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 distinguishes the probability implied by an option model from uncertainty about whether that model is correct. Under a specified risk-neutral measure and pricing model, an in-the-money probability is a model-implied quantity; gamma does not…
The document introduces speed, the change in gamma as the underlying moves, and color, the change in gamma over time, then asks how traders might capture such higher-order sensitivities. Its answer focuses on gamma scalping: establish a delta-neutral,…
The document compares computational approaches for long-dated American options. One answer recommends implicit or Crank–Nicolson finite-difference methods: these can use larger time steps on a given price grid than a binomial tree and can restrict the…
The document frames a stress-testing problem for a portfolio exposed to an equity index through short autocallables and other worst-of basket products, short exchange-traded puts, and long underlying holdings. It lists potential risk drivers: falling prices,…
The document raises a question about principal component analysis applied to several fund return series. The author reports that the first component has much higher volatility than the funds used to construct it and wonders whether the component might…
The document explores whether implied volatility represents the market’s expected volatility, why volatility often rises as equity prices fall, and why out-of-the-money puts tend to have higher implied volatility than calls. One answer notes that…
The document asks how to estimate local volatility with Dupire’s formula using finite differences in option prices across maturity and strike. Its answer stresses that local volatility is a function of time and the underlying price, rather than a volatility…
The document explains the rationale for setting the up and down multipliers in a binomial asset-price tree to exponential functions of volatility and the square root of the time step. The scaling makes the size of each log-price move proportional to…
The document considers how default changes the probability distribution of a stock price at a future date. It proposes representing the probability mass associated with survival through a maturity-dependent factor multiplying a standard lognormal price…
The document asks why the delta of a standardized, perpetually at-the-money, 30-day put on SPY became less negative during the 2008 financial crisis. The described series uses OptionMetrics data, with standardized option prices derived by linear…
The document explains the common decomposition of an option’s market price into intrinsic value and time value. Intrinsic value is the immediate exercise payoff; time value is the residual after subtracting intrinsic value from the option price. For a call,…
The document describes a proposed tree-based approach for valuing an American call with two possible exercise references: a fixed strike and a floating strike tied to a 10-day moving average. The proposed model tracks both the underlying price and the moving…
The document examines a local relationship between daily stock returns and trading volume. The author ranks each day’s return and volume within a rolling 30-day window, then compares the resulting distributions for same-day observations and for next-day…
The problem considers a European payoff that pays a fixed amount below a strike and declines as the inverse square of the asset price above it. Under geometric Brownian motion with continuous dividends, the proposed value splits into two discounted…
The discussion explains why Gatheral's SVI smile is parameterized in implied variance rather than volatility or total variance. Variance is a natural scale because the model is linear in variance in the far strike wings. This makes the asymptotic slopes,…
The discussion asks whether the Feller condition, expressed through the Heston variance parameters, should be enforced when calibrating option prices. The motivating examples reportedly place the condition close to its boundary, with some parameter sets…
The document investigates why simulated GARCH(1,1) returns can have sample kurtosis well below a theoretical population value. The explanation centers on highly persistent squared returns and the possibility of rare, very large volatility shocks. When the…
The document asks how to recover the SABR parameters for the lognormal case, where beta equals one, from the local Taylor expansion of implied volatility around the forward. It labels the constant term as at-the-forward volatility, the linear coefficient as…
The document explains why vega is less straightforward in stochastic volatility models than in Black–Scholes, where volatility is a direct input. It describes two common bump-and-reprice interpretations: a model Greek, formed by perturbing model parameters,…
The note introduces the Heston stochastic volatility model, in which the stock price and its variance evolve together, with correlated shocks. It gives the risk-neutral characteristic function for log stock price and defines its component expressions. These…
This explanation motivates common boundary conditions for pricing a European call under the Heston stochastic-volatility model. At maturity, the option equals its payoff; when the underlying price is zero, the call is worthless. As the underlying price…
This discussion outlines possible reasons a value-at-risk estimate may be too high or too low when evaluated against later portfolio outcomes. It emphasizes that VaR depends on the assumed distribution and method, and that in-sample market behavior may…
The document sets up a comparison between two ways to price European calls under the Merton jump-diffusion model: the analytical Poisson-weighted Black–Scholes series and a numerical Carr–Madan Fourier inversion. It outlines the characteristic function, the…
The document explains intuitive bounds on VIX futures using forward-starting volatility and variance swaps. A volatility swap reflects expected future volatility, while a variance swap reflects expected squared volatility. Because the square-root function is…