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 example considers a riskless asset and one risky asset whose dynamics are driven by two Brownian motions, while the information available to traders is generated by both. Although the asset equation can be rewritten using their scaled sum, that sum alone…
The document presents a closed-form expression for a zero-coupon bond price under an extended Hull-White short-rate model with time-varying drift, mean-reversion, and volatility parameters. The bond price is exponential-affine in the current short rate: one…
The document explains the distinction between Markit’s assumed and real recovery rates for credit default swaps. The assumed rate supports quote conversions between running spread and upfront payment, while the real rate represents the recovery used in…
The document explains how a pension fund can use interest rate swaps within liability-driven investing to reduce the gap between asset and liability duration. Its example assumes a long-dated liability valued using a government bond yield, with only part of…
The document asks how to approximate a down-and-out call price when the underlying follows the Heston stochastic-volatility model. It states the option payoff with a barrier condition, contrasts the problem with a Black–Scholes closed-form expression, and…
The document raises a distinction among probability measures used in quantitative finance. It describes risk-neutral probability as the framework used to price derivatives under no-arbitrage conditions, while identifying physical probability as the relevant…
The document discusses using importance sampling to estimate prices for far out-of-the-money barrier options under local volatility. It describes changing the distribution of simulated inputs from the original density to a shifted normal density, then…
The document explains how the time increment in a simulated Heston model relates to the number of steps per year: once the time unit is fixed, choosing one determines the other. Its main example is the Euler–Maruyama discretization of the stock and variance…
The document asks how replication should be understood for an American option in a pricing model under a risk-neutral measure. A self-financing portfolio with an adapted strategy has a martingale value process, while an American option’s payoff process…
The document describes a simulation question about the Heath–Jarrow–Morton interest-rate framework. The author simulates forward rates with constant volatility and antithetic normal draws, then compares averages of simulated rates with corresponding values…
The document explains how to represent dual-curve pricing for caps and caplets in QuantLib Python. The curve handle supplied to the pricing engine is used to discount cash flows, while the curve associated with the Ibor index supplies forward rates.…
The document outlines a validation approach for a finite-difference implementation of the Libor Market Model. It recommends comparing European swaption prices across at-the-money and in-the-money or out-of-the-money strikes with prices from a zero-shift SABR…
The discussion examines when at-the-money (ATM) implied volatility can approximate the risk-neutral expectation of average integrated variance. For very short maturities, it motivates the approximation by noting that instantaneous volatility near the start…
The discussion asks whether implied volatility (IV) produces better option prices than historical volatility (HV), noting that the common forward-looking argument for IV is not enough by itself to establish better pricing performance. It cites an experiment…
The Longstaff–Schwartz method prices American options by simulating possible asset paths and using regression to estimate continuation values. The discussion notes that simulation can accommodate options whose value depends on multiple factors, so the method…
The question describes pricing a fixed coupon bond in QuantLib with a flat yield curve built from semiannual zero rates. The user encounters a negative-time error because the curve’s first date comes after the evaluation date. The accepted answer explains…
The document examines a density that appears when expressing forward implied variance as an average of local variance. It derives the weighting from the risk-neutral distribution of the underlying and a factor involving spot squared and Black-Scholes gamma.…
The question examines the interest term in a derivation of the value change for a call option hedged by shorting delta shares. The reader reasons that buying the call requires borrowing its price, while short-sale proceeds from the shares are lent, implying…
The document clarifies that an option-adjusted spread is not a return that a hedge will realize on every interest-rate path. OAS is found by projecting an MBS’s cash flows across modeled rate paths, discounting them with the yield curve plus a spread, and…
The document compares two ways to obtain future distributions and price claims from an Itô process. A Fokker–Planck equation evolves the probability density through a deterministic partial differential equation, while Monte Carlo estimates outcomes from…
The document distinguishes several USD interest-rate curves that may appear in equity option pricing. A discount curve supplies discount factors for cash flows, but its name alone does not specify the benchmark or collateral convention used to construct it.…
The document explains two special cases of the SABR model by conditioning on a realized volatility path. With beta equal to one, the asset follows a multiplicative diffusion, so its logarithm has a normal distribution and the asset level is lognormally…
The document clarifies that normal and lognormal implied volatility refer to different option pricing conventions, not necessarily to the distribution assumed for the underlying in the SABR model. A normal implied volatility is the value used with Bachelier…
The document explains why a Double Heston model uses two stochastic variance processes. A second factor adds flexibility to the relationship between asset returns and volatility, allowing that correlation to change as the factors’ relative contributions…