Figure 2.1: MPE - Black-Scholes Environment
This figure plots the MPE for a range ofγ’s in the Black-Scholes environment for European call options with different combinations of moneyness and maturity. The vertical dashed line corresponds toγ =−0.8. For each γ, the MPE of a given option is calculated as the average across 5000 simulations of the percentage pricing error of the generalized entropic estimator using 200 returns from the physical distribution of the model. S/X represents the option moneyness, andT is the time to maturity in years.
Figure 2.2: MPE Surface - Black-Scholes Environment
This figure plots the MPE surface for a range ofγ’s and moneynesses (S/X) in the Black-Scholes environment for different maturities (T, in years). For eachγ, the MPE of a given option is calculated as the average across 5000 simulations of the percentage pricing error of the generalized entropic estimator using 200 returns from the physical distribution of the model.
Figure 2.3: MAPE Surface - Black-Scholes Environment
This figure plots the MAPE surface for a range of γ’s and moneynesses (S/X) in the Black-Scholes environment for different maturities (T, in years). For each γ, the MAPE of a given option is calculated as the average across 5000 simulations of the absolute percentage pricing error of the generalized entropic estimator using 200 returns from the physical distribution of the model.
Figure 2.4: MPE - SVCJ Environment
This figure plots the MPE for a range ofγ’s and four different risk-neutral distributions in the SVCJ environment for European call options with different combinations of moneyness and maturity. Each risk-neutral distribution (RN1-RN4) gives a different theoretical option price as benchmark. For each γ and risk-neutral distribution, the MPE of a given option is calculated as the average across 5000 simulations of the percentage pricing error of the generalized entropic estimator using 200 returns from the physical distribution of the model. S/X represents the option moneyness, andT is the time to maturity in years.
Figure 2.5: MAPE Surface - SVCJ Environment
This figure plots the MAPE surface for a range of γ’s and moneynesses (S/X) in the SVCJ environment for different maturities (T, in years), considering the risk-neutral parametrization Q2. For each γ, the MAPE of a given option is calculated as the av-erage across 5000 simulations of the percentage absolute pricing error of the generalized entropic estimator using 200 returns from the physical distribution of the model.
Figure 2.6: MPE - Generalized Entropic Estimators and Black-Scholes with Historical Volatility
This figure plots the MPE for each moneyness and maturity category of the generalized entropic estimators and the Black-Scholes (BS) model estimated with historical volatility.
The sample ranges from January 4, 1996 to June 28, 2019.
Figure 2.7: MAPE - Generalized Entropic Estimators and Black-Scholes with Historical Volatility
This figure plots the MAPE for each moneyness and maturity category of the generalized entropic estimators and the Black-Scholes model estimated with historical volatility. The sample ranges from January 4, 1996 to June 28, 2019.
Figure 2.8: MPE - Generalized Entropic Estimators and Black-Scholes with ATM Implied Volatility
This figure plots the MPE for each moneyness and maturity category of the general-ized entropic estimators and the Black-Scholes (BS) model estimated with ATM implied volatility. The sample ranges from January 4, 1996 to June 28, 2019.
Figure 2.9: MAPE - Generalized Entropic Estimators and Black-Scholes with ATM Implied Volatility
This figure plots the MAPE for each moneyness and maturity category of the generalized entropic estimators and the Black-Scholes model estimated with ATM implied volatility.
The sample ranges from January 4, 1996 to June 28, 2019.
Figure 2.10: MPE - Generalized Entropic Estimators and GARCH
This figure plots the MPE for each moneyness and maturity category of the generalized entropic estimators and the GARCH option model. The sample ranges from January 4, 1996 to June 28, 2019.
Figure 2.11: MAPE - Generalized Entropic Estimators and GARCH
This figure plots the MAPE for each moneyness and maturity category of the generalized entropic estimators and the GARCH option model. The sample ranges from January 4, 1996 to June 28, 2019.
Figure 2.12: MPE - Generalized Entropic Estimators and GJR GARCH
This figure plots the MPE for each moneyness and maturity category of the generalized entropic estimators and the GJR GARCH option model. The sample ranges from January 4, 1996 to June 28, 2019.
Figure 2.13: MAPE - Generalized Entropic Estimators and GJR GARCH
This figure plots the MAPE for each moneyness and maturity category of the generalized entropic estimators and the GJR GARCH option model. The sample ranges from January 4, 1996 to June 28, 2019.
Chapter 3
Endogenous Wealth and the Pricing Kernel Puzzle
We provide a unifying framework for the literature that calculates empirical pricing ker-nels (EPKs) as the ratio of the option-implied state-price density and the historical return distribution of the market index. Such EPKs are often a U-shaped function of market returns, which is puzzling under the assumption of a complete market where the index proxies for wealth. We propose to estimate the pricing kernel by minimizing a convex dis-crepancy function subject to correctly pricing the underlying asset and observed options.
The projection of our estimated pricing kernel onto market returns is the EPK identified by the literature. However, by duality, we are also able to obtain the pricing kernel as a function of the endogenous wealth of the marginal investor pricing the index and index options. This implies that the U-shaped EPK can be rationalized as the projection of a pricing kernel that is monotonically decreasing in the endogenous wealth. That is, there is no puzzle when we recognize that options also constitute investment opportunities. The U-shaped pattern of the EPK arises because the marginal investor in the index and index options sells protection against large movements in the index.
This chapter is co-authored with Caio Almeida.