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 discussion explains how to solve the Black–Scholes partial differential equation directly with a finite-difference scheme, without first transforming it into the heat equation. A European option’s payoff at maturity supplies the terminal condition;…
The question examines whether an American option on a deterministic, zero-volatility asset can have time value. Its proposed reasoning treats the option’s value as the best payoff reached along the asset’s deterministic path and argues that early exercise…
The document presents a C++ Monte Carlo setup for pricing a call under the SABR model and comparing simulated prices with Hagan and first-order Paulot approximations. It describes Euler and Milstein schemes as intended approaches, and supplies code that…
The document considers a put payoff on a geometric Brownian motion when exercise occurs at an independent exponential time. It explains that replacing the random exercise time with its mean is generally invalid because the payoff is nonlinear in both the…
The document explains how to model a USD-denominated convertible bond whose conversion feature delivers shares priced in euros. If conversion delivers a fixed number of shares and all relevant thresholds are USD-based, the conversion value is a composite USD…
The document derives the price of a call option that pays the square of the underlying price divided by the strike when the underlying finishes at or above the strike, and otherwise pays nothing. Under a Black–Scholes lognormal model, the direct approach…
The discussion explains why a floating-strike lookback call can have zero gamma at inception under a homogeneous underlying model, yet develop nonzero gamma later. Its payoff depends on the terminal price relative to the minimum price reached over the…
The document explains why options on one stock can have different implied volatilities across strikes and maturities. Implied volatility is a way to express an option price using Black–Scholes assumptions; it does not mean the market expects each option’s…
The document addresses whether Dupire’s local volatility formula can be expressed using put prices as well as call prices. Under the stated zero-rate, zero-dividend setup, or when working with undiscounted options on forwards, the formula uses the option’s…
The document presents a Python implementation intended to price a European option under the Kou double-exponential jump-diffusion model. The author reports a call value that differs from a reference value and asks for help locating the error. The code…
The document works through a European cash-or-nothing call that pays a fixed amount if the underlying price finishes above the strike. It outlines the risk-neutral stock process when the asset pays a continuous dividend yield, identifies the market price of…
The document asks how traders should select strikes and expiries for option positions such as protective puts and volatility strangles, while expressing concern that choices based only on backtest performance may be overfit. The response offers a practical…
The question examines why the interest-rate sensitivity of a European option on futures can differ from the rho formula for an option on an asset with a continuous yield. The explanation identifies two rate exposures: discounting the expected payoff and the…
The document describes an implementation problem with Least Squares Monte Carlo (LSM) pricing of American vanilla puts. The questioner follows Longstaff and Schwartz’s setup, including simulated paths with antithetic variates and regressions using weighted…
The discussion explains how a forward used to delta hedge a vanilla option can be represented synthetically with options. Under put–call parity, a short forward position corresponds to a short call and a long put at the same strike and expiry. The example…
The discussion answers where Python strategies can send live orders across markets, distinguishing execution connectivity from Python’s role in research or backtesting. It names Interactive Brokers’ APIs and tools, including IBPy, IB Gateway, and Trader…
The document explains two Brownian motion calculations relevant to barrier option pricing: the midpoint distribution conditional on the terminal value, and the probability of crossing a barrier between fixed endpoints. For the midpoint, it uses the joint…
The document addresses why extending an option’s time to expiry does not always increase a call’s value when the underlying has a positive dividend yield. Its explanation invokes the Black–Scholes call formula and put–call parity: as the horizon grows,…
The document explains two ways a long option position can benefit from volatility: a rise in implied volatility can increase the option’s resale value, while large underlying price moves can benefit a long gamma position. A long straddle combines exposure to…
The document presents a Carr and Wu option-pricing identity for a Markov model in which the underlying asset price and time describe the state. A call expiring at a later date can be represented as an integral of calls expiring at an intermediate date,…
The discussion examines a Monte Carlo pricing expression that represents VIX through the risk-neutral expectation of the log of the S&P 500’s future level relative to its current level. The question is why simulated values might make the expression under the…
The document discusses a proposed solution to a modified Black–Scholes equation for a European put when volatility depends on the underlying asset price. The proposed expression uses associated Laguerre polynomials and is evaluated as a series. A numerical…
The document investigates why a short put’s hedge delta increased even as the underlying price rose slightly. It reports that volatility also declined and two days elapsed, with a long time remaining to expiration. The response identifies charm, or delta…
The document raises practical questions about modeling American options with Black–Scholes–Merton, including early exercise, changing bid–ask spreads, intraday time to maturity, volatility estimation, historical risk-free rates, strike listings, and…