## Estimation of the risk-neutral density function from option prices

2018-01-01
Zhou, Sen
Lisheng Hou
##### Organizational Units
Organizational Unit
Mathematics
##### Abstract

The risk-neutral density function (RND) is a fundamental concept in mathematical finance and is heavily used in the pricing of financial derivatives. The estimation of a well-behaved RND is an ill-posed problem and remains to be a mathematical and computational challenge due to the limitations of data and complicated constraints. Both parametric and non-parametric methods for estimating the RND from option prices have been developed and used in the literature and industry. In this dissertation we propose and study more effective non-parametric methods.

We develop the methods under the framework of linear programming and quadratic programming in combination with Support Vector Regression (SVR).

Under the framework of linear programming, we propose two methods with different penalty schemes. i) The first one named LPSVR uses a general kernel, the log-logistic function, with the standard ε-insensitive loss function to formulate the estimation process into a semi-infinite linear programming optimization problem. We prove the solution of this optimization problem is global by the Cutting Plane Method (CPM). Monte-Carlo simulations are conducted to evaluate the performance of LPSVR. Compared to the benchmark method SML, LPSVR improves both the accuracy and stability. ii) The second one named εi-LPSVR modifies LPSVR with the εi-insensitive loss function and also formulates the estimation process into a semi-infinite linear programming optimization problem. We may similarly prove the globalness of the solution by CPM. Monte- Carlo simulations are also conducted. Compared to LPSVR, εi-LPSVR maintains the stability level and improves the accuracy level by the modified penalty scheme. Overall εi-LPSVR outperforms LPSVR.

Under the framework of quadratic programming, we also propose two methods with different penalty schemes. i) The first one named QPSVR uses the RBF kernel with the ε-insensitive square loss function to formulate the estimation process into a semi-infinite quadratic programming optimization problem. We prove the solution of this optimization problem is global by CPM. Moreover, we prove uniqueness of the solution by the approximation theory. Simulations show that QPSVR maintains the accuracy level as LPSVR. Compared to LPSVR and εi-LPSVR, QPSVR improves the stability level by the uniqueness of the solution. Overall QPSVR outperforms LPSVR and εi-LPSVR. ii) The second one named εi-QPSVR modifies QPSVR with the εi-insensitive square loss function and also formulates the estimation process into a semi-infinite quadratic programming optimization problem. We may similarly prove the globalness and uniqueness of the solution by this scheme. Simulations show that εi-QPSVR improves both accuracy and stability over LPSVR. Compared to εi-LPSVR, εi-QPSVR maintains the accuracy level and improves the stability level by the uniqueness of the solution. Compared to QPSVR, εi-QPSVR maintains the stability level and improves the accuracy level by the modified penalty scheme. Overall εi-QPSVR outperforms LPSVR, εi-LPSVR and QPSVR.