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 explains how daily leveraged and inverse exchange-traded products can lose value through compounding when the underlying index is volatile. It compares short sequences of positive and negative index returns and shows that the inverse product’s…
The document discusses pricing a zero-coupon bond under the Cox-Ingersoll-Ross short-rate model and asks how to choose its long-run rate, mean-reversion speed, and volatility. One response expresses bond value as the expected discount factor over the bond’s…
The document considers a one-period market with two risky assets and three possible future states. After arguing that the market is arbitrage-free, the question is how to establish that every contingent claim can be replicated without solving explicitly for…
The document asks how to select an Esscher change-of-measure parameter for an exponential Lévy asset so that the discounted asset price is a martingale under the new measure. It writes the Radon–Nikodym density as an exponential tilt and uses conditional…
The document explains how correlation between asset returns and volatility innovations affects the relationship between Black–Scholes volatility and a stochastic-volatility model such as Heston. Negative correlation is reported as typical for equity indices…
The document surveys R packages for option pricing and related analysis. It names RQuantLib and NMOF for model implementations, including Black–Scholes–Merton, jump-diffusion, Heston, and characteristic-function approaches. It also points to RND for…
The document shows how to check that a Black-Scholes call price satisfies the Black-Scholes partial differential equation by substituting expressions for delta, gamma, and theta. The PDE relates the option value to its time sensitivity, underlying-price…
The document describes several uses of options pricing formulas even when markets are broadly efficient. Models can help traders assess possible mispricing, value contracts that are newly issued or lack a direct market quote, and infer quantities such as…
The document explains how to adapt a collateralized forward-pricing relationship for an inflation-linked bond with a coupon paid before forward settlement. The nominal bond proceeds method is adjusted by applying an index ratio to the spot value, interim…
The document examines whether a European call can be replicated when the stock follows an arithmetic Brownian motion and the bank account grows at a constant rate. It outlines an argument for market completeness by finding a change of measure that makes the…
The document explains why operating a gas-fired power plant creates exposure that resembles being long electricity and short natural gas. The plant’s gross margin depends on the electricity price received minus the fuel and emissions costs, adjusted for…
The document explains how to interpret the variance-rate assumption in the original Black–Scholes model. For a diffusion process, the instantaneous variance rate is the square of the coefficient multiplying the Brownian motion. In geometric Brownian motion…
The document explains how to derive an implied volatility surface for a currency pair from surfaces quoted for two related pairs. Its example combines EURUSD and USDJPY data to price EURJPY options. It first expresses the target option payoff in terms of the…
The document describes an implementation attempt for Haug’s double-barrier binary option formulas and explains why the asymmetric case produced incorrect results. The key correction is to treat the final term involving the asset’s position between the lower…
The document poses a derivation problem in currency derivatives. It defines stochastic dynamics for a foreign-market stock and the domestic/foreign exchange rate, alongside a deterministic domestic money account. The proposed approach converts the foreign…
The answer introduces the relationship between a chosen numeraire and the probability measure used for pricing. Its central idea is that, under the measure associated with a numeraire, the price of another asset divided by that numeraire has no expected…
The document explains why binomial-tree formulas for the up factor can differ across constructions. In the Cox–Ross–Rubinstein tree, the up and down factors are set symmetrically around zero log drift, with volatility determining their size. The answer notes…
The document asks how to calculate the variance of integrated variance in the Heston stochastic-volatility model. It connects the problem to the closely related Cox-Ingersoll-Ross square-root process, whose integrated state variable has a known…
The document explains how to screen listed-option observations before calibrating a pricing model. It describes filters based on low average prices, wide bid-ask spreads, extreme moneyness, and very short or long times to expiry. These rules aim to remove…
The note explains why an asset’s volatility over a fixed horizon matches the volatility of a prepaid forward contract on that asset with the same expiry. Both are measured as the standard deviation of a log return over the period. At the start, the asset…
The document rearranges European put-call parity to explain the components of put value when there are no dividends. It frames the put's intrinsic value as the strike minus the stock price, with time value reflecting both the call's upside protection and the…
The document asks how to obtain an analytical expression for a process defined as a power of geometric Brownian motion. It states the geometric Brownian motion model, presents an application of Itô’s lemma for the powered process, and rewrites the dynamics…
The document presents Python code for pricing a European call or put with an implicit finite-difference grid. It asks why the coefficient matrix is modified at its first and last rows and why boundary grid values at each time step are extrapolated from…
The document asks when a fixed-strike arithmetic Asian call can be worth more than a vanilla call with the same maturity. Its answer compares their maturity payoffs: the Asian call pays on the average underlying price, while the vanilla call pays on the…