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 document examines claims that call options have limited downside and that their value variation is concentrated on the upside, then asks how these properties relate to model risk. A long call’s expiry payoff cannot fall below zero, so after paying the…
The document explains how to express option prices generated by the Heston stochastic-volatility model as implied volatility. The proposed convention is to invert a Black–Scholes pricing formula to find the volatility that reproduces the given price. In this…
The document explains the appearance of the standard normal cumulative distribution function in the Black–Scholes price of a European call. The model assumes the stock follows geometric Brownian motion, so its terminal log price is normally distributed.…
The document describes a numerical attempt to compute local volatility from a smoothed implied volatility surface using Dupire’s equation. The author compares the resulting local volatility with smoothed implied volatility and asks whether local volatility…
The discussion compares ways to estimate implied volatility at expiries that fall between dates where SABR parameters have been calibrated. For rough estimates, it recommends linear interpolation of total variance at a fixed strike. Interpolating the SABR…
The document asks how option time value appears in quantitative pricing when a pricing expression focuses on the underlying price and strike. The answer links time value to the convex payoff of an option and to the fact that settlement occurs at a future…
The document offers an intuition for why the risk-neutral probability that a call finishes in the money, represented in Black–Scholes by the normal cumulative value at d2, tends toward zero as volatility grows. The explanation focuses on the terminal stock…
The document asks how to price a stock call under stochastic interest rates using a maturity-matched bond as numeraire. It starts from the change-of-numeraire relation and rewrites the discounted option value as the bond price times an expectation under the…
The discussion outlines ways to account for dividends when applying Black–Scholes to stock options. A continuous dividend yield is presented as a simple extension, especially suitable for European options. More detailed approaches can represent dividends as…
The discussion explains how CME Eurodollar option premiums are quoted and compares a displayed quote of 0.0375 with a calculator value of 3.75. The answer identifies the calculator value as 3.75 basis points for the described option, apparently a September…
The document asks how to create a derivative that pays a fixed cash amount when the underlying asset’s price at maturity lies within a specified interval, and pays nothing outside it. The proposed construction uses two call spreads: buy one around the lower…
The document addresses a discrepancy in the third adjustment term used when reproducing the CBOE SKEW Index. The adjustment accounts for the difference between the at-the-money strike and the forward level. The questioner reports that their first two…
The document explains how the normal cumulative terms in Black–Scholes pricing relate to different contingent payoffs. It corrects the geometric Brownian motion solution by including the half-variance adjustment in the exponent, and notes that Brownian…
The document derives the value dynamics of a self-financing portfolio that holds an underlying asset and a money-market account. With a single risk-free rate, the cash position earns that rate, while the risky holding contributes the asset’s drift and…
The document explains how to choose a risky-asset weight for a one-period portfolio when the investor has power utility, a risk-free return, and a probability distribution for the risky asset’s return. The return distribution is represented by possible…
The document poses a portfolio construction question about equally weighting short put positions by option price across different stocks with a shared maturity. The author describes Monte Carlo results in which portfolios involving more highly correlated…
The document compares binomial trees with finite difference methods (FDM) for option valuation, especially American puts. It explains that a recombining binomial tree is a special case of an explicit FDM scheme, while more general FDM approaches can handle…
The document gives a concise relationship for European calls and puts with the same strike and expiration: their price difference corresponds to the value of a forward position. In standard pricing terms, call price minus put price represents the present…
The document raises a valuation question about European options or warrants that dilute existing shareholders. It outlines two proposed adjustments: scaling a Black–Scholes value by the inverse of one plus a dilution ratio, or first adjusting the share price…
The document explains how to interpret and estimate the expected growth parameter μ in a geometric Brownian motion. Under a real-world probability measure, μ represents the expected stock return. If returns are assumed stationary, it can be estimated by…
The document explains how to determine whether a firm’s equity is in or out of the money by applying the option interpretation of corporate equity. Shareholders hold a residual claim on the firm’s assets after debt is repaid, so equity can be represented as…
The document presents an early-exercise question for American call and put options on a dividend-paying stock. It specifies a nine-month term, two scheduled dividends, a strike, a continuously compounded risk-free rate, a current stock price, and quoted…
The document asks how the money-market account appears in a Black–Scholes delta-hedging portfolio and its relationship to a hedging-error expression involving gamma and a difference between assumed and realized variance. The answer identifies the cash…
The document concerns reproducing inflation cap prices with the Bachelier formula as a step toward calibrating a SABR model. The questioner says they have normal volatility quotes by strike and maturity, a forward-rate curve, and cap prices expressed in…