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.
10,263 documents
The document discusses how to infer the perturbation factor in a discrete-time Ho–Lee interest-rate tree when the initial bond term structure, risk-neutral up probability, and short rates are known. It gives a relationship between the logarithm of the…
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 document presents a perpetual coupon bond valuation problem in a diffusion setting. Its pricing equation relates the required return times the bond price to the coupon and the price’s drift and curvature terms, with a boundary condition at a stopping…
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 derives the foreign-exchange forward pricing relationship using the cost-of-carry framework. It treats the domestic interest rate as the opportunity cost of holding the currency and the foreign interest rate as the benefit earned by investing in…
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 why a hedge does not automatically reduce exposure at default (EAD) for risk-weighted asset calculations. Basel capital requirements separate counterparty credit, market, and operational risk; an ordinary market hedge may offset market…
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 document examines why par swap quotes reconstructed from a zero-coupon curve may fail to match the quotes used to build that curve. It describes bootstrapping a curve from deposit and swap helpers, extracting zero rates, then constructing swaps against a…
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 note explains that a standard QuantLib bond yield calculation returns yield to maturity based on the bond reaching its stated maturity, even when it has callable dates. It shows a basic fixed-rate bond example that calculates yield from price and then…
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 raises a question about the risk-neutral valuation of a zero-coupon bond when interest rates vary over time. It compares a bond-pricing expression involving the ratio of discount factors with a general risk-neutral pricing formula, asking…
The document examines a currency-hedged excess-return formula attributed to Black and Litterman, focusing on why the hedge adjustment includes a factor of one plus the domestic short rate. The responses derive the hedge payoff by combining the foreign…
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 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…