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.
10,263 documents
The document explains why hedging a mortgage-backed security with a swap is complicated when the underlying mortgages can be prepaid. Because prepayments depend in part on future interest rates, the security’s outstanding principal and the matching swap…
The document frames option leverage as delta divided by option price and asks which European strike maximizes that ratio. It supplies Black–Scholes call and put prices and deltas, then presents a numerical exploration across strikes for selected fixed…
The document compares the classical LIBOR Market Model (LMM) with a stochastic-volatility extension called SABR-LMM. LMM models a family of forward rates jointly, allowing their correlation structure to inform the pricing of products that depend nonlinearly…
The document explains how to assess completeness in a one-step trinomial stock model with a risk-free bond. An equivalent martingale measure assigns positive probabilities to all terminal states and makes the discounted stock price a martingale. The stock…
The document explains why the mean of a Black–Scholes log return is (μ − ½σ²)Δt rather than μΔt. Starting from geometric Brownian motion for the asset price, Itô’s lemma yields a log price whose drift includes the negative half variance term. Brownian…
The document explains the formula (1 − exp(−sT))/s by modeling a bond that pays continuously at a unit rate until maturity or default. With zero interest rates, no recovery, and a flat default intensity equal to the spread, the probability of surviving to…
The document introduces basis-function selection in Longstaff–Schwartz least-squares Monte Carlo pricing for options on multiple underlyings. In this approach, continuation values are approximated with basis functions during backward induction, which…
The document asks how to express the value of a European call at an intermediate node of a Cox–Ross–Rubinstein binomial tree. It defines the node by the time step and the number of upward moves, and describes a backward-induction recurrence that discounts…
The document explains why differentiating the closed-form geometric Brownian motion solution as if it were an ordinary deterministic exponential gives the wrong drift. Because the exponent includes Brownian motion, the appropriate tool is Itô’s lemma, which…
The document raises a theoretical question about a consistency property in the Vanna–Volga method. Starting from a price curve constructed by interpolation and extrapolation across strikes, the cited paper claims that selecting three different strikes on…
The document shows how to generate joint spot and variance paths from QuantLib's Heston process in Python. It constructs the process from initial variance, mean-reversion, long-run variance, volatility-of-variance, correlation, spot, and rate inputs, then…
The document asks whether information criteria or divergence measures can guide SABR calibration. The response describes a common rates-market workflow: preset beta, solve alpha to match the at-the-money volatility, then optimize rho and nu against smile…
The document examines why a bisection-style search for Black–Scholes implied volatility can fail for a put even when it works for a call. The key diagnosis is that the put price in the example is above the greatest value a European put can reach under the…
The response outlines a pricing approach for an FX-linked payoff when payment is made in a third, quanto currency. It defines the exchange rate with the foreign currency as the numerator and gives its drift under the quanto currency measure. The drift…
This document asks why a leveraged WTI exchange-traded note and its inverse counterpart appear to trace different slopes during large price moves. One note is described as providing leveraged long exposure and the other as leveraged inverse exposure. The…
The document answers whether the standard European put-call parity relation also applies when the underlying is modeled with Heston stochastic volatility. Its central point is that parity follows from the option contracts and the ability to construct…
The document addresses how to summarize implied volatility across an option chain rather than relying only on the at-the-money option. It describes a model-free variance-swap approximation built from option prices over a range of strikes: calls above the…
The document asks for the stochastic differential of an exponential built from an integral involving a diffusion process. The questioner applies Itô’s formula to the exponential, but assumes the integral has differential equal only to the integrand at the…
The example asks how a company can exploit a forward contract offered at an unfavorable price. It assumes a non-dividend-paying stock, a continuously compounded interest rate, and a transfer of an existing long forward position for a fee before maturity. The…
The document outlines several ways to think about pricing options on a volatility index such as VIX. One approach starts from the option portfolio used to replicate variance and adds a convexity adjustment that accounts for the square root linking variance…
The document asks what changed in Monte Carlo methods for pricing American options between 2005 and 2016, following earlier machine-learning work on optimal stopping. One answer argues that the period brought few groundbreaking Monte Carlo advances and that…
The document frames three ways to convert a fixed-rate bond position into floating-rate cash flows while seeking to isolate the issuer spread: a par-par asset swap, a same-notional payer swap, and a payer swap sized to match the bond’s DV01. It explains the…
The document derives a simple FX forward relationship from covered interest parity. For two currencies, it compares converting at spot and investing at the domestic rate with investing at the foreign rate and locking in the maturity conversion rate through a…
The document considers a finite-difference approximation to the Black–Scholes equation and asks how coefficient sign conditions can establish a maximum-norm bound. The proposed argument rewrites each value at one time level using the neighboring values and…