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 asks for the expectation and variance of a stock price modeled as an exponential function of drift and standard Brownian motion, evaluated at twice a given time. Its key lesson is that the exponential moment of a normal random variable must be…
The document derives the expected contribution of the jump component in Merton’s jump diffusion model. Conditional on the Poisson count, independent identically distributed jump sizes with mean k make the expected product of jump multipliers equal to (1+k)…
The document explains calibration in the context of using the SABR stochastic volatility model to price foreign exchange call options. Calibration means choosing model parameters so that model outputs match observed market quotes as closely as possible. The…
The document considers a contingent claim that pays the positive excess of one stock’s terminal price over a strike, provided that a second stock finishes inside a specified price interval. The question proposes pricing it from the two terminal marginal…
The document explains why gamma exposure is expressed using the square of a change in the underlying price. It uses dimensional analysis: option-price changes are measured in dollars, while gamma measures the second derivative of option value with respect to…
The note asks why a textbook calculation integrates a differential involving a stochastic process using what looks like an ordinary Riemann integral. The answer distinguishes integrals with a variable integrand from the integral of a differential alone. For…
The note asks why American option valuation is expressed as the best conditional expected discounted payoff over stopping times rather than over fixed exercise dates. A fixed date is a stopping time, so optimizing over all stopping times cannot produce a…
The document asks why the variance accumulated by a geometric Brownian motion over a time horizon is represented by the integral of instantaneous variance, and how time averaging enters the stated realized-variance expression. The answer builds the result…
The document asks how to price American options when the underlying follows risk-neutral dynamics with volatility that varies with spot. The response identifies this as a special case of a local volatility model, where volatility may depend on time and spot,…
The document asks whether an equity’s implied volatility skew can reveal the probability of an equity move conditional on a change in implied volatility. It defines skew as the difference between implied volatilities at two option deltas and explains why…
The discussion addresses why a Carr–Madan fast Fourier transform calculation can return complex-valued quantities when pricing caplets. The accepted answer says the complex form is expected in the method and that the option price is obtained from the real…
The response points to an intuitive route to understanding Gyöngy’s theorem through European call prices and generalized derivatives. It starts from an asset whose drift follows a risk-neutral specification and whose volatility may depend on the adapted…
The discussion compares QuantLib pricing approaches for geometric and arithmetic Asian options. It reports that the finite-difference Black–Scholes Asian engine supports discrete arithmetic averaging, not continuous geometric averaging. A discrete geometric…
The document addresses an apparent mismatch between simulated discount factors in a Hull–White one-factor Monte Carlo model and the market discount curve at the initial time. Its central point is that the model’s no-arbitrage setup is intended to reproduce…
The question concerns how to infer an equity’s local volatility when its drift is the short rate and that rate is stochastic. The author proposes adjusting market implied volatility on a forward for short-rate volatility, then asks whether cumulative…
The document explains GSR as the Gaussian Short Rate model and gives its risk-neutral short-rate dynamics in mean-reverting form, with time-varying reversion, level, and volatility. It places GSR within the broader family of affine short-rate models, whose…
The document investigates why changing a Euribor index tenor appears not to alter zero rates built from swap quotes in a QuantLib example. The explanation focuses on single-curve construction: the same curve is used both to estimate floating-leg forwards and…
The document considers how to value currency options when a central bank caps an exchange rate, using the Swiss National Bank’s franc ceiling against the euro as context. Its answer argues that standard models calibrated to historical market behavior may be…
The note examines a claim that pays one dollar when a stock first reaches a specified upper level, under a zero-interest-rate assumption. Although a zero-drift stock model makes the stock price a martingale, equating its initial expected value with its value…
The note explains how a finite-difference valuation of an American option should handle a discrete dividend at an ex-dividend date. The stock-price jump is represented by mapping the option value just after the dividend to the value just before it,…
The discussion explains how overnight index swaps and Libor–OIS basis swaps relate to interest-rate curve construction. An OIS floating coupon compounds daily overnight fixings over the accrual period and annualizes the resulting compounded return. A basis…
The document offers an intuitive explanation for why standard call and put options have positive gamma in the Black–Scholes setting. As the underlying price rises, a call’s delta rises toward one because the option behaves more like the underlying. A put’s…
The document explains why analyzing option Greeks is not expected to improve the payout of a standard box spread. A box combines calls and puts at two strikes to create a payoff similar to a loan or zero-coupon bond. Its return reflects the risk-free rate…
The post works through a derivation of call-option gamma by differentiating an option-price expectation and using the Dirac delta to evaluate the resulting integral. The key step is to recognize the density of the transformed variable as lognormal, apply the…