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 addresses fitting SABR to market volatility smiles quoted in normal and lognormal conventions. It distinguishes the volatility convention used to translate SABR prices into implied volatility: normal market quotes should be compared with normal…
The document explains why a continuously monitored knock-out option’s value approaches zero as the underlying price reaches its barrier. For driftless Brownian motion, the reflection principle relates the probability of crossing a nearby barrier before…
The document raises a pricing question about autocallable products: why their value can depend on volatility of volatility and the forward smile, and why a local stochastic volatility model may be appropriate. The questioner offers a hedging intuition for…
The discussion asks how to price options with several expiration dates when volatility figures are supplied for successive years. The replies distinguish two interpretations: separate vanilla options observed in 2015, or forward volatility periods used to…
The discussion asks why an American option’s price should be continuous around an ex-dividend date and sketches an argument based on when early exercise is optimal. During periods when exercise is not optimal, the American option behaves like a European…
The document works through a Ho-Lee short-rate model with a Brownian short rate and derives the zero-coupon bond price, its stochastic differential equation, and the discounted bond’s dynamics. It then constructs the forward-measure density from the bond…
The document asks whether a European call’s delta must stay at or below one in an equity model with zero rates and dividends. Its answers show that this is not a general model-free bound: delta depends on how the option price changes with the current…
The document poses a one-period option-pricing problem in which three calls share an underlying and maturity, and their strikes are evenly spaced. It assumes the middle-strike call costs more than the average price of the two outer-strike calls, then asks…
The document explains why the prices of two calls on the same underlying and maturity cannot differ by more than the discounted difference between their strikes. Its main method combines put-call parity with the no-arbitrage ordering of option prices: a call…
The document introduces the uncertain volatility model (UVM), which represents volatility as unknown within specified lower and upper bounds, and asks how it compares with stochastic volatility models. Its appeal is a tractable way to incorporate volatility…
The document explains the typical effect of decreasing time to expiration on option delta, holding strike and other conditions comparable. As time value declines, an in-the-money option tends to behave more like its underlying, so its delta moves toward one…
The note contrasts two ways to represent volatility when modeling option prices. A local volatility model makes volatility a deterministic function of the underlying asset level and time. A stochastic volatility model instead allows volatility to evolve…
The document concerns recovering implied volatility from option prices when options are out of the money, where a direct closed-form inversion of the Black–Scholes price is unavailable. It points readers toward numerical methods and published work, including…
The document addresses negative prices produced for low-priced, out-of-the-money calls while calibrating a stochastic-volatility model. Its central explanation is numerical: if option prices are computed by numerical integration, approximation error can…
The document compares option deltas under sticky-strike and sticky-delta assumptions. With sticky strike, implied volatility is held fixed at a strike as spot changes, making the option’s spot sensitivity the usual Black–Scholes delta in that setup. With…
The document gives a proposed way to estimate the aggregate vega exposure of volatility-linked exchange-traded products. For an individual product, the answer scales its shares outstanding by share price and leverage, then divides by the one-month…
The document considers three cointegrated stocks with known portfolio weights and a mean-reversion view: when the weighted portfolio falls unusually low, a trader may expect it to recover. It first describes matching the portfolio’s directional exposure by…
The document asks how market makers handle heavy customer selling in short-dated, deep out-of-the-money options. Its example describes a dealer buying puts, delta-hedging, and accumulating long exposure to volatility skew and tail convexity. The question…
The document explores which derivatives can be described by the standard Black-Scholes PDE. Its main explanation is that the PDE applies when the derivative’s value can be represented using time and the current underlying price as the changing state…
The discussion presents the Black formula for a caplet and asks how to differentiate its price with respect to the forward rate and volatility. It states the familiar forward delta as the discounting and accrual factors multiplied by the normal cumulative…
The discussion derives the generalized Fourier transform of a European put payoff, defined as the positive part of strike minus the underlying price, when the underlying is represented in log-price coordinates. Integrating only over log prices below the log…
The discussion asks whether ATM swaption volatilities indexed by expiry and tenor must satisfy no-arbitrage conditions. It explains that there are no direct no-arbitrage restrictions across such quotes because each swaption is written on a different…
The document explains why delta hedging a cash-or-nothing binary option becomes impractical near expiry. Under the stated lognormal pricing model, the option value is discounted and proportional to the cumulative normal probability of the terminal price…
The document examines whether a steeper downside volatility skew raises or lowers the price of an at-the-money binary call expiring the next day. It explains that a digital call’s value is tied to the risk-neutral probability that the underlying finishes…