Stochastic Discount Factor Research Articles

Demand in the Option Market and the Pricing Kernel

Demand in the Option Market and the Pricing Kernel

Market-Wide Illiquidity and the Distribution of Non-Parametric Stochastic Discount Factors

Market-Wide Illiquidity and the Distribution of Non-Parametric Stochastic Discount Factors

Projections of the Stochastic Discount Factor with Applications to Bubbles

Projections of the Stochastic Discount Factor with Applications to Bubbles

A generalized Esscher transform for option valuation with regime switching risk

A generalized Esscher transform is introduced for option valuation in a Markov regime-switching model. It is intended that the generalized Esscher transform might provide novel insights into pricing regime switching risk. A new pricing kernel and the related martingale condition are derived which might provide a convenient way to price both diffusion and regime switching risks. Numerical studies are provided to illustrate the proposed method.

Arbitrage pricing theory, the stochastic discount factor and estimation of risk premia from portfolios

The arbitrage pricing theory (APT) attributes differences in expected returns to exposure to systematic risk factors. Two aspects of the APT are considered. Firstly, the factors in the statistical asset pricing model are related to a theoretically consistent set of factors defined by their conditional covariation with the stochastic discount factor (SDF) used to price securities within inter-temporal asset pricing models. It is shown that risk premia arise from non-zero correlation of observed factors with SDF and the pricing errors arise from the correlation of the errors in the statistical model with SDF. Secondly, the estimates of factor risk premia using portfolios are compared to those obtained using individual securities. It is shown that in the presence of pricing errors consistent estimation of risk premia requires a large number of not fully diversified portfolios. Also, in general, it is not possible to rank estimators using individual securities and portfolios in terms of their small sample bias.

Realized bipower variation, jump components, and option valuation

AbstractWe develop a new option pricing model that captures the jump dynamics and allows for the different roles of positive and negative return variances. Based on the proposed model, we derive a closed‐form solution for option pricing under the condition of a nonmonotonic pricing kernel. Our results indicate that the new model has superior option pricing performance to its nested models, including the jump model of Christoffersen et al. (2015) and affine realized semivariance model of Feunou and Okou (2019). The models accommodating jumps, high‐frequency information, and accounting for variance risk premium perform well compared with traditional benchmark models.

A discrete-time hedging framework with multiple factors and fat tails: On what matters

This article presents a quadratic hedging framework for a general class of discrete-time affine multi-factor models and investigates the extent to which multi-component volatility factors, fat tails, and a non-monotonic pricing kernel can improve the hedging performance. A semi-explicit hedging formula is derived for our general framework which applies to a myriad of the option pricing models proposed in the discrete-time literature. We conduct an extensive empirical study of the impact of modelling features on the hedging effectiveness of S&P 500 options. Overall, we find that fat tails can be credited for half of the hedging improvement observed, while a second volatility factor and a non-monotonic pricing kernel each contribute to a quarter of this improvement. Moreover, our study indicates that the added value of these features for hedging is different than for pricing. A robustness analysis shows that a similar conclusion can be reached when considering the Dow Jones Industrial Average. Finally, the use of a hedging-based loss function in the estimation process is investigated in an additional robustness test, and this choice has a rather marginal impact on hedging performance.

Hedging and Evaluating Tail Risks via Two Novel Options Based on Type II Extreme Value Distribution

Tail risk is an important financial issue today, but directly hedging tail risks with an ad hoc option is still an unresolved problem since it is not easy to specify a suitable and asymmetric pricing kernel. By defining two ad hoc underlying “assets”, this paper designs two novel tail risk options (TROs) for hedging and evaluating short-term tail risks. Under the Fréchet distribution assumption for maximum losses, the closed-form TRO pricing formulas are obtained. Simulation examples demonstrate the accuracy of the pricing formulas. Furthermore, they show that, no matter whether at scale level (symmetric “normal” risk, with greater volatility) or shape level (asymmetric tail risk, with a smaller value in tail index), the greater the risk, the more expensive the TRO calls, and the cheaper the TRO puts. Using calibration, one can obtain the TRO-implied volatility and the TRO-implied tail index. The former is analogous to the Black-Scholes implied volatility, which can measure the overall symmetric market volatility. The latter measures the asymmetry in underlying losses, mirrors market sentiment, and provides financial crisis warnings. Regarding the newly proposed TRO and its implied tail index, economic implications can be offered to investors, portfolio managers, and policy-makers.

Analytic Correlation Risk for CDS

We consider a rates credit hybrid model with the rates and the credit intensity governed by Hull-White and Black-Karasinski short-rate models, respectively. We report on and make use of a highly accurate analytic representation of a pricing kernel for European-style options and/or protection payments in powers of the credit spread (not of its lognormal volatility). We show how this can be applied to calculate analytically the impact of rates credit correlation on the pricing of credit default swaps (CDS), extinguishing interest rate swaps, and survival-contingent capped LIBOR flows. Very simple expressions are obtained in all cases for present values (PVs). Highly favorable comparison is found between even the first-order approximations and Monte Carol simulation.

State price density estimation with an application to the recovery theorem

Abstract This article introduces a model to estimate the risk-neutral density of stock prices derived from option prices. To estimate a complete risk-neutral density, current estimation techniques use a single mathematical model to interpolate option prices on two dimensions: strike price and time-to-maturity. Instead, this model uses B-splines with at-the-money knots for the strike price interpolation and a mixed lognormal function that depends on the option expiration horizon for the time-to-maturity interpolation. The results of this “hybrid” methodology are significantly better than other risk-neutral density extrapolation methods when applied to the recovery theorem.

Foreign exchange option pricing under regime switching with asymmetrical jumps

In this study, we investigate the valuation of foreign exchange (FX) options by identifying a two-factor Markov-modulated stochastic volatility model with double exponential jumps to capture long- and short-term stochastic volatility and asymmetrical jumps in the underlying spot FX rate. Furthermore, the dynamics of domestic/foreign instantaneous forward interest rates are governed by a Markov-modulated Heath-Jarrow-Morton model. The dynamic measure change technique is employed to determine a pricing kernel for deriving the FX option pricing formula. Finally, numerical illustrations are provided and analyzed.

Ross Recovery and the Contemporaneous Pricing Kernel

This chapter investigates the contemporaneous pricing kernel obtained by applying the recovery theorem to options on the S&P500 index. Similarly to the literature on the pricing kernel puzzle, I find evidence that the recovered pricing kernels are U-shape, but only some of the time. The different shapes observed for the recovered kernels appear to be clustered in time, indicating that they are a genuine feature of risk preferences and not just an artifact created by noisy pricing kernel estimates. Although the recovered contemporaneous kernels cannot significantly predict future realized returns over the whole distribution, they are successful for the first 3 quartiles. A practitioner needing an instantaneous measure of downside risk could therefore obtain such information by applying the recovery theorem to options data.

Semi-equilibrium Pricing: The CAPM Formula

We present a new type of pricing method, the semi-equilibrium pricing method, which solves the optimal portfolios of investors first. This stage is consistent with the equilibrium pricing method, but the second stage of the semi-equilibrium pricing method uses only a part of the market clearing conditions rather than the entirety. Based on this, we show that the pricing formula by capital asset pricing model (CAPM) is not an equilibrium pricing formula but a semi-equilibrium pricing formula: When the CAPM formula is valid, the market may not be cleared. Only when the total market value of risky securities is given can the CAPM formula completely determine the price vector of the risky securities. Additionally, we point out that the stochastic discount factor (SDF) pricing formula is the result of investors' portfolio optimization, and the market clearing conditions are not involved at all. The SDF pricing formula is presented in the form of a linear pricing function and can be used as a foundation template to establish pricing functions.

Structuring Mortgages for Macroeconomic Stability

ABSTRACTWe study mortgage design features aimed at stabilizing the macroeconomy. We model overlapping generations of borrowers and an infinitely lived risk‐averse representative lender. Mortgages are priced using an equilibrium pricing kernel derived from the lender's endogenous consumption. We consider an adjustable‐rate mortgage with an option that during recessions allows borrowers to pay only interest on their loan and extend its maturity. The option stabilizes consumption growth over the business cycle, shifts defaults to expansions, and enhances welfare. The cyclical properties of the contract are attractive to a risk‐averse lender so that the mortgage can be provided at a relatively low cost.

Entangled risks in incomplete FX markets

We introduce the concept of risk entanglement in a preference-free setting to jointly explain the exchange rate volatility, cyclicality, and currency risk premia in the data. Risk entanglement specifies a subset of incomplete market models, in which nondiffusive or nonlog-normal shocks to exchange rates are not fully spanned by asset returns. When risks are entangled, there exist multiple pricing-consistent exchange rates, but none of them are equal to the ratio of the stochastic discount factors (SDFs) or their projections. Decoupling the exchange rate from the SDFs allows us to address key FX market patterns that are puzzling in international finance.

Efficient Estimation of Pricing Kernels and Market-Implied Densities

This paper studies the nonparametric identification and estimation of projected pricing kernels implicit in European option prices and underlying asset returns using conditional moment restrictions. The proposed series estimator avoids computing ratios of estimated risk-neutral and physical densities. Instead, we consider efficient estimation based on the conditional Euclidean empirical likelihood or continuously-updated GMM criterion, which takes into account the informativeness of option prices of varying strike prices beyond observed conditioning variables. In a second step, we convert the implied probabilities into predictive densities by matching the informative part of cross-sections of option prices. Empirically, pricing kernels tend to be U-shaped in the S&P 500 index return given high levels of the VIX, and call and ATM options are more informative about their payoff than put and OTM options.

Structuring Mortgages for Macroeconomic Stability

We study mortgage design features aimed at stabilizing the macroeconomy. We model overlapping generations of mortgage borrowers and an infinitely lived risk-averse representative mortgage lender. Mortgages are priced using an equilibrium pricing kernel derived from the lender's endogenous consumption. We consider an adjustable-rate mortgage (ARM) with an option that during recessions allows borrowers to pay only interest on their loan and extend its maturity. We find that this maturity extension option stabilizes consumption growth over the business cycle, shifts defaults to expansions, and is welfare enhancing. The cyclical properties of the maturity extension ARM are attractive to a risk-averse lender so the mortgage can be provided at a relatively low cost. Institutional subscribers to the NBER working paper series, and residents of developing countries may download this paper without additional charge at www.nber.org.

Direct Semi-Parametric Estimation of the State Price Density Implied in Option Prices

We present a model for direct semi-parametric estimation of the state price density (SPD) implied by quoted option prices. We treat the observed prices as expected values of possible pay-offs at maturity, weighted by the unknown probability density function. We model the logarithm of the latter as a smooth function, using P-splines, while matching the expected values of the potential pay-offs with the observed prices. This leads to a special case of the penalized composite link model. Our estimates do not rely on any parametric assumption on the underlying asset price dynamics and are consistent with no-arbitrage conditions. The model shows excellent performance in simulations and in applications to real data.

Dynamically complete markets under Brownian motion

This paper investigates how continuous-time trading renders complete a financial market in which the underlying risk process is a Brownian motion. A sufficient condition, that the instantaneous dispersion matrix of the relative dividends is non-degenerate, has been established in the literature for single-commodity, pure-exchange economies with many heterogenous agents where the securities’ dividends as well as the agents’ utilities and endowments include flows during the trading horizon which are analytic functions. In sharp contrast, the present analysis is based upon a different mathematical argument that assumes neither analyticity nor a particular underlying economic environment. The novelty of our approach lies in deriving closed-form expressions for the dispersion coefficients of the securities’ prices. To this end, we assume only that the pricing kernels and dividends satisfy standard growth and smoothness restrictions (mild enough to allow even for options). In this sense, our sufficiency conditions apply irrespectively of preferences, endowments or other structural elements (for instance, whether or not the budget constraints include only pure exchange).

Risk premia in electricity derivatives markets

This study examines the prices of options contingent on electricity futures traded on the European Energy Exchange, with the aim to recover the probability density functions and risk premia. After we extract the risk-neutral probability density functions from prices of such options, we transform the risk-neutral densities into real-world densities using both parametric and non-parametric statistical calibration methods and investigate the evolution of risk premia and pricing kernels. We find that both risk-neutral and real-world option-implied densities accurately forecast realized futures electricity prices. Positively skewed densities suggest that there is an inverse (or positive) leverage effect in the electricity market, meaning that a higher probability of large price increases in electricity has been incorporated in the traded option prices. In addition, we find that the state price densities are mostly increasing, implying that investors are more risk-averse to high electricity prices. Over a period of 15 years, our results provide evidence of negative market price of risk and risk premia in this new market.