Stochastic Discount Factor Research Articles

Treasury option returns and models with unspanned risks

We document the phenomenon that average excess returns of out-of-the-money puts and calls on bond futures are negative, both unconditionally and conditionally on economic states. To explain these findings, we develop economically motivated restrictions in the context of a theory in which the pricing kernel is a general diffusion process with spanned and unspanned components. Our reconciliation is a framework that introduces market incompleteness and priced unspanned volatility risks, allowing for time-varying downside and upside futures risk premiums. The estimated model shows consistency with data on bond yields, yield volatilities, bond futures return volatilities, option prices, and option risk premiums.

Analysis of Option Butterfly Portfolio Models Based on Nonparametric Estimation Deep Learning Method

The option butterfly portfolio is the commonly option arbitrage strategy. In reality, because the distribution of the option state price density (SPD) function is not normal and unknown, so the nonparametric deep learning methods to estimate option butterfly portfolio returns are proposed. This paper constructs the single-index nonparametric option pricing model which contains multiple influencing factors and presents the nonparametric estimation form for option butterfly portfolio returns. The empirical analysis shows that the SPD function estimated by using single-index nonparametric option model can effectively calculate the option butterfly portfolio returns with the minimum option strike price interval and provide an effective reference tool for risk-averse investors with limited risk preferences.

The Pricing Kernel in Options

The empirical option valuation literature specifies the pricing kernel through the price of risk, or defines it implicitly as the ratio of risk-neutral and physical probabilities. Instead, we extend the economically appealing Rubinstein-Brennan kernels to a dynamic framework that allows path- and volatility-dependence. Because of low statistical power, kernels with different economic properties can produce similar overall option fit, even when they imply cross-sectional pricing anomalies and implausible risk premiums. Imposing parsimonious economic restrictions such as monotonicity and path-independence (recovery theory) achieves good option fit and reasonable estimates of equity and variance risk premiums, while resolving pricing kernel anomalies.

OPTIMAL INVESTMENT UNDER PARTIAL INFORMATION AND ROBUST VAR-TYPE CONSTRAINT

This paper extends the utility maximization literature by combining partial information and (robust) regulatory constraints. Partial information is characterized by the fact that the stock price itself is observable by the optimizing financial institution, but the outcome of the market price of the risk [Formula: see text] is unknown to the institution. The regulator develops either a congruent or distinct perception of the market price of risk in comparison to the financial institution when imposing the Value-at-Risk (VaR) constraint. We also discuss a robust VaR constraint in which the regulator uses a worst-case measure. The solution to our optimization problem takes the same form as in the full information case: optimal wealth can be expressed as a decreasing function of state price density. The optimal wealth is equal to the minimum regulatory financing requirement in the intermediate economic states. The key distinction lies in the fact that the price density in the final state depends on the overall evolution of the estimated market price of risk, denoted as [Formula: see text] or that the upper boundary of the intermediate region exhibits stochastic behavior.

Asset pricing with costly and delayed firm entry

AbstractSurvey evidence tells us that stock prices reflect the risks investors associate with long-run technological change. However, there is a shortage of models that can rationalize long-run risks. Unlike the previous literature assuming a fixed number of products, our model allows for new product varieties that appear in the form of new firms which face entry costs and delay in the entry process. The fixed variety model has a significant limitation in translating macroeconomic volatility into asset return volatility. Our model with growing varieties induces endogenous low-frequency fluctuations in productivity driving large, persistent variations in consumption growth and asset prices. It also changes the valuation of assets through the increase in the volatility of the pricing kernel (with a positive long-run component) and leads to higher excess returns. Our model is motivated by a simple recursively identified VAR model containing quarterly US data 1992Q3-2018Q4 with the following list of variables: total factor productivity, output, a measure of firm entry, and the excess return on stocks.

Bowley Insurance with Expected Utility Maximization of the Policyholders

This article studies the Bowley solution for a sequential game within the expected utility framework. We assume that the policyholders are expected utility maximizers and there exists a representative policyholder who faces a fixed loss with given probability and no loss otherwise. This policyholder selects the optimal indemnity function in response to the pricing kernel set by the insurer. Knowing the policyholder’s choice of indemnity function, the insurer adjusts the pricing kernel to maximize its expected net profit. This pricing kernel is of our central interest in this article, and in our setting the pricing kernel can be evaluated via the safety loading factor in an expected value premium principle. For a wide class of utility functions, we show that the optimal safety loading factor increases with respect to both the policyholder’s risk aversion level and the probability of zero loss. We also show that the insurance contract corresponding to the Bowley solution is Pareto dominated in the sense that both parties’ interests can be further improved, which shows the inefficiency of the Bowley solution. Some numerical examples are presented to illustrate the main results, and it is shown that both the policyholder and insurer can strictly benefit from the Bowley solution.

Currency Risk Premiums Redux

Abstract We study a large currency cross-section using asset pricing methods that account for omitted-variable and measurement-error biases. First, we show that the pricing kernel includes at least three latent factors that resemble (but are not identical to) a strong U.S. “dollar” factor and two weak high Sharpe ratio “carry” and “momentum” slope factors. Evidence for an additional “value” factor is weaker. Second, using this pricing kernel, we find that only a small fraction of the over 100 nontradable candidate factors considered have a statistically significant risk premium, mostly relating to volatility, uncertainty, and liquidity conditions, rather than macro variables. Authors have furnished an Internet Appendix, which is available on the Oxford University Press Web site next to the link to the final published paper online.

Option Valuation with Nonmonotonic Pricing Kernel and Embedded Volatility Component Premiums

This article develops a nonmonotonic pricing kernel with long-run and short-run variance risk premiums for option valuation, with a proposed pricing kernel retaining a U-shaped pattern that significantly improves the fitting ability for index options pricing and implied volatility. The estimation results show that the long-run volatility component is critical in generating the negative risk premium. In the in-sample and out-of-sample tests, the model with the new pricing kernel has more accurate predictions, especially the year around the financial crisis, wherein there is a decrease of an average of 35% root mean square error relative to the benchmark. Considering the bull and bear market states, our model improves implied volatility root mean square error by 23% on average.

Pricing Kernels and Risk Premia implied in Bitcoin Options

Bitcoin Pricing Kernels (PKs) are estimated using a novel data set from Deribit, the leading Bitcoin options exchange. The PKs, as the ratio between risk-neutral and physical density, dynamically reflect the change in investor preferences. Thus, the PKs improve the understanding of investor expectations and risk premiums in a new asset class. Bootstrap-based confidence bands are estimated in order to validate the results. Investors are heterogeneous in their risk profiles and preferences with respect to volatility and investment horizon. The empirical PKs turn out to be U-shaped for short-dated instruments and W-shaped for long-dated instruments. We find that investors are willing to pay a substantial risk premium to insure themselves against short-term price movements. The risk premium is smaller for longer-dated instruments and their traders are risk averse. The shape of the empirical PKs reveals the existence of a time-varying risk premium. The similarity between the shape of empirical PKs for Bitcoin and other markets that represent aggregate wealth shows that Bitcoin is becoming an established asset class.

Extrapolative asset pricing

This paper studies implications of return extrapolation in a consumption-based asset pricing model. We show that return extrapolation has strong implications for the pricing kernel. The time variation in the agent's return expectations is mainly reflected in the short rate and little in return volatility and equity premium. Return extrapolation causes return volatility and equity premium to be lower than the rational counterparts. In addition to the risk premium, the equity premium can include a sentiment premium that is due to dividend expectation bias rather than return extrapolation bias. Thus, time-varying dividend (rather than return) expectation bias helps produce a volatile equity premium. These results show that return extrapolation exacerbates asset pricing puzzles and fundamental extrapolation helps resolve puzzles.

Market-wide illiquidity and the distribution of non-parametric stochastic discount factors

Employing out-of-sample non-parametric estimation techniques, we show that market-wide liquidity risk matters for asset pricing independently of the specific functional form of the stochastic discount factor (SDF) and, therefore, of the asset pricing model specification. Market-wide illiquidity significantly affects the distribution of the SDF. Specifically, it boosts up the volatility of the SDF, causing minor effects on higher moments of its distribution. This finding is robust to the use of different sets of test assets in the estimation of the SDF, including equity and corporate bond portfolios, and the use of a high-dimensional data estimation procedure.

Credit risk pricing in a consumption‐based equilibrium framework with incomplete accounting information

AbstractWe present a consumption‐based equilibrium framework for credit risk pricing based on the Epstein–Zin (EZ) preferences where the default time is modeled as the first hitting time of a default boundary and bond investors have imperfect/partial information about the firm value. The imperfect information is generated by the underlying observed state variables and a noisy observation process of the firm value. In addition, the consumption, the volatility, and the firm value process are modeled to follow affine diffusion processes. Using the EZ equilibrium solution as the pricing kernel, we provide an equivalent pricing measure to compute the prices of financial derivatives as discounted values of the future payoffs given the incomplete information. The price of a zero‐coupon bond is represented in terms of the solutions of a stochastic partial differential equation (SPDE) and a deterministic PDE; the self‐contained proofs are provided for both this representation and the well‐posedness of the involved SPDE. Furthermore, this SPDE is numerically solved, which yields some insights into the relationship between the structure of the yield spreads and the model parameters.

Covariance dependent kernels, a Q-affine GARCH for multi-asset option pricing

This paper introduces a class of multivariate GARCH models that extends the existing literature by explicitly modeling correlation dependent pricing kernels. A large subclass admits closed-form recursive solutions for the moment generating function under the risk-neutral measure, which permits efficient pricing of multi-asset options. We perform a full calibration to three bivariate series of index returns and their corresponding volatility indexes in a joint maximum likelihood estimation. The results empirically confirm the presence of correlation dependance in addition to the well known variance dependance in the pricing kernel. The model improves both the overall likelihood and the VIX-implied likelihoods, with a better fitting of marginal distributions, e.g., 15% less error on one-asset option prices. The new degree of freedom is also shown to significantly impact the shape of marginal and joint pricing kernels, and leads to up to 53% differences for out-of-the-money two-asset correlation option prices.

The global factor structure of exchange rates

We propose a model-free methodology to estimate international stochastic discount factors (SDFs) that jointly price cross-sections of international stocks, bonds, and currencies in markets with frictions. We theoretically establish a SDF decomposition into one global factor and a currency basket. We show that our global factor prices a large cross-section of international asset returns, not just in- but also out-of-sample, across different currency denominations. Moreover, the pricing ability of the global factor is largely independent of the market structure or the size and type of market friction.

When Do Cross-Sectional Asset Pricing Factors Span the Stochastic Discount Factor?

When Do Cross-Sectional Asset Pricing Factors Span the Stochastic Discount Factor?

When Do Cross-Sectional Asset Pricing Factors Span the Stochastic Discount Factor?

When Do Cross-Sectional Asset Pricing Factors Span the Stochastic Discount Factor?

Recovering Market Beliefs with Ordinal Stochastic Discount Factor Constraints

Recovering Market Beliefs with Ordinal Stochastic Discount Factor Constraints

Optimal investment of DC pension plan under a joint VaR-ES constraint

<abstract><p>In this paper, we investigated an optimal investment problem of a defined contribution (DC) pension plan under a joint Value-at-Risk (VaR) and an expected shortfall (ES) constraint. By using a martingale method, we transformed a dynamic optimization problem to a static pointwise optimization problem and derived the closed-form representations of the optimal wealth and portfolio processes in terms of the state price density. Numerical results showed that in comparison to only an ES constraint or a VaR constraint, the joint VaR-ES constraint can not only improve risk management for the bad economic states but also lower the volatility of the optimal terminal wealth.</p></abstract>

Characterizing the Conditional Pricing Kernel: A New Approach

Characterizing the Conditional Pricing Kernel: A New Approach

International Yield Curves and Currency Puzzles

ABSTRACTThe currency depreciation rate is often computed as the ratio of foreign to domestic pricing kernels. Using bond prices alone to estimate these kernels leads to currency puzzles: the inability of models to match violations of uncovered interest parity and the volatility of exchange rates. This happens because of the FX bond disconnect, the inability of bonds to span exchange rates. Incorporating innovations to the pricing kernel that affect exchange rates but not bonds helps resolve the puzzles. This approach also allows one to relate news about cross‐country differences between international yields to news about currency risk premiums.