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 asks how allowing a stock to finish unchanged affects no-arbitrage pricing in a one-period model. The stock starts at 100 and may end at 50, 100, or 150; the European call has a strike of 150. Since the call pays zero in every listed state, its…
The document explores how the standard Heston characteristic function might be expressed using an affine forward variance curve. It lays out a transform representation involving a Riccati equation and a convolution with the initial forward variance term,…
The document presents a poorly fitting SABR calibration and asks what is wrong with the implementation. The Python function calculates a forward value from a variable associated with the volatility parameter, while the fitting routine estimates SABR…
The document examines a proposed options trade: if a trader believes volatility is known and constant, a Black–Scholes price below the market price might appear to imply an unusually high return from delta hedging. It asks why traders use implied volatility…
The document collects recommendations for an undergraduate mathematics student seeking an accessible introduction to Black-Scholes theory and stochastic calculus. The suggestions include texts that introduce stochastic calculus and derivative pricing,…
The document compares two ways to value a structured payoff that rewards moderate upward or downward moves but expires worthless after a larger move. One method represents the payoff as an up-and-out call combined with a down-and-out put, then prices each…
The document asks how to enforce a minimum terminal wealth in a continuous-time consumption and investment problem with CRRA utility. It presents a proposed construction: scale down the unconstrained Merton strategy and use the remaining initial capital to…
The note describes a theoretical way to express changes in implied volatility skew across maturities using a calendar spread of digital puts. A digital option’s price depends on the strike slope of implied volatility, so comparing digitals across expiries…
The document separates the effect of an isolated implied-volatility increase from what may happen during a real VIX spike. Holding other inputs constant, a call’s positive vega means a rise in implied volatility raises its price. Vega itself changes with…
The document considers a European call when the underlying is modeled as unable to rise above a fixed upper price. It proposes integrating the discounted lognormal payoff only between the strike and the bound, with a separate expression for the chance of…
The document asks why the interest-rate sensitivity, or rho, of calls and puts appears to have different signs under Black–Scholes and Black-76. It reports Black-76 formulas in which call and put rho are both negative, then contrasts them with the familiar…
The document raises questions about an argument linking the skew of an asset distribution, risk-neutral valuation of a binary call, and a call’s delta. It asks whether risk-neutral probabilities compensate for greater payoff potential in one tail, whether a…
The problem compares a call and put with the same strike and maturity against a binomial stock-price model and a stated borrowing rate. Because the call-minus-put position replicates a forward payoff, its cost can be compared with the value implied by the…
The discussion suggests research topics for a finance thesis at the beginning of master’s level. One direction is to study how option implied volatility and skewness can help estimate market betas or expected returns. The cited examples cover equity risk…
The note identifies finite-difference methods as a classical approach to pricing discrete barrier options. For a local-volatility model, it points to a PDE solved with Crank–Nicolson time stepping and Rannacher startup steps. For a stochastic…
The document explores whether a one-step binomial model must use the option’s underlying stock in a replicating portfolio. The response explains replication from the seller’s perspective: construct and, where possible, rebalance a portfolio so its future…
The document surveys possible approaches to valuing a loan prepayment option. One response frames prepayment as an option on a bond and recommends calibrating an interest rate model, with tree and simulation methods raised as candidate valuation techniques.…
The document explains a constraint on implied volatility across option maturities under a Black–Scholes term-structure assumption. It considers overnight and two-week at-the-money implied volatilities that are equal, and asks how to bound the one-week…
The document describes a way to estimate an option market’s implied financing rate when call and put quotes are available across strikes for a fixed expiry. Using bid and ask combinations in put-call parity, it calculates a range of implied forward values at…
The document derives a scalar Riccati ordinary differential equation for a coefficient in the affine characteristic function of the Heston stochastic-volatility model. Starting from the stated coupled equations, the questioner observes that one coefficient…
The document asks whether SABR can be used for foreign-exchange and equity underlyings when beta, estimated from a regression of at-the-money implied volatility against the forward, falls below zero or above one. The responses distinguish parameter fitting…
The document examines a position consisting of an index purchase, a put with the same strike, and borrowing to finance part of the initial cost. The asset and put together create a floor: at expiration, the position’s value is bounded below by the strike and…
The document explains how to handle future discrete dividends when pricing options with a recombining tree. Declared payments can be entered directly, while dividends beyond the announced period must be estimated. The answer describes using market data…
The document gives a way to choose the volatility bump used to estimate option vega by finite differences in a binomial pricing model. A very small bump can amplify pricing precision errors when subtracting nearby prices, while a large bump increases the…