Stochastic Discount Factor Research Articles

Machine learning private equity returns

In this paper, we use two machine learning techniques to learn the aggregated return time series of complete private capital fund segments. First, we propose Stochastic Discount Factor (SDF) model combination to determine the public factor exposure of private equity. Here, we describe our theoretical motivation to favor model combination over model selection. This entails that we apply simple coefficient averaging to obtain multivariate SDF models that mimic the factor exposure of all major private capital fund types. As a second step, we suggest componentwise L2 boosting to estimate the error-term time series associated with our factor models. The simple addition of the public factor model returns and the error terms then yields the total return time series. These return time series can be applied for proper integrated public and private risk management or benchmarking.

The role of health in consumption and portfolio decision-making: Insights from state-dependent models

In this paper, we study an optimal consumption and asset allocation problem accounting for the fact that individuals’ utility differs across various health states. Our study is based on the assumption that an individual’s health status evolves through a semi-Markov process, where the transition probabilities are contingent on both the present state and the duration spent in that state. The optimal form of stock investment and of consumption is determined analytically. The optimal consumption level is significantly shaped by the integration of health-related factors and can be represented as the inverse of the marginal utility function with respect to time and state price density. Additionally, we introduce a Lagrangian multiplier that can be derived by solving a fixed point problem. While our optimal solutions are applicable to a wide range of utility functions, we provide numerical illustrations specifically using the power utility function in a model encompassing three distinct health states. Transition probabilities are estimated from real data collected by the China Insurance Regulatory Commission (CIRC), using a high-order polynomial Perks formula. We find that the investor’s consumption is higher when health is good, but lower when care is needed.

Persistent and transient variance components in option pricing models with variance-dependent Kernel

This paper examines theoretically and empirically a variance-dependent pricing kernel in the continuous-time two-factor stochastic volatility (SV) model. We investigate the relevance of such a kernel in the joint modeling of index returns and option prices. We contrast the pricing performance of this model in capturing the term structure effects and smile/smirk patterns to discrete-time GARCH models with similar variance-dependent kernels. We find negative and significant risk premium for both volatility factors, implying that investors are willing to pay for insurance against increases in volatility risk, even if it has little persistence. In-sample, the component GARCH model exhibits a slightly better fit overall and across all maturity buckets than the two-factor SV model. However, the two-factor SV model reduces strike price bias, giving rise to the model’s ability in reconciling the physical and risk-neutral distribution. Out-of-sample, the two-factor SV model has better fit to data.

Portfolio optimization beyond utility maximization: the case of driftless markets

This paper presents a novel perspective on portfolio optimization by recognizing that prices can be expressed as a scaled likelihood ratio of state price densities. This insight leads to the immediate conclusion that the optimal portfolio has a simple representation in terms of the likelihood ratio between the agent-defined physical measure and the risk-neutral measure, eliminating the need for utility maximization. The agent only needs to specify her choice of the physical measure, and we demonstrate both frequentist and Bayesian approaches for this selection. Utility maximization can be seen as a specific method for choosing the physical measure. The resulting likelihood ratio is log-utility optimal with respect to all benchmarks, aligning our approach with finding the growth optimal portfolio described in the literature. Notably, the expected log return corresponds to the relative entropy between the physical and risk-neutral measures, establishing a fundamental link to information theory. As a proof of concept, we explore previously unexplored territory in portfolio optimization, specifically addressing perceived mean reversion in specific driftless markets, such as foreign exchange (FX) markets.

Assessing network risk with FRM: links with pricing kernel volatility and application to cryptocurrencies

The Financial Risk Meter (FRM) employs Quantile-LASSO regression to identify systemic financial risk and dependencies among tail events across financial assets. This paper establishes, both theoretically and empirically, a meaningful economic relationship between the FRM index, derived from the penalization parameter in quantile LASSO regression, and the volatility of assets' pricing kernels, the attainable maximal Sharpe ratio, and market volatility. Despite the rapid growth of the crypto market and its increasing integration with traditional financial markets, there remains a dearth of risk measures in this space. FRM @ Crypto exhibits robust predictive capabilities in anticipating future market risk, potentially filling a critical void in this market.

News and Asset Pricing: A High-Frequency Anatomy of the SDF

Abstract Utilizing real-time newswire data, together with a robustly estimated intraday stochastic discount factor (SDF), we identify and quantify the economic news that is priced. News related to monetary policy and finance on average accounts for most of the variation in the SDF, followed by news about international affairs and macroeconomic data. We also document nontrivial temporal variation in the relative importance of the news, along with marked differences in the estimated news risk premiums in the “factor zoo.” To further highlight the economic mechanisms at work, we associate the different news effects with interest rate, growth, and risk premium shocks. (JEL C58, G12, G14)

Testing the boundaries of applicability of standard Stochastic Discount Factor models

We provide a joint non-parametric test to gather insights on the boundaries of applicability of Stochastic Discount Factor (SDF) models. We find that a non-trivial class of models cannot price the U.S. stock market equally weighted portfolio, implying non-monotonic SDFs, especially over the last 50/60 years in (recessionary) periods characterized by higher market volatility. Stocks responsible for this rejection mostly belong to the smallest NYSE market cap decile, are characterized by high idiosyncratic risk, and typically cannot be priced via SDF models where the aggregate level of risk aversion is bigger then 9 or 10. Excluding these stocks increases the ability to explain the cross-section of returns without impairing the ability to span the mean–variance frontier.

Pricing derivatives on foreign assets using Markov-modulated cojump-diffusion dynamics

In this study, cross-currency derivatives pricing under stochastic interest rates is analyzed when Markov-modulated cojump-diffusion dynamics with both idiosyncratic and simultaneous jumps drive the underlying foreign equity price and foreign exchange rate processes. More specifically, the noise dynamics of these two assets are captured by the correlated Wiener processes, and the two types of jump events are described as two independent compound Poisson processes with log-normal jump amplitudes. Cojumping addresses the correlation between these two asset price jumps. In an incomplete market setting, the dynamic Esscher transform approach is applied to determine a pricing kernel. According to the resulting dynamics, explicit analytical formulas for evaluating values of European foreign equity call options are provided. Numerical experiments are also described.

The Term Structure of Covered Interest Rate Parity Violations

ABSTRACTWe quantify the impact of risk‐based and nonrisk‐based intermediary constraints (IC) on the term structure of covered interest rate parity (CIP) violations. Using a stochastic discount factor (SDF) inferred from interest rate swaps, we value currency derivatives. The wedge between model‐implied and observed derivative prices reflects the impact of nonrisk‐based IC because our SDF incorporates risk‐based IC. There is no wedge at short horizons, while the wedge accounts for 40% of long‐term CIP violations. Consistent with IC theory, the wedge correlates with the shadow cost of intermediary capital, and the SDF‐implied interest rate is a weighted average of collateralized and uncollateralized interest rates.

Factor correlation and the cross section of asset returns: A correlation-robust machine learning approach

This paper investigates high-dimensional factor models for cross-sectional asset returns, with a specific focus on robust estimation in the presence of (highly) correlated factors. Factor correlations can significantly compromise the robustness and credibility of commonly employed analytical methods. To address this, we utilize the stochastic discount factor (SDF) and integrate it with a recently developed Machine Learning methodology (Figueiredo and Nowak, 2016). This novel approach allows us to select factors while accounting for factor correlations and to disentangle correlated factors without imposing rigid assumptions. Our empirical findings consistently highlight the paramount role of the ‘market’ factor in driving cross-sectional asset returns. In contrast, other benchmarks, including the LASSO, the Elastic-Net, and the Fama–MacBeth regression, are adversely impacted by factor correlations, rendering the ‘market’ factor redundant. Additionally, our findings underscore the importance of ‘profitability’, ‘momentum’, and ‘liquidity’-related factors in driving cross-sectional asset returns.

Empirical Asset Pricing with Many Test Assets

Abstract We formulate the problem of estimating risk prices in a stochastic discount factor (SDF) model as an instrumental variables regression. The IV estimator allows efficient estimation for models with non-traded factors and many test assets. Optimal instruments are constructed using a regularized sparse first stage regression. In a simulation study, the IV estimator is close to the infeasible GMM estimator in a setting with many assets. In an empirical application, the tracking portfolio for consumption growth appears strongly correlated with consumption news. It implies that consumption is a priced factor for the cross-section of excess equity returns. A similar regularized regression, projecting the SDF on test assets, leads to an estimate of the Hansen–Jagannathan distance, and identifies portfolios that maximally violate the pricing implications of the model.

Informational Updates and the Derivative Pricing Kernel

It is common in financial markets for market makers to offer prices on derivative instruments even though they are uncertain about the underlying asset’s value. This paper studies the mathematical problem that arises as a result. Derivatives are priced in the risk-neutral framework, so as the market maker acquires more information about the underlying asset, the change of measure for transition to the risk-neutral framework (the pricing kernel) evolves. This evolution takes a precise form when the market maker is Bayesian. It is shown that Bayesian updates can be characterized as additional informational drift in the underlying asset’s stochastic process. With Bayesian updates, the change of measure needed for pricing derivatives is two-fold: the first change is from the prior probability measure to the posterior probability measure, and the second change is from the posterior probability measure to the risk-neutral measure. The relation between the regular pricing kernel and the pricing kernel under this two-fold change of measure is characterized.

Functional Stochastic Discount Factors in the Stochastic Volatility Model

Functional Stochastic Discount Factors in the Stochastic Volatility Model

Non-Monotonic Pricing Kernel and an Extended Class of Mixture of Distributions For Option Pricing

Non-Monotonic Pricing Kernel and an Extended Class of Mixture of Distributions For Option Pricing

Forecasting Chinese stock market volatility with option-implied risk aversion: Evidence from extended realized EGARCH-MIDAS approach

This paper investigates the role of option-implied risk aversion (IRA) in forecasting the Chinese stock market volatility. We derive the IRA from the SSE 50ETF option prices using behavioral pricing kernel theory. We extend the realized EGARCH-MIDAS (REGARCH-MIDAS) model to incorporate the IRA. Furthermore, we propose the REGARCH-MIDAS-IRA-ES model accounting for the extreme IRA information to model and forecast the Chinese stock market volatility. Empirical results based on the extended REGARCH-MIDAS models show that the IRA has a significantly positive impact on the long-term volatility of Chinese stock market, and the extremely positive IRA has a greater impact on the long-term volatility compared to the extremely negative and normal IRA. Moreover, the IRA possesses predictive value for the Chinese stock market volatility. In particular, we observe that the extreme IRA can provide more valuable information to forecast the Chinese stock market volatility. Our proposed REGARCH-MIDAS-IRA-ES model outperforms a variety of competing models in terms of out-of-sample volatility forecast. We confirm that the superior forecasting performance of the model is robust to different out-of-sample evaluation approach, different out-of-sample forecast windows, different MIDAS lags and different volatility states. Finally, a volatility-timing strategy demonstrates that incorporating the IRA, and in particular the extreme IRA, leads to economic gains for a mean–variance utility investor.

Equilibrium pricing of European crude oil options with stochastic behaviour and jump risks

In this paper, we investigate the pricing of European crude oil options under nonlinear dynamics with stochastic behaviour and jump risks, incorporating the features of arising convenience yield of crude oil and potential extreme fluctuation in the dynamics of crude oil prices. We present a closed-form solution to European crude oil option prices under an incomplete market setting, after deriving the pricing kernel with the equilibrium method and determining the risk neutral dynamics of crude oil prices. We extend our model to a mean-reverting stochastic volatility case, which also admits an analytical formula for the equilibrium price of European crude-oil options. Finally, our model is shown to be consistent with a number of interesting facts documented in the recent literature.

A multi-agent incomplete equilibrium model and its applications to reinsurance pricing and life-cycle investment

This paper develops an incomplete equilibrium model with multi-agents' different risk attitudes and heterogeneous income/payout profiles. Particularly, we apply its concrete and computationally tractable model to reinsurance derivatives pricing and life-cycle investment, which are important for insurance and asset management companies in practice. In numerical experiments, we explicitly obtain endogenously determined expected returns of the risky asset in equilibrium, agents' specific reinsurance prices with their stochastic discount factors (SDF) and optimal life-cycle trading strategies. Moreover, we investigate how each agent's degree of risk aversion and income/payout profile, and correlations between an insurance or economic factor and the risky asset price affect reinsurance claims pricing and optimal portfolios in life-cycle investment.

Volatility-dependent probability weighting and the dynamics of the pricing kernel puzzle

In order to estimate volatility-dependent probability weighting functions, we obtain risk neutral and physical densities from the Pan (J Financ Econ 63(1):3–50, 2002. https://doi.org/10.1016/S0304-405X(01)00088-5) stochastic volatility and jumps model. Across volatility levels, we find pronounced inverse S-shapes, i.e. small probabilities are overweighted, and probability weighting almost monotonically increases in volatility, indicating higher skewness preferences and crash aversion in volatile market environments. Moreover, by estimating probabilistic risk attitudes, equivalent to the share of risk aversion related to probability weighting, we shed further light on the pricing kernel puzzle. While pricing kernels estimated from the Pan (J Financ Econ 63(1):3–50, 2002. https://doi.org/10.1016/S0304-405X(01)00088-5) model display the typical U-shape as documented in the literature, pricing kernels—net of probability weighting—are strictly monotonically decreasing and thus in line with economic theory. Equivalently, we find risk aversion to be positive across wealth levels. Our results are robust to alternative maturities, wealth percentiles, alternative functional forms, a nonparametric empirical setting and variations of the Pan (J Financ Econ 63(1):3–50, 2002. https://doi.org/10.1016/S0304-405X(01)00088-5) coefficient estimates.

Equilibrium multi-agent model with heterogeneous views on fundamental risks

This paper investigates an equilibrium-based multi-agent optimal consumption and portfolio problem incorporating uncertainties on fundamental risks, where multiple agents have heterogeneous (conservative, neutral, aggressive) views on the risks represented by Brownian motions. Each agent maximizes its expected utility on consumption under its subjective probability measure. Specifically, we formulate the individual optimization problem as a sup–sup–inf problem, which is an optimal consumption and portfolio problem with a choice of a probability measure. Moreover, we provide an expression of the state-price density process in a market equilibrium, which derives the representations of the interest rate and the market price of risk. To the best of our knowledge, this is the first attempt to investigate the multi-agent model with heterogeneous views on the risks by considering a market equilibrium and solving sup–sup–inf problems on the choice of a probability measure. We emphasize that the setting, where each agent has heterogeneous views on different risks, incorporates a special case where each agent has only conservative or neutral views on risks with different degrees of conservativeness. Also, the setting includes the case where the agents have aggressive views on risks, commonly observed as bullish sentiments in the financial markets in the monetary easing after the global financial crisis and particularly in the COVID-19 pandemic. Finally, we present numerical examples of the interest rate model, which show how heterogeneous views of the multiple agents on the risks affect the shape of the yield curve.

Reference dependence and endogenous anchors

AbstractIn a complete market, we find optimal portfolios for an investor whose satisfaction stems from both a payoff's intrinsic utility and its comparison with an endogenous reference as modeled by Kőszegi and Rabin. In the regular regime, arising when reference dependence is low, the marginal utility of the optimal payoff is proportional to a twist of the pricing kernel. High reference dependence leads to the anchors regime, whereby investors reduce disappointment by concentrating significant probability in one or few fixed outcomes, or “anchors.” Multiple equilibria arise because anchors may not be unique. If stocks follow geometric Brownian motion, the model implies that investors with longer horizons choose larger stocks holdings.