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 clarifies that a probability calculated from option implied volatility is not automatically a real-world forecast. Under Black–Scholes assumptions, the probability of finishing below a strike differs depending on the chosen pricing measure:…
The question asks whether the normal or shifted log-normal model is the industry standard for pricing swaptions during negative-rate conditions, with the stated purpose of supporting credit valuation adjustment work. The response does not choose between…
The discussion addresses why discount factors inferred from put-call parity across strikes may appear nonconstant. The key missing input is dividends: the basic parity equation without dividend treatment does not describe the relationship correctly when the…
The note shows how to establish that a call and put have the same vega when they share the same forward underlying, strike, maturity, and volatility inputs. It starts from put-call parity, which expresses the call-minus-put price difference as the forward…
This thesis-related question asks how to choose bounds for the Heston model parameters kappa, theta, sigma, and rho when generating synthetic data to train a neural network for calibration. It compares one published set of ranges, described as excluding…
This discussion considers a European call with an early-exercise feature that lets the holder exercise a portion and receive that portion’s current Black–Scholes value. The proposed replication is to hold the call and sell the corresponding fraction whenever…
The document introduces a lookback option whose value depends on the underlying asset price and its running maximum. It defines the underlying as a diffusion process and presents the running maximum as the limit of an integral-based approximation using…
The document addresses why option pricing under the risk-neutral measure uses the risk-free rate rather than a stock’s real-world expected return. It begins with a discrete-time replication argument: a call payoff can be matched by a portfolio of stock and…
The document explains how an FX options collar can hedge a foreign-currency receivable while preserving some favorable exchange-rate movement. It contrasts the structure with a forward, which locks in a conversion rate, and with leaving the receivable…
The document asks how to compute an option-value expectation expressed as a path integral. The central practical answer is to use Monte Carlo simulation: generate many possible realizations of the stochastic process and estimate the expectation with a…
The document explains how to estimate a variance risk premium when the available inputs are probability densities rather than volatility estimates. It defines the premium, loosely, as the difference between objective and risk-neutral variance of future…
Implied volatility is the volatility input that makes an option pricing model, such as Black–Scholes, match an option’s market price. Because each strike, maturity, and option type is a distinct contract, each can trade at a different price and therefore…
The document asks whether two assets with equal return volatility can be combined into a product with greater percentage-return volatility, without simply using cash leverage. It considers a long-only linear combination and derives its variance from the…
The document discusses foresight, or look-ahead, bias in least-squares Monte Carlo valuation of American options. In the Longstaff–Schwartz approach, regression on the same simulated paths used for valuation can overfit continuation values and make estimated…
The question challenges the idea that an American option should always be worth more than its intrinsic value, making sale preferable to early exercise. The key correction is that at an optimal exercise point, the option’s value equals its exercise value.…
The discussion considers how to build an equity volatility surface when American and European options are listed on the same underlying and provide separate implied volatility observations at each maturity. One practical preference is to use European options…
The document explains how to infer implied correlation from an observed Margrabe option price. Given the two assets’ volatilities, prices, yields, and time to maturity, correlation determines the option’s relative volatility. The response notes that this…
The document clarifies the proof that the first time an American derivative’s value equals its payoff is an optimal exercise time in a finite-period binomial model. The price process is constructed as the Snell envelope: at each node it is the greater of…
The document describes how to identify the underlying asset, strike, and maturity of French traded warrants when only their ISINs are available. Its suggested workflow uses Bloomberg: look up the ISIN in the terminal to open the security record, then view…
The discussion examines why Black–Scholes can produce implausibly high values for very long-dated options. One explanation is that the model holds interest rates and volatility constant, even though both can change over time. The more detailed argument…
The document explains why derivative pricing in a risk-neutral framework discounts expected future payoffs at the risk-free rate. Its central idea is that changing from real-world probabilities to risk-neutral probabilities accounts for risk premia in the…
The document asks whether the Black–Scholes risk-neutral probability that a call expires in the money, expressed using N(d2), can be adapted to estimate the real-world probability. The reply says this is possible if the asset's real-world drift is known. The…
The document examines how to relate options written on reciprocal currency pairs. It asks whether the delta of a USDJPY put corresponds to the delta of a JPYUSD call at the reciprocal strike and with the same maturity. The answer derives the terminal payoff…
The document asks how to derive an option-pricing probability expression in Gatheral’s framework by directly performing a complex inverse Fourier integral. It compares that route with the Gil-Pelaez inversion theorem and focuses on the boundary condition at…