earn from rebalancing as the…
The document addresses a one-factor Hull–White calibration that fits mean reversion and a single volatility parameter to market normal-volatility quotes for at-the-money swaptions. The reported fit is poor across quotes, raising the question of how to…
The discussion asks whether Heston option prices have a convenient approximate Black–Scholes implied volatility formula, comparable to formulas used with other stochastic volatility models. Heston European option prices can be computed accurately through…
The document addresses how to estimate implied volatility for expiries shorter than the first available maturity on a volatility surface, with very short dated binary options as the motivating use case. One proposed method is to hold volatility flat at the…
The document outlines a framework for choosing how often to rebalance a delta hedge during gamma scalping. Under geometric Brownian motion, it approximates the hedge adjustment as gamma times the underlying price move, then estimates trading costs by…
The document addresses how to adjust an implied-volatility surface when its quoted option prices contain small butterfly arbitrage violations. The specific issue is a discrete set of call prices across strikes that fails convexity, which can undermine later…
The document asks how to price an American-style Asian option whose exercise payoff uses the average underlying price over a recent rolling window. It explains why early exercise and path dependence complicate standard approaches: Monte Carlo and tree…