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 considers how to value American calls with five-to-ten-year maturities on an accumulation fund that has no readily available exchange price or option quotes. It notes that such long-tenor options may be available over the counter, but…
The document outlines how a parabolic pricing PDE can be related to an expectation over paths of the associated stochastic process. Applying Itô’s lemma to the solution along that process produces a drift term matching the PDE and a stochastic integral. With…
The document explains two ways to measure an option’s one-day theta when implied volatility varies across expiries. Fixed-expiry theta holds the option’s implied volatility constant during the time bump, matching the Black–Scholes convention. Fixed-tenor…
The document derives the dynamics of a log forward price in a Schwartz model where the logarithm of the spot price follows a mean-reverting diffusion. The forward price is given as an explicit function of time and the current log spot. Applying Itô’s lemma…
The document examines a lower bound for the variance swap strike, expressed as a risk-neutral average of Black-Scholes implied variance over a standard normal coordinate. It uses a Taylor expansion around the center of that coordinate. The linear, or skew,…
The note explains the negative sign in the first-order relationship between a bond’s price change and its yield change. Modified duration is conventionally positive for an ordinary bond, while bond prices and yields move in opposite directions: a yield…
The document diagnoses a small discrepancy between a fixed-rate bond price from QuantLib and prices obtained from a spreadsheet or another pricer. The example specifies a semiannual coupon bond, an evaluation date, a yield, and an Actual/Actual day-count…
The document asks how to compare the predictive pricing errors of three option valuation models against Black–Scholes prices, using both in-sample and out-of-sample results. It raises the Diebold–Mariano test as a possible method and questions whether its…
The document illustrates a common failure in Monte Carlo control variates: the target and control estimates were simulated with separate random draws, so their covariance did not provide the intended variance reduction. In the displayed put and call example,…
The document asks how control variates might improve Monte Carlo pricing for European and American down-and-in basket puts under a Heston model simulated with the QE scheme. It considers using a vanilla put as a control for a barrier payoff, while noting…
The document explores whether a trader can neutralize an options portfolio’s sensitivity to implied volatility, or vega, while retaining directional exposure. Combining long and short options can offset vega because options carry vega whereas linear…
The document weighs the practical value of studying complex analysis for quantitative finance. Its clearest application is derivative pricing: some models have conditional characteristic functions that enable quasi-analytical European option prices through…
The document explains why the term risk-neutral probability is used in derivative pricing, even though the measure is not a direct statement of real-world event odds. The question notes that risk attitudes are reflected in the pricing measure and that, under…
The document considers a European call option on the VIX, expressing its payoff as the positive part of the square root of a variable representing squared VIX, minus the strike. It states a Fourier transform involving the error function and asks how that…
The document asks why an overnight indexed swap (OIS) par rate cannot simply be converted into a discount factor using a single-period formula. The response explains that discount factors at longer maturities must remain consistent with discount factors…
The document presents a question about iteratively estimating volatility under the Heston–Nandi model from simulated price data. The author simulates a price process using a separate GARCH-generated volatility series, then applies a proposed recursion to…
The document outlines mathematical preparation for studying stochastic volatility models and volatility arbitrage. It identifies calculus, differential equations, linear algebra, and probability as a sufficient foundation for engaging with standard textbooks…
The document explains how to price fixed-rate bonds in QuantLib when the discounting term structure is non-flat. It distinguishes a curve built from zero rates, which requires maturities and corresponding yields, from one built directly from discount…
The document discusses a sovereign credit default swap pricing formula that produces negative values when the market spread is below the contract’s fixed coupon. The question focuses on contracts with a stated coupon, for which this outcome occurs often, and…
The document considers how to add EURUSD European options to a one-day Monte Carlo VaR framework already using correlated historical shocks for FX and commodity returns. The proposed method applies each simulated spot return, reduces time to expiry by one…
The document asks how equivalent martingale measures (EMMs) differ from equivalent local martingale measures (ELMMs) in Brownian financial-market models, and whether existence of an ELMM guarantees an EMM. It highlights that an EMM is also a local martingale…
The example tests whether a stock, call, and put with the same strike are priced consistently. It proposes buying the put and the stock while selling the call. For the stated prices, the combined position costs less than its payoff at expiration, which the…
The document considers an option that pays one dollar if a non-dividend-paying stock, initially priced at one dollar, reaches an upper barrier. A geometric Brownian motion assumption under risk-neutral pricing gives the questioner a value of the reciprocal…
The document sets up the Cox–Ingersoll–Ross short-rate model, with mean-reverting drift and volatility proportional to the square root of the rate. It applies an exponential-affine form to bond value and gives the resulting differential equation for the…