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 asks whether the variance of a price can be derived from the variance of its daily returns, including for use in Bollinger Bands. It explains that treating the previous price as known can yield a conditional relationship for one step, but that…
The document asks how useful implied volatility remains when the Black–Scholes assumptions do not fit the market. It uses stock-borrowing fees as an example: different borrowing and lending rates can change the pricing equation, so an option price must be…
The document clarifies the relationship between normally distributed log returns and the expected value of a geometric Brownian motion price. If log price follows a Brownian process with drift μ and volatility σ, exponentiating it yields a lognormal price…
The document examines whether a portfolio that is neutral to vega is also neutral to gamma and theta. One response gives a specific result for portfolios of calls and puts sharing the same maturity: under the Black–Scholes framework, gamma neutrality and…
The document considers whether the leading principal component of historical interest-rate changes can serve as the volatility curve for calibrating a Hull–White or Black–Derman–Toy tree. It contrasts estimating a separate standard deviation at each tenor…
The document introduces an ATM implied-volatility correction in a free-boundary extension of SABR, motivated by pricing under negative rates. It describes a small-time heat-kernel approximation and gives a first-order correction for the at-the-money case…
The discussion explores estimating one-day downside risk for a portfolio that combines linear assets with short option exposure. It contrasts fitting a generalized Pareto distribution to the tail of repriced historical portfolio returns with simulating…
The document raises a mathematical question about deriving an asymptotic option-price expression used in the Hagan SABR implied volatility formula. It focuses on a partial differential equation for option value, with terms involving the forward price,…
The document compares two ways to extract risk-neutral information from option prices. Breeden-Litzenberger (BL) uses the curvature of call prices across strikes to estimate an entire risk-neutral density. Bakshi-Kapadia-Madan (BKM) methods estimate…
The document discusses hedging a perpetual up-and-out call with a strike of 110 and barrier of 120. One proposed static hedge buys stock in an amount intended to cover the option’s maximum intrinsic payoff at the barrier. This is a buy-and-hold superhedge…
The document raises the question of how to incorporate GARCH volatility forecasts into a Markowitz mean-variance portfolio optimization. The investor’s existing process estimates expected returns and a variance-covariance matrix, then seeks a portfolio with…
This discussion explains how to interpret at-the-money swaption volatility quoted under normal and Black conventions. Normal volatility is expressed in rate units, commonly basis points, while Black volatility is relative to the forward swap rate. The…
The document raises a derivatives calibration question by analogy with joint modeling of the S&P 500 and VIX. Since the VIX is derived from S&P 500 options, the author asks whether the Euro Stoxx 50 and VSTOXX should likewise be calibrated together. The…
The document clarifies competing conventions for dollar gamma. It distinguishes dollar gamma, commonly expressed as gamma multiplied by the square of the underlying price, from the half-dollar-gamma coefficient used in the second-order approximation of…
The document discusses how phoenix and snowball autocallable notes can be valued and hedged. It describes the note price as a combination of a zero-coupon bond, coupon exposure, a short down-and-in put that may be terminated by autocall, and issuer profit.…
The document asks how to interpolate implied volatility between listed maturities to construct a volatility surface for a Black–Scholes delta-hedging experiment. It considers bilinear interpolation and seeks guidance on methods that can fill values between…
The document explains how to express a view that an option volatility skew is too flat or too steep while limiting exposure to the overall volatility level. Since option prices move with implied volatility, a trader can combine options at different strikes…
The discussion addresses why an at-the-money call can appear to have a delta near one when volatility is close to zero. One reply offers an intuition based on positive carry: with a risk-free rate above the dividend yield, a spot-at-the-money option can be…
The document explains why option volatility surfaces may use forward moneyness instead of raw strike. Moneyness measures a strike’s distance from a reference price; forward moneyness compares the strike with the forward price for that option’s maturity. This…
The document compares holding a cryptocurrency investment for a fixed period with periodically withdrawing gains while restoring the original principal. It describes a Monte Carlo example using simulated daily returns, then reports average outcomes for both…
The document asks how a risk surface can turn sensitivities from a model’s implied volatility function into sensitivities to market quoted volatilities. The described method adds an interpolated surface of risk bumps to the model volatility. At the base…
The document asks how to size a four leg iron butterfly and warns that expiration payoff calculations alone do not capture risks while the position is open. The response highlights execution costs, liquidity, assignment, margin, and the possibility of forced…
The document discusses estimating the volatility and mean-reversion parameters of a Hull–White short-rate model when the market has daily overnight rates and a swap-based zero-coupon curve but no actively traded options. It distinguishes estimation from…
The document asks whether negative short-term risk-free rates require changes to the Black-Scholes option pricing formula. Its answer says the standard model does not require the rate to be positive, so a negative rate can be used directly. Under the stated…