Option pricing and CVA calculations using the Monte Carlo-Tree (MC-Tree) method

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2022-07-18
Trinh, Yen Thuan
Publisher
University College Cork
Abstract
The thesis introduces a new method, the MC-Tree method, for pricing certain financial derivatives, especially options with high accuracy and efficiency. Our solution is to combine Monte Carlo (MC) method and Tree method by doing a mixing distribution on the tree, and the output is the compound distribution on the tree. The compound distribution in the tree output (after a logarithmic transformation of the asset prices) is not the ideal Gaussian distribution but has entropy values close to the maximum possible Gaussian entropy. We can get closer using entropy maximization. We introduce two correction techniques: distribution correction and bias correction to improve the accuracy and completeness of the model. The thesis presents an algorithm and numerical results for calculations of CVA on an American put option using the MC-Tree method. The MC-Tree method with the distribution correction technique significantly improves accuracy, resulting in practically exact solutions, compared to analytical solutions, at the tree depth $N=50$ or $100$ and MC-drawings $M=10^5$. The bias-correction technique makes the resulting tree model complete in the sense of financial mathematics and obtains the risk-neutral probability. Besides, we have obtained new formulae for the calculations of the entropy and the Kullback-Leibler divergence for rational densities and approximate entropy of finite Gaussian mixture.
Keywords
Entropy , Kullback-Leibler (KL) divergence , Optimization , MC-Tree method , Option pricing , Credit valuation adjustment , Pricing models , Finite Gaussian mixture
Citation
Trinh, Y. T. 2022. Option pricing and CVA calculations using the Monte Carlo-Tree (MC-Tree) method. PhD Thesis, University College Cork.