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 solution to a geometric Brownian motion whose drift and volatility vary with time. Applying Itô's lemma to the logarithm of the asset price produces a drift term adjusted by half the instantaneous variance, alongside a stochastic…
Bootstrapping a yield curve from market instruments produces discount factors or zero rates at the instruments’ maturities. These observed maturities are curve nodes; they do not by themselves specify values at every date. An interpolation method fills in…
The document asks how to implement time-shifted power-law kernels used in a path-dependent volatility model. The model forms volatility from a baseline plus weighted past returns and weighted squared returns. Separate kernels assign the lag-dependent weights…
The discussion explains how a local-volatility model relates an implied-volatility surface to an instantaneous volatility function of spot and time. Its central clarification is that local volatility is calibrated from the current implied surface; if that…
The document explains the one-period binomial model’s no-arbitrage condition by comparing the stock’s up and down factors with the risk-free growth factor. If the stock’s return in both states is no better than the risk-free asset, shorting stock to buy the…
The document frames a one-month portfolio problem involving cash, a stock, and a six-month European put. Returns are evaluated under the physical measure, but the option’s future market value depends on risk-neutral pricing. The author explains why valuing…
The document asks whether hedging strategies should be compared using the absolute values of their errors or the squared errors. One response favors squared-error summability, noting that it is a weaker requirement than absolute summability and that squaring…
The document explains how state prices and a stochastic discount factor express the same pricing rule in different forms. In a discrete set of possible future states, a state price is the current price of a claim paying one unit only in a particular state.…
The document asks how to price an Arrow security that pays one unit if a specified event occurs at a future date. It frames the question through the stochastic discount factor: the current value is expressed as the expected product of that factor and the…
The document concerns Greeks for the Bachelier model, which assumes normally distributed price changes. The author says delta, vega, and gamma are straightforward to derive, then focuses on deriving rho and theta when the interest rate is nonzero. They…
The document introduces a diffusion model that can interpolate between arithmetic Brownian motion and geometric Brownian motion. In the proposed stochastic differential equation, the diffusion term is proportional to the state raised to a power γ. Setting γ…
The document explains how the Stochastic Volatility Inspired (SVI) parameterization can represent implied total variance across option strikes. Its formula describes a smile using five parameters: a level, slope scale, skew, horizontal shift, and curvature…
The document asks how to select a yield curve for discounting cash flows when pricing bonds, swaps, swaptions, equity options, FX forwards, and FX options. It distinguishes government, swap, overnight-indexed, and reference-rate curves, and asks whether the…
The document addresses how option vega relates to gamma and explains a formula attributed to Nassim Taleb’s Dynamic Hedging. Under Black–Scholes, gamma is expressed using the asset price, volatility, time to expiration, and the normal density at d1; vega…
The document derives the conditional variance of a price process that changes through Poisson-timed jumps with independent, lognormally distributed multipliers. It first expresses the price at the horizon as its current value multiplied by all jump factors…
The document examines an interpolation rule for a Wiener process whose volatility is piecewise constant between known endpoint values. The proposed rule scales the endpoint increment by the square root of a ratio involving accumulated variance terms. The…
The discussion considers how to obtain a nine-month LIBOR fixing when that tenor is unavailable. It emphasizes that interpolation depends on the intended use. In a loan whose contract required a nine-month rate, the parties agreed to create a synthetic…
The document works through pricing a zero-coupon bond under a Vasicek-style short-rate model with zero mean reversion level. It starts from the rate’s stochastic-integral solution, inserts that solution into the discounted payoff, and changes the order of…
The document explains why QuantLib-Swig exposes separate names such as PiecewiseLogCubicDiscount, PiecewiseLinearZero, and PiecewiseCubicZero while QuantLib 1.17 C++ does not. In C++, these curve variants are generated from the PiecewiseYieldCurve class…
The document asks why an FX 25-delta butterfly is commonly expressed using the average of 25-delta call and put volatilities adjusted by the at-the-money straddle volatility. It also asks what the 25-delta label means in this construction, placing the focus…
The document explores how to shock selected points on an on-the-run Treasury yield curve, propagate those shocks to other maturities, and reprice an off-the-run bond. The proposed method applies principal component analysis to historical yield changes, then…
The document asks how to build a function that returns implied FX volatility for an arbitrary strike and maturity from quoted market points. The available inputs are 10-delta and 25-delta put and call volatilities, along with zero-delta straddle or…
The document explains how to construct a forward swap rate from discount factors or forward Libor rates, then connect it to swaption valuation. It derives the swap rate as the ratio of the swap’s floating-leg value to its fixed-leg annuity. Given Libor…
The document outlines pricing approaches for derivatives with general payoff functions under European, American, and perpetual exercise rules. For European claims, it gives the risk-neutral discounted expected payoff and explains that a specified underlying…