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 post demonstrates a numerical implementation problem in recovering normal, or Bachelier, implied volatility from a call option price. The author prices a call under the Bachelier model with stated forward, strike, expiry, and volatility inputs, then…
The document asks whether a call and put with the same underlying, strike, and maturity must have identical implied volatility. For European options under idealized frictionless conditions, put-call parity links their prices. When both prices are interpreted…
The document discusses how to infer volatility from an observed option price when the pricing model can handle American exercise and dividends. Its central numerical method is bisection: choose lower and upper volatility bounds, price the option at their…
The discussion explains how a lender's fixed cash flows from a prepayable loan can be understood through swap and swaption positions. When a borrower prepays, the lender receives principal and must reinvest it at prevailing market rates, changing the value…
The document explains why OTC vanilla options are commonly discussed and negotiated using implied volatility rather than a currency premium. Volatility gives counterparties a shared quote for comparing offers even when their pricing systems or market inputs…
A Bermudan option permits exercise on specified earlier dates, giving its holder choices that a comparable European option, exercisable only at maturity, does not have. At each permitted date, the holder can exercise or continue holding the contract. Since…
This discussion compares ways to obtain U.S. listed options data for scanning many contracts. The question describes the tradeoffs among broker feeds, a streaming broker API with symbol limits, and commercial services that offer broader access but differ in…
The discussion examines whether buying an at-the-money option and delta hedging can profit when a trader expects realized volatility to exceed implied volatility. It explains that the outcome is not guaranteed: delta-hedging profits depend on the path of the…
The document works through the connection between variance swaps, log contracts, and the VIX calculation. Under risk-neutral assumptions, integrated variance is related to the expected log return. A static replication identity then expresses the log payoff…
The document explains how to value a reverse knock-out call in a finite, discrete price tree. The option expires worthless if the underlying has crossed an upper barrier at any point before maturity. The suggested procedure is to assign zero payoff to paths…
The document asks how to interpret an inequality relating prices of three European calls with the same expiration and increasing strikes. The accepted explanation invokes convexity of the call price as a function of strike: the price at an intermediate…
The document examines an apparent inconsistency in the one-period binomial option model. For a call spread whose strikes lie between the model’s down and up outcomes, the payoff slopes at those two endpoints can both be zero. Differentiating the model value…
The document explains how to convert an option delta quoted under a Black model into the corresponding strike, given the forward price, volatility, and time to expiry. It gives separate expressions for calls and puts, using the inverse standard normal…
The document asks why a contingent claim can be valued as the risk-neutral expectation of its discounted payoff. The answers connect this formula to the money-market account: expressing asset values relative to that account discounts cash flows for the…
The document considers whether an exotic option’s Greeks can be estimated by dividing historical option price changes by historical changes in the underlying. It explains that an option’s observed price movement is generally influenced by several factors at…
The document proves that a European put’s value is nondecreasing as its strike increases, assuming standard risk-neutral pricing and no-arbitrage conditions. For two ordered strikes, the put payoff at the higher strike is at least as large in every terminal…
The document examines why repeated Monte Carlo estimates of a European call price can vary, even when the same calculation is implemented in different programming languages. It compares sample estimates with the known analytical price and argues that the…
The document explains how to represent a piecewise-linear butterfly payoff as a portfolio of vanilla options. The payoff is flat outside two outer strikes, rises between the lower and middle strikes, and falls between the middle and upper strikes. The…
The document considers why retail platforms often show live options quotes but little historical price data for individual contracts. For directional trades, the underlying asset’s chart can guide timing because the option’s value depends on the underlying.…
The document explains why time to expiration in Black–Scholes and binomial option models is conventionally measured in years rather than seconds. The answers describe this as a practical convention aligned with annualized interest rates and volatility,…
The note identifies Bloomberg fields for retrieving equity option implied volatility through its API. For real-time data, it gives the mid, ask, and bid volatility field variants; for historical data, it gives the corresponding non-real-time variants. This…
The document addresses how to plot the value of a bull call spread before expiration. It distinguishes an option’s expiration payoff from its current theoretical value: before expiry, the Black-Scholes model prices each call using the current stock price,…
The discussion examines whether a self-financing options portfolio has Greeks equal to the sum of the Greeks of its holdings. For a fixed portfolio, sensitivities of total value are linear combinations of component sensitivities. With a changing portfolio,…
The document presents ways to convert between lognormal Black implied volatility and normal Bachelier implied volatility, a useful distinction in interest rate markets where quoted volatility conventions vary. It gives an approximation attributed to Hagan…