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 discusses how to express the Heston stochastic volatility pricing equation using a forward price, log-moneyness, time to expiry, and an undiscounted option value. It gives the spot and variance processes, their correlated Brownian shocks, and a…
The document explains why the Black–Scholes delta of an in-the-money call can first decline and later increase as volatility rises. Interpreting delta as the risk-neutral probability of exercise, it separates two effects: greater volatility makes the strike…
The document discusses how to convert daily US Treasury volatility to an hourly measure and cautions against assuming that trading activity is evenly distributed across the full day. Treasury trading is described as over the counter, without official…
The document explains how to find the probability mass under a normal distribution between a lower and upper bound. It uses the cumulative distribution function (CDF), which gives the accumulated probability from negative infinity up to a chosen value.…
The note explains how delta-hedging error arises when an option is valued using an assumed volatility that differs from the underlying asset's true volatility. The option's Black–Scholes pricing equation links its theta to its price, delta, gamma, interest…
The note asks why options on leveraged exchange-traded funds can appear cheaper than options on an unleveraged fund tracking the same index and expiry, despite an expectation that leverage should multiply implied volatility. It explains that a daily…
The document compares two numerical approaches for pricing European calls under the Heston–Nandi GARCH model using an inversion integral. One implementation evaluates the integrand through adaptive quadrature, while another precomputes…
The post asks how to select conditional-mean terms when fitting GARCH-family models to South African, Australian, and US equity index returns. The author compares autocorrelation plots, information criteria, ARCH tests, and the significance of terms in…
The document explains why jump intensity cannot be interpreted independently of the assumed jump-size distribution. To estimate a model’s jump frequency, it recommends fitting intensity and the other model parameters jointly, using maximum likelihood or…
The document investigates how to reproduce CBOE’s discontinued single-name 30-day normalized skew field using historical options data. It describes the target as the 25-delta put implied volatility minus the 25-delta call implied volatility, divided by…
The example shows how put-call parity and relationships across option strikes can be used to infer some option prices from quoted market data. Ignoring discounting, the response applies parity at the 70 strike to the given futures level and straddle price,…
The document explains how options systems represent time remaining on expiration day. Rather than using whole-day buckets, pricing and risk software generally represents expiry tenors fractionally and may measure time internally in years, seconds, or…
The document poses a pricing problem for a path-dependent option on a geometric Brownian motion asset. Its European payoff depends on the running maximum and a time-averaged quantity formed from the integral of the squared asset price. The question asks how…
The document explains how to value a European put that knocks in at a lower barrier and knocks out at an upper barrier. It expresses the payoff using the terminal put payoff and indicators for whether the underlying has crossed the lower barrier and remained…
The document asks whether the effective parameters in a time-decay SABR model can be expressed by one closed-form formula across all values of the decay start time. It presents two results from Sander Willems’s backward-looking SABR work: one for decay…
The document explains that there is no standard rule for how many past days to use when estimating historical volatility. It distinguishes historical, or realized, volatility from implied volatility: option prices reflect the market’s view of future…
The note explains that options on volatility control indexes exist, even though they may not appear as exchange-listed contracts. It says these options are generally embedded in structured notes and are mainly traded over the counter. This provides a reason…
The note discusses numerical instability when using the Bjerksund–Stensland 2006 option pricing model at very low volatility, especially when solving for implied volatility on out-of-the-money options. It offers a practical parameter choice: use a minimum…
The document describes calculating Black–Scholes delta and vega in R for a sample of executive compensation contracts containing multiple options. The author derives the Black–Scholes call sensitivities and reports that computed vega values are zero,…
The document asks how practitioners estimate equity volatility surfaces when listed option prices are scarce or unavailable. The answer describes a market-based approach rather than a formal model: index volatility is informed by active interdealer markets,…
The document describes a question about measuring volatility for five equal-weighted portfolios drawn from S&P 500 constituents and grouped by ESG ratings. Constituents are updated monthly as index membership changes, while ESG group assignments are…
The document raises a notation question about up-and-out and down-and-out call payoffs. It defines the terminal call payoff using the strike and multiplies it by an indicator based on whether the price path remains on the permitted side of a barrier. The…
The document examines why paired leveraged and inverse ETFs can have different daily returns even when their stated goals are equal and opposite multiples of an index’s daily performance. It distinguishes the index level, the fund’s net asset value based on…
The document raises a practical question about maintaining a long volatility hedge by rolling spot variance swaps monthly. It describes replicating spot VIX with a portfolio of out-of-the-money calls and puts, weighted by the inverse square of each strike,…