Recovering Market Beliefs with Ordinal Stochastic Discount Factor Constraints

46 Pages Posted: 30 Nov 2023 Last revised: 4 Dec 2023

See all articles by Daniel Grosshans

Daniel Grosshans

University of Zurich - Department Finance

Date Written: November 16, 2023


Investor beliefs about the future dynamics of a financial market are a key determinant of market prices. Yet, these beliefs cannot easily be identified from prices as they are altered by stochastic discounting. Under conditions described by Ross (2015), beliefs and the discount factor can be jointly and uniquely disentangled from market prices of financial derivatives.
The approach has been widely criticized for its poor performance on empirical prediction tasks. In this paper, I show how to impose ordinal constraints on the stochastic discount factor to mitigate the method's fragility.
Using simulations, I show that the original recovery method fails at approximately identifying true probability distributions from prices, while the constrained version produces reliable predictions. Finally, using data on futures and its European options prices on the S&P 500 between 2007-2021, I compare different methods to predict the distribution of future returns, including known variants of Ross recovery. The results suggest that ordinal constraints on the implied stochastic discount factor make the method a competitive forecasting instrument.

Keywords: distributional forecasting, market beliefs, stochastic discount factor, shape constraints, Ross recovery

JEL Classification: C14, C58, G12, G13, G17, G41

Suggested Citation

Grosshans, Daniel, Recovering Market Beliefs with Ordinal Stochastic Discount Factor Constraints (November 16, 2023). Available at SSRN: or

Daniel Grosshans (Contact Author)

University of Zurich - Department Finance ( email )

Plattenstrasse 32
Zürich, 8032

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