KDE Distributionally Robust Portfolio Optimization And Option Pricing

Portfolio optimization and option pricing are the fundamental problems of financial math-ematics,which have extensive and important applications in the fields of asset management and risk control.The classical mean-variance optimization model and Black-Scholes-Merton option pricing model depend on the special probability distribution,which are sensitive to the mod-el parameters.In recent years,the robust optimization and distributionally robust optimization(DRO)model,which are immune to the uncertainty of model parameters and distribution,are gotten more and more attentions,and a lot of research achievements are obtained.The DRO model can be divided into the moment based one and the distance based one according to the difference of the definition of the distributional uncertainty set(DUS).The distance based DRO model can be reformulated into the tractable convex optimization problem for the several spe-cial cases.For the more general case,the several KDE-DRO models are proposed in this paper,where the DUS is the ?-divergence "ball" in the finite dimensional probability distribution space that is spanned by the weighted kernel density estimation(KDE).The corresponding tractable reformulations and convergence are derived.Moreover,the distributionally robust option pricing problem is investigated.(1)KDE and ?-divergence based distributionally robust mean-CVaR(Conditional Value at Risk)portfolio optimization model is proposed.In order to overcome the so-called "curse of dimensionality",we consider the one-dimensional probability distribution of the port-folio return,rather than the joint probability distribution of the assets return vector.The two issues of "the distribution dependent on the decision variables" and "the distance based DUS for the continuous distribution" are efectively addressed by using the finite dimensional KDE based probability distribution.Under the mild conditions of the ker-nel function and ?-divergence function,the tractable reformulation of the corresponding DRO model is derived by Lagrange duality theory.Moreover,the convergence of optimal value and solution set of the KDE mean-CVaR distributionally robust portfolio optimiza-tion problem to those of the corresponding stochastic optimization model with the real distribution is proved.(2)KDE-?-divergence based distributionally robust mean-HMCR(Higher Moment Coherent Risk)portfolio optimization model is proposed.Comparing with the linearity of the KDE based distributionally robust mean-CVaR portfolio model with respect to the weight vec-tor,the model presents the nonlinearity of the power function.Under the mild conditions of the kernel function and ?-divergence function,the corresponding DRO model can be re=formulated into the tractable convex optimization problem with the geometric mean cone constraints by Fenchel duality theory.Moreover,the similar convergence is proved.(3)KDE-?-divergence based distributionally robust mean-EVaR(Entropic Value at Risk)portfolio optimization model is proposed.Comparing with the nonlinearity(the power function)of the KDE based distributionally robust mean-HMCR portfolio model with re-spect to the weight vector,the model presents the nonlinearity of the logarithmic function.Under the mild conditions of the kernel function and ?-divergence function,the corre-sponding DRO model can be reformulated into the tractable convex optimization problem with the exponential cone constraints by Fenchel duality theory.Moreover,the similar convergence is also obtained.(4)The distributionally robust option pricing model is first proposed in which the DUS is the Wasserstein ball centered at the empirical distribution of the cumulative return of the underlying asset The tractable reformulation of the corresponding DRO model is derived by Lagrange duality theory.Specially,for 2-norm in Wasserstein distance,the SOCP reformulation can be obtained.The lifting-penalty method is proposed for the extension of the SOCP.KDE distributionally robust portfolio optimization model and Wasserstein distributionally robust option pricing model are tested by the data sets from Kenneth R.French's Web site,the historical data of stocks from Hang Seng Index of Hong Kong Stock Exchange and the historical data of the underlying asset of the SSE 50 ETF options.Primary empirical test results show that the proposed model is meaningful.