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 derives the density ratio between two forward measures by applying the change-of-numéraire principle. A numéraire is a strictly positive price process used as the unit for valuing other assets; changing the numéraire changes the associated…
The document raises a question about the payoff represented by Black’s model for an option on a futures contract. It distinguishes the futures price agreed when the option is exercised from the later spot price at the futures contract’s own maturity. The…
The document clarifies the distinction between a probability measure on entire paths and the distribution of a process observed at a fixed time. Knowing the pathwise Radon–Nikodym process does not generally give the new one-time density by evaluating it at a…
The document uses a Gram–Charlier approximation to connect the distribution of log returns with the implied volatility curve. It standardizes log returns, approximates their density using a normal density adjusted for skewness and kurtosis, prices a call…
The discussion separates a futures contract’s initial market value from its price, economic exposure, and the obligations created by taking a position. At a market transaction price, the contract typically starts with zero market value, while the long still…
The note explains how changing from the bank-account numeraire to a zero-coupon bond numeraire changes the drift of the Vasicek short rate. In a one-factor Brownian setting, the bank account has zero diffusion, while the bond price has diffusion proportional…
A digital call pays when the underlying finishes above its strike, so its discounted price equals the risk-neutral probability of finishing in the money. The document connects this price to the strike derivative of an ordinary call price. When ordinary call…
The document compares duration and convexity calculated for a zero-coupon bond represented in QuantLib as either a dedicated zero-coupon instrument or a fixed-rate bond with a zero coupon. With the same maturity, settlement date, and zero yield, the reported…
This exchange concerns transforming the Black–Scholes equation from an underlying spot price to a forward or futures price to obtain an equation for options on futures. The question gives the relationship between spot and forward prices and presents the…
This question examines when conditioning on the time-zero information set in a financial model gives the same expectation as an unconditional expectation. It highlights a potential tension: if the initial sigma algebra is trivial, every measurable random…
This discussion explains why an option can have a curved value before expiration even though its payoff at expiration is piecewise linear. Before expiry, an out-of-the-money option retains value because the underlying may still move enough for the option to…
The document asks how much smoothness a function needs for Itô’s formula to apply, using the positive-part payoff as its motivating example. The answer points to generalized functions, whose derivatives may be interpreted in the distributional sense, and…
The document clarifies the distinction between an instantaneous short rate and a continuously compounded spot rate for a particular maturity. In a short-rate model such as Vasicek, the instantaneous rate is a theoretical rate over an infinitesimal interval.…
The document asks how to implement the Vellekoop–Nieuwenhuis interpolation approach for pricing American options with discrete cash dividends. The author refers to a 2006 paper on derivatives with discrete dividends and provides a Cox–Ross–Rubinstein…
The document poses a question about how an open-ended factor certificate can deliver a fixed multiple of the underlying asset’s daily percentage move. It contrasts that target with a simple financed position: subtracting a growing loan from the underlying…
The document explains how to supply the initial value to QuantLib’s BlackProcess when pricing European options. The accepted answer clarifies that the process should receive the spot price; the analytic pricing engine derives the forward price from that…
The document examines Malliavin integration by parts for estimating option sensitivities in a diffusion model. It asks whether a Delta weight concentrated up to the first payoff observation loses relevant Brownian information, and how to extend the method to…
The document discusses why derivatives are often valued under a risk-neutral measure and how this connects to no-arbitrage pricing and hedging. One response frames risk-neutral valuation as an abstract link between a replicable payoff and the cost of the…
The document clarifies the valuation signs behind an asset swap spread. In the described arrangement, the investor pays the price of a risky fixed-rate bond and receives its fixed cash flows, while entering a swap that converts those payments into…
The document presents an approximation for pricing a geometric Asian call under Heston stochastic volatility. It replaces the random variance path with its deterministic expected variance, then integrates that expected variance over the option horizon to…
The document explains why an American call on a non-dividend-paying stock has the same value as its European counterpart, even though a binomial pricing algorithm may compare early exercise value with continuation value at each node. The key distinction is…
The document explains the coupon effect: fairly priced bonds with the same maturity but different coupons can have different yields to maturity when the spot-rate curve is not flat. Yield to maturity compresses a bond’s cash flows into a single rate and…
The document explains how to obtain Black–Scholes implied volatility for a European option when the underlying follows a stochastic volatility jump model. Since the model lacks a closed-form log-return density and a direct analytical option price, the answer…
The document outlines post-crisis swap valuation using overnight indexed swap rates for discounting, followed by adjustments for counterparty credit, funding, and other costs. It contrasts this approach with older conventions that used government yields or…