Yannick Dillschneider

Asset Pricing | Financial Econometrics

Option Prices and Option-Implied Probabilities in Incomplete Markets

Abstract: This paper develops a methodology to quantify the uncertainty about option prices that persists in (statically) incomplete option markets due to finite strike and maturity grids as well as bid-ask price quoting. Uncertainty measures are provided for European option prices and their slopes (i.e., tail probabilities), each derived from model-free bounds that are consistent with a finite sample of observed bid and ask option prices and elementary no-arbitrage constraints in a multi-period setting. We obtain novel explicit expressions for these bounds, which are simple and efficient to compute. Moreover, we suggest a decomposition of the bounds and uncertainty measures into contributions of strike, maturity, and price quote incompleteness. In an empirical analysis of S&P 500 index options, we quantify the empirical uncertainty about option prices. We document sizable uncertainty levels for option prices and tail probabilities. Our findings challenge the common belief that such "option-implied" (tail) information can be measured at sufficiently high precision, which has implications for many applications that use this information.
Presented at: University of Amsterdam 2025, SoFiE 2025 (Paris)

What can you really tell from option prices? (with Oleg Bondarenko, Paul Schneider, and Fabio Trojani)

Abstract: Option-implied risk-neutral moments are widely used throughout the asset pricing literature. In this paper, we characterize the information content of such optionimplied moments, accounting for the market incompleteness inherent in empirical option markets. Our analysis relies on novel model-free bounds for a large class of risk-neutral moments built from a given option cross-section, which allow us to quantify the associated valuation uncertainty. We show that these bounds are infinite for almost all popular risk-neutral moments used in the literature, including the VIX, such that these moments cannot be reliably determined in practice. The lack of robustness in the specification of risk-neutral moments has effects that are profound both quantitatively and qualitatively, with ramifications for many well-known theories and empirical findings based upon them. As an alternative, we propose new robust moments and investigate their implications for the literature.
Presented at: QFFE 2024 (Marseille), SoFiE 2024 (Rio), ESEM 2024 (Rotterdam), CDI Conference 2024 (Montreal), FinEML 2024 (Lugano), EWMES 2024 (Mallorca), Erasmus University Rotterdam 2025, Tilburg University 2025

Functional Estimation of Option Pricing Models (with Evgenii Vladimirov) [SSRN]

Abstract: In this paper, we develop a novel estimation procedure for parametric option pricing models specified under the risk-neutral measure. We set up our estimation strategy to minimize the distance in a functional sense between the option-implied and model-implied logarithm of conditional characteristic functions. Within a broad class of option pricing models, for which the characteristic function is marginally affine, the model's latent state vector can be concentrated out in closed form. As a result, our estimation procedure is computationally fast and easy to implement, while at the same time exploiting all distributional information contained in an option panel about the risk-neutral dynamics of the underlying asset price. We establish the asymptotic properties of the parameter and state estimators and investigate the finite-sample performance of our method in Monte Carlo simulations. In an empirical application, we illustrate the usefulness of our estimation procedure based on by far the largest option panel, both in the time-series and cross-sectional dimensions, considered in the related literature.
Presented at: FinEML 2024 (Lugano) (poster), EWMES 2024 (Mallorca), Financial Risks International Forum 2025 (Paris), University of Amsterdam 2025, Erasmus University Rotterdam 2025, SoFiE 2025 (Paris), IAAE 2025 (Turin), FMA Conference on Derivatives and Volatility 2025 (Chicago), EC2 Conference 2025 (Lugano)(poster)

Transform-Based Moments of Derivatives Prices in Stochastic Volatility Models (with Raimond Maurer) [SSRN]

Abstract: Derivatives, especially equity and volatility options, contain valuable and oftentimes essential information on state dynamics in stochastic volatility models. To capture this information, this paper develops a novel and unified methodology to determine moments involving derivatives prices within a setting that jointly accounts for stock returns as well as prices of equity and volatility options. Using transform-based techniques, we derive analytically tractable expressions for exact moments and devise a computationally attractive approximation procedure. For the latter, we provide numerical results that support its effectiveness. Our methodology has a broad spectrum of possible applications, among others in estimation and diagnostic procedures. To illustrate these applications, we empirically investigate moments of equity and volatility options and devise a GMM estimation procedure that incorporates moments obtained from our methodology.
Presented at: EEA 2020, Virtual Derivatives Workshop 2020, SoFiE Summer School 2021, RCEA 2022, ESAM 2022, EWMES 2022 (Berlin), AFFI 2023 (Bordeaux), IAAE 2023 (Oslo), CFE-CMStatistics 2023 (Berlin)

Standard Transform Analysis for Asset Pricing and Parameter Estimation [SSRN]

Abstract: This paper provides tractable expressions for broad classes of standard transforms for the case of affine jump diffusions. Extending and unifying existing results, these include expressions for exponential, polynomial-log-linear, and polynomial transforms of a state vector in both conditional and unconditional form. We cover the usual single-period setting, but also extensions to multi-period settings, non-stationarity, and time-inhomogeneous state dynamics. Using these standard transforms, we discuss applications in asset pricing and parameter estimation.

Pricing Dividend Derivatives in Reduced-Form and Structural Models [SSRN]

Abstract: In this paper, we develop a general pricing framework for dividend derivatives, which covers different specifications for state-space models, pricing kernels, and dividends. For certain special cases within an affine jump-diffusion setting, we derive transform-based pricing formulas for general dividend derivatives. Using our framework, we then obtain specific pricing formulas for dividend futures and options within both reduced-form and structural models.

A Machine Learning Framework for Asset Pricing [SSRN]

Abstract: This paper introduces a unified machine learning framework for solving general asset pricing problems. Building on representations of asset prices in discrete-time and continuous-time models, we develop a solution strategy using neural networks and further machine learning techniques to determine asset prices. We exemplify and numerically study our methodology for consumption-based asset pricing models.

Generalized Transform Analysis for Asset Pricing and Parameter Estimation [SSRN]

Abstract: In this paper, we extend the existing generalized transform analysis in a way that allows us to propose a novel GMM approach for estimating asset pricing models. Our methodology is capable of incorporating a broad class of assets within both reduced-form and structural models, while avoiding the drawbacks of competing approaches. To formulate moment conditions, we derive expressions for moments involving transform-based asset prices. Our theory yields analytically tractable expressions for exact moments and an additionally computationally efficient strategy to obtain approximate moments, whose convergence we establish under standard conditions. Exact and approximate moments induce exact and approximate GMM estimators, respectively, for which we discuss asymptotic properties. Finally, we exemplify our methodology with a worked-out estimation problem involving equity options.
Presented at: University of Amsterdam, Esade Business School, AFFI 2021, SoFiE 2021 Pre-Conference, IAAE 2021, AMES 2021, CMES 2021, ESEM 2021

Functional Ross Recovery: Theoretical Results and Empirical Tests (with Raimond Maurer) [SSRN] [JEDC]

Journal of Economic Dynamics and Control, 2019, 108, 103750:1-32
Abstract: Recently, Ross (2015) showed that the real-world probability distribution of a discrete Markovian state variable can be recovered from observed option prices. The so-called recovery theorem follows from Perron-Frobenius matrix theory when the pricing kernel is transition independent. In this paper, we generalize the recovery theorem to continuous state spaces using Perron-Frobenius operator theory. Building on our theoretical results, we devise a nonparametric estimation approach to empirically recover the pricing kernel and real-world probability density in closed form. Using S&P 500 index options, we analyze recovered pricing kernels empirically and find evidence that Ross recovery is misspecified.
Presented at: AFFI 2018 (Paris), CDI Conference 2019 (Montreal), DGF 2019 (Essen)

Ross Recovery in Pricing Semigroups [SSRN]

Abstract: Ross recovery specifies conditions under which it is possible to recover the real-world probability measure as well as the pricing kernel from a family of pricing operators. In this paper, we study Ross recovery theoretically within a semigroup framework using Perron-Frobenius operator theory. Thereby, we derive sufficient conditions guaranteeing existence and uniqueness of a transition independent pricing kernel consistent with the pricing semigroup. Our resultant theory provides a unified perspective on various existing recovery approaches. Building on our theoretical results, we also briefly address the estimation of a pricing semigroup and the corresponding (Markov) transition semigroup from a sample of option prices.
Presented at: AFFI 2019 (Quebec City)

Dynamic Portfolio Choice

Optimal Annuitization Under Stochastic Interest Rates (with Raimond Maurer and Peter Schober) [SSRN] [ASB]

ASTIN Bulletin, 2024, 54(3), 626-651
Abstract: The decision about when and how much to annuitize is an important element of the retirement planning of most individuals. Optimal annuitization strategies depend on the individual's exposure to annuity risk, meaning the possibility of meeting unfavorable personal and market conditions at the time of the annuitization decision. This paper studies optimal annuitization strategies within a life-cycle consumption and portfolio choice model, focusing on stochastic interest rates as an important source of annuity risk. Closing a gap in the existing literature, our numerical results across different model variants reveal several typical structural effects of interest rate risk on the annuitization decision, which may however vary depending on preference specifications and alternative investment opportunities: When allowing for gradual annuitization, annuity risk is temporally diversified by spreading annuity purchases over the whole pre-retirement period, with annuity market participation starting earlier in the life cycle and becoming more extensive with increasing interest rate risk. Ruling out this temporal diversification possibility, as embedded in many institutional settings, incurs significant welfare losses, which are increasing with higher interest rate risk, together with larger overall demand for annuitization.

Generalized Euler Equation Errors for Discrete Time Dynamic Portfolio Choice Models (with Raimond Maurer and Peter Schober) [SSRN]

Abstract: The solution to dynamic portfolio choice models can be formulated in terms of a value function by the Bellman principle of optimality, which reduces the multi-period optimal policy choice problem to a sequence of one-period maximization problems. For two adjacent periods, economists compute the error of numerically obtained policies by measuring how much these policies violate the intertemporal first order conditions for the optimal policy choice problem --- so-called Euler equation errors. In this paper, we derive generalized Euler equation errors for the solution to a broad class of discrete time dynamic portfolio choice models where the policies are continuous choice variables. Our key precondition is that the gradient of the value function with respect to the state variables can be approximated. This is, for example, the case when a global polynomial basis or B-spline basis functions are used for the approximation of the value function within the discrete time dynamic programming approach. We apply our theoretical results to exemplary, representative dynamic portfolio choice models.

Page updated

Google Sites

Report abuse