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 distinguishes a yield curve as a general representation of market interest rates across maturities from a swap curve as one curve built from swap market rates. It explains that other instruments, including government bonds and interbank rates,…
The document derives the expected terminal stock value conditional on the stock finishing above the strike, a term used in the Black–Scholes call price. It starts from geometric Brownian motion, converts the condition that the terminal price exceeds the…
The document asks how much a short straddle could earn in the long run under an efficient market and Black–Scholes assumptions. It assumes the implied volatility used to price options matches future realized volatility, which remains constant over the…
The document explains why a QuantLib schedule moves a swap's Saturday effective date to the following Monday. The schedule uses a business-day convention for date adjustment; the example uses Modified Following, which shifts a non-business date according to…
The document asks how to value a perpetual American put in the Black–Scholes–Merton framework when exercise occurs the first time the underlying reaches a chosen threshold below the strike. It sets up the discounted payoff and attempts to transform its…
The document asks how to express the Ho-Lee interest-rate model as a stochastic differential equation after deriving its short rate from the instantaneous forward-rate expression. The central clarification is that the short rate follows a Brownian diffusion…
The document asks how to summarize implied volatility and Greeks for a bull put spread, formed by buying a lower-strike put and selling a higher-strike put. It considers averaging each metric across the legs versus taking the long leg’s value minus the short…
The post investigates a persistent difference between simulated returns on an option account and a delta-hedged account when the Black–Scholes model is run at very high volatility. It asks whether the discrepancy comes from confusing the volatility of log…
The document explains how to price currency forwards across different maturities and reconcile a theoretical GBP/USD forward with traded futures. Its central approach expresses the forward as the current exchange rate multiplied by the foreign discount…
The document surveys instruments traders use to infer market expectations for central bank policy rates. For the United States, it describes extracting an implied move from one-month Fed Funds futures under the simplifying assumption that the overnight rate…
The document examines why a basic Black–Scholes calculation may produce theoretical call prices outside observed bid–ask quotes. It points out that the example omits the continuous dividend yield in the pricing formula, but argues that correcting this alone…
The document outlines checks for whether a SABR-fitted option surface is free of static arbitrage. For a fixed maturity, it proposes deriving the risk-neutral density from the second strike derivative of call prices, then checking that the density is…
The document considers whether changing the mean and variance used in a geometric Brownian motion simulation to control how often paths cross a structured product threshold conflicts with arbitrage free pricing. Its answer distinguishes the physical measure,…
The document explores how to represent variance swap exposure using options. It starts from the risk neutral replication identity linking expected squared returns to a log contract and a continuum of calls and puts weighted by inverse strike squared. The…
The document addresses the apparent paradox of a digital call's delta growing without bound as expiration approaches the strike. A payoff capped at one unit may seem unable to have a price sensitivity greater than that amount, but delta measures a local…
The document introduces replication of common option strategies, including vertical spreads, butterflies, straddles, and strangles. Its central idea is to build a target payoff by combining simpler instruments, particularly zero-coupon bonds and European…
The discussion explains that counting Brownian drivers and risky assets alone does not establish market incompleteness. In the example, the two assets depend on three named Brownian motions, but their exposures can be rewritten using only two independent…
The document compares financial mathematics with risk theory and risk management, noting that their methods can overlap while their scope and probability conventions differ. Financial mathematics is presented as quantitatively focused, with applications such…
The document derives a strong solution for a stochastic differential equation whose drift is inversely proportional to the state and whose diffusion is proportional to the state. One approach applies Itô’s formula to the squared process, turning it into a…
The document discusses the exposure profile of a buyer of credit protection in a single-name credit default swap. It explains the contract’s value as the protection leg minus the premium leg: as time passes, fewer premium payments remain, while the…
The document asks how a positively skewed stock-price distribution affects Black–Scholes–Merton option valuations across strikes. It includes two brief responses: one links right-skewed outcomes to greater probability of high terminal prices and a higher…
The document clarifies several measures of interest-rate sensitivity whose names are used inconsistently. Bond DV01 is described as the price change for a one basis point move in the bond’s own yield, commonly its yield to maturity. This differs from…
The document considers how to calculate option deltas across strikes when implied volatility varies with strike. It frames the key issue as selecting the implied volatility input to Black–Scholes when the volatility surface is skewed, especially for hedging…
The document addresses the intuition behind the Black-Scholes call price as volatility grows without bound. An accepted explanation states that the call approaches the discounted value of the underlying forward, which is the spot price when dividends and…