Many tasks in Quantitative Finance—pricing early-exercise options, computing hedge ratios, calibrating volatility models—reduce to evaluating a conditional expectation 𝔼[Y| X=x] from simulated data. The market standard, least squares Monte Carlo, estimates this by regression and is accurate in the bulk of the distribution but fragile in the tails and for the hedge sensitivities. We introduce GMMR, a data-driven and model-free (‘model free’ in the sense that the proposed algorithm is not specific to a given model or class of models) approach that instead approximates the joint law of (X,Y) by a fitted Gaussian Mixture Model (GMM), an established tool from statistical learning, see [1]. The only numerical step is the mixture fit; the conditional expectation, its variance and the associated hedge then follow from closed-form Gaussian formulas, so the estimates are fast and smooth even in the tails and for multi-modal distributions. Choosing the underlying as the hedge proxy recovers the time-discrete minimal-variance delta analytically, and further state variables can be added as control variates. We apply GMMR to the pricing and hedging of (multi-dimensional) exotic Bermudan options and to the calibration and pricing of Stochastic Local Volatility models, using data generated from challenging models—rough Bergomi, Bates and Cheyette—in one and several dimensions.
In dieser Notiz untersuchen wir fünf verschiedene Definitionen einer stetigen Zufallsvariable, welche so in der (Schul‑)Literatur und im Internet zu finden sind. Wir zeigen, dass diese nicht notwendigerweise von einander abgeleitet werden können und daher nicht die selben Objekte bezeichnen. Einige Argumente benötigen Vorkenntnisse der Maßtheorie und werden teilweise in vereinfachter Form präsentiert. Auf diese Weise soll ein Beitrag zur Stochastikausbildung (von zukünftigen Lehrer:innen) geleistet werden.
