Research On Pricing Financial Derivatives Based On Non-monotonic Pricing Kernel

The pricing kernel captures investor preference for payoffs over different states of the world and forms the backbone of asset pricing theory.Accurate understanding of the pricing kernel's monotonicity and shape is crucial to evaluate the financial assets.Also known as stochastic discount factor or marginal rate of substitution,pricing kernel provides the connection between asset prices and fundamental economic principles and plays an important role as a link between economics and finance.In standard models of financial economics,the pricing kernel is proportional to the marginal utility of a representative investor.And under the assumption of perfect market,risk aversion and correct belief,the pricing kernel should be decreasing.Namely,the pricing kernel or the marginal utility of a representative investor decreases with the increase of total wealth.Since the early 2000 s,many empirical research point out that the relationship between pricing kernel and total wealth not only is monotonically decreasing but also has increasing parts.This phenomenon that the pricing kernel is non-monotonic,is known as the pricing kernel puzzle.Therefore,how to explain and describe the non-monotonic feature of pricing kernel has theoretical and practical significance for us to understand and model the dynamics of financial asset price.In addition,it is meaningful research to answer whether a non-monotonic pricing kernel exists in China's financial market,according to our own market characteristics.And if it exists,how to explain and characterize this phenomenon in equilibrium and how to apply it to the pricing of financial derivatives,is also a very valuable research work.This paper is organized as follows.First,we extract the pricing kernel's information under the risk-neutral measure and physical measure through using China's market data and build formal test to examine whether it exists the stylized facts of non-monotonic pricing kernel.Based on the empirical evidence,we then explain the non-monotonic pricing kernel puzzle through the lens of variance risk under an equilibrium framework.Finally,we build a class of new models that incorporate the non-monotonic pricing kernel and stylized facts of underlying returns' dynamics to price derivatives and compare these model's performance.The main contents and findings of each part are as follows:(1)Non-monotonic pricing kernel: evidence from the Chinese financial market.This chapter first derive the risk-neutral probabilities of asset returns under the generalized Black-Scholes model.Then considering the jump risk of asset price in the extreme economic condition,we use the Jump-GARCH process to model the asset return's dynamics and extract its physical density using kernel density estimation method.Finally,we analysis the shape of pricing kernel and construct a statistical test to test the monotonicity of pricing kernel for the first time in Chinese market.By extracting the monthly density distribution of the50 ETF option contracts over the period from 2015 through 2017 under the risk neutral measure,we find that the risk-neutral probability distributions of returns have obviously time-varying characteristics.Under the physical measure,we model the returns' dynamics using a Jump-GARCH model with a time-varying jump intensity and extract the physical density distribution with kernel density estimation approach.The empirical results show that the physical density has time-varying characteristics,too.Compared with the risk-neutral probability distribution,the physical probability distribution has obvious peak characteristics.Based on the asset pricing theory,we also present the shape of the pricing kernel,and find that the pricing kernel presents obviously non-monotonic characteristics,and its shape is approximately U shaped.Furthermore,the formal monotonicity test is constructed to test the monotonicity of the pricing kernel.The piece-wise test results suggest that the pricing kernel is non-monotonic again.(2)Explaining the non-monotonic pricing kernel puzzle through the lens of variance risk.This chapter first model the 50 ETF returns and i VX under the physical measure and risk-neutral measure,respectively.The standard maximum likelihood function estimation method is used to estimate the conditional variance under physical measure and risk neutral measure,and then the variance risk premium is calculated.Based on the empirical study of the 50 ETF return from October 9,2006 to December 29,2017 and the i VX index from February 9,2015 to December 29,2017,we find that significant negative variance risk premium is existed in China.Next,we incorporate the variance risk under the Epstein-Zin-Weil recursive utility function and assume that the logarithmic growth rate of consumption is an affine form of logarithmic price change and variance change.By solving and analyzing the model,we find that the equilibrium model can explain the non-monotonicity of the pricing kernel in the market and fully describe the features of the U-shaped pricing kernel.As the risk aversion level of the agent increases,the logarithmic pricing kernel moves up as a whole and the Convexity of curve of the pricing kernel also increases.Fixing the intertemporal elasticity of substitution coefficient,we find that the agents will prefer to solve the uncertainty earlier with the increase of risk aversion level.Fixing the risk aversion coefficient,we find that when the intertemporal elasticity of substitution coefficient is less than 1,the pricing kernel presents a "U" shape.The pricing kernel moves down and its convexity of curve becomes smaller,with the decrease of the intertemporal elasticity of substitution coefficient.Conversely when the intertemporal elasticity of substitution coefficient is greater than 1,the logarithmic pricing kernel curve displays an inverted "U" shape which indicates that there is a positive variance risk premium in the market and is not inconsistent with empirical evidence(variance risk premium is negative).The equilibrium results can not only help investors and regulators to understand the phenomenon of non-monotonic pricing kernel in the market,but also provide a theoretical basis for the pricing of financial derivatives to incorporate the risk of variance.(3)Forecasting i VX Index with conditional skewness and non-monotonic pricing kernel.As the first volatility index to measure the sentiment of investors in the Chinese market,accurate modeling and prediction of the i VX index has important regulatory and investment implications.Considering the non-normality of asset returns,we model the underlying asset dynamics of i VX index based on the inverse Gasussian(IG)GARCH process.In addition,we adopt a non-monotonic pricing kernel that captures variance risk premium to risk-neutralize the return's dynamics and model the i VX Index under the risk-neutral measure.The empirical application is based on the Shanghai Stock Exchange(SSE)50ETF returns over the period from February 23,2005 through February 14,2018 and SSE Volatility Index from February 9,2015 to February 14,2018.The empirical results suggest that the IG GARCH has a better performance in returns and i VX Index fitting and the out-of-sample forecasting results also favor the IG GARCH model over the benchmark model.Equivalently important,the empirical results also show that the Chinese market has significantly negative variance risk premium and non-monotonic pricing kernel.(4)VIX Futures pricing with non-monotonic pricing kernel and affine Jump-GARCH dynamics.After 2008 financial crisis,Volatility Index(VIX)futures are among the most actively traded contracts at the Chicago Board Options Exchange,in response to the growing need for protection against volatility risk.This paper develops a new closed-form VIX futures pricing model,in which the underlying asset returns follow the time-varying jump intensity GARCH model,and which is risk-neutralized by a non-monotonic pricing kernel.To test whether time-varying Jump-GARCH models yield improvements over the Heston-Nandi(HN)model in returns fitting and futures pricing,and whether the non-monotonic pricing kernel can significantly affect pricing performance,we jointly fit the S&P500 Index returns for the period from January 5,1995 to December 30,2016 and VIX futures from January 3,2007 to December 21,2016.The empirical results suggest that the GARCH model with a jump component can better describe non-normality characteristics of the underlying asset,and has a larger log-likelihood value.The Jump-GARCH model significantly outperform the GARCH model in terms of VIX futures valuation.In addition,the joint estimation results show that the significant negative variance risk premium and the non-monotonic pricing kernel are significantly existed in both the Jump-GARCH model and the HN-GARCH model.(5)Option valuation with non-monotonic pricing kernel and double exponential jumps.This paper develops a new class of closed-form option pricing models that allow for jump and variance risk premium.Based on the stylized facts:the underlying assert price could have upward(downward)jumps when good(bad)news arrives.The models assume the jump processes follow symmetric or asymmetric double exponential distribution with time-varying jump intensity.In addition,we risk-neutralize the asset return dynamics using a non-monotonic pricing kernel that includes the jump risk,based on the non-arbitrage pricing principle.Based on the empirical results of the first option in China,50 ETF option,we find that under the physical measure,the time-varying jump intensity and variance of the underlying asset returns are significant,and the model with asymmetric double exponential jumps can fully describes the characteristics of the underlying asset returns.Under the risk neutral measure,we find that the market risk prices of normal random shocks and jump random shocks can be significantly estimated by directly using the option's information.In addition,in-sample and out-of-sample pricing performance shows that the model with double exponential jump and non-monotonic pricing kernel is a relatively good choice for pricing options.Compared with the existing literature,this paper mainly has the following three innovations.(1)This paper tests the monotonicity of pricing kernel in Chinese market by extracting the information implied in the option prices of50 ETF and the returns on underlying assets.In the literature known to author,there is no study on the monotonicity of pricing kernel in China by constructing formal test statistics and using 50 ETF option data.Extracting pricing kernel from option data is helpful to understand the characteristics of pricing kernel in China.The findings of this paper provide empirical evidence from the Chinese market for the non-monotonic pricing kernel.(2)The non-monotonic pricing kernel are inconsistent with the assumption of the classical asset pricing theory.Considering the stylized facts of non-monotonic pricing kernel,we introduce variance risk into the affine equilibrium framework to explain the phenomenon of non-monotonic pricing kernel.Under the affine equilibrium,macro growth rates(such as consumption growth rates)are assumed to follow affine processes.The equilibrium interpretation of pricing kernel anomaly is helpful to deepen our theoretical understanding of pricing kernel and provides an economic theoretical basis for considering non-monotonic pricing kernel in asset pricing model.(3)After considering the different dynamic characteristics of underlying assets,we use non-monotonic pricing kernel and construct a series of new financial derivatives pricing models.First,as the underlying asset returns have negative skewness,we adopt a GARCH process that has an inverse Gaussian innovation.Based on the non-monotonic pricing kernel,we risk-neutralize the IG-GARCH process to model and forecast the i VX index.Then,considering the jump risk,we develop a closed-form VIX futures pricing formula based on a non-monotonic pricing kernel and a Jump-GARCH process with time-varying jump intensity.Finally,as good and bad news affect the asset price differently,we extend the Jump-GARCH model by assuming the jump size follows the asymmetric double exponential distribution and compute the option prices analytically with a non-monotonic pricing kernel.These results not only enrich the research on the statistical characteristics of the capital market,but also provide investors and regulators with more accurate pricing methods for financial derivatives,which is of great significance to the construction of investment strategies and the regulation of the financial market.