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 SABR calibration on Ethereum option data where the fitted alpha parameter changes substantially when strikes and the forward are expressed at a different scale, even though the implied volatilities are unchanged. The questioner…
The document clarifies how to approach an implicit finite-difference exercise for a European call when volatility varies over time. The stated example uses a linear volatility function, while the exercise also provides a Black–Scholes approximation that…
The document explains the Carr–Madan formula as a way to represent a sufficiently smooth payoff as a linear position in the underlying, a cash amount, and integrals of European put and call payoffs across strikes. The decomposition expands the payoff around…
The document shows how put–call parity can identify an arbitrage across European options with two different strikes, even when the current stock price is unknown. Applying parity separately at each strike and subtracting the equations removes the stock-price…
The document discusses extracting implied volatility from American option prices using a Cox–Ross–Rubinstein binomial tree. The tree prices the option by working backward from terminal payoffs, comparing continuation value with early exercise value at each…
The document explains how to simulate stock paths when periodic log returns are modeled with a Normal Inverse Gaussian (NIG) distribution, with the aim of pricing options by Monte Carlo. It warns that an NIG random variable cannot simply replace the normal…
The document questions a boundary condition for a travel-time option whose state includes a mean-reverting process with additive Brownian noise and its time integral. In the familiar geometric Brownian motion example, zero is absorbing: once the underlying…
The document presents two parametric ways to describe equity implied volatility across strikes and maturities. One uses ATM forward volatility plus linear skew and quadratic curvature on a log-moneyness scale normalized by volatility and the square root of…
The document addresses whether a martingale-corrected Andersen discretization for the Heston model needs adjustment when the model has a nonzero constant interest rate. The answer endorses adding the risk-free accrual over each time step to the exponent…
The document asks whether vanilla options on two assets, together with the assets themselves, can create insurance that costs money when correlation is high and pays when correlation falls. The motivation is to hedge a strategy that benefits from the assets…
The document compares three ideas used to explain why the Black–Scholes–Merton option-pricing formula can hold: profitable trading opportunities, market efficiency, and market equilibrium. It asks why equilibrium might be considered the most important of…
The document raises the problem of checking whether prices produced by models such as Black–Scholes or Heston are accurate. It distinguishes a priori validation, which includes choosing a model and fitting it to market data, from a posteriori validation…
The document discusses how to choose the spot-price bump when approximating call-option Greeks numerically and comparing them with Black–Scholes analytical values. It favors central differences, which reprice at spots on both sides of the current level, over…
This note derives the delta of a European cash-or-nothing call that pays when the terminal asset price exceeds a strike. Under the Black–Scholes framework, the option value is the discounted probability of finishing above the strike, and differentiating it…
This note explains why a delta-hedged position that is short options can lose when the underlying price moves. A zero net delta at the hedge point describes the position’s immediate sensitivity; it does not mean the option price falls when spot rises. For a…
The document raises a methodology question about selecting near- and next-term options when calculating intraday CBOE VIX values under the post-2014 maturity rules. It describes the eligible terms as more than 23 and fewer than 37 days to expiration. As time…
The document explains how to convert an annually compounded interest rate into the continuously compounded rate used in the Black–Scholes model. Equating the accumulated value under each convention over the same time horizon gives the conversion: take the…
This discussion examines the commonly used SABR at-the-money volatility approximation that keeps only the leading term, alpha divided by the forward price raised to one minus beta. The fuller Hagan expression includes correction terms involving expiry, the…
The document asks whether extensions to the SABR stochastic-volatility model, often motivated by rates that can become negative, offer benefits for options on non-negative underlyings such as equities and foreign exchange. The response argues against using…
The document addresses a performance question about calculating option prices or Greeks across many inputs in R. Its answer explains that the fOptions package's GBSGreeks function already accepts vector inputs, so a separate compiled-loop package is not…
The document explains why risk-neutral probabilities can be useful even though they are not forecasts of real-world frequencies. It uses a currency example: a terminal probability of a specified euro decline against the dollar can be viewed as an implication…
The note asks how to size option positions so a strategy reaches a chosen long-run daily volatility. It considers strangles and risk reversals, and compares sizing from the historical P&L volatility of one unit, from implied-volatility and vega estimates, or…
The document formulates a mean-variance approach to pricing a European call in an incomplete market. It seeks an equivalent martingale measure that minimizes the expected squared difference between the terminal value of a dynamically managed portfolio and…
The note clarifies what volatility implied by VIX options represents. VIX itself reflects an estimate of S&P 500 volatility over a forward 30-day period, but a VIX option does not simply measure the future volatility of today’s spot VIX. VIX options expire…