Risk Relation Matrix Decomposition for Asset Pricing
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Solution Overview
Problem
Current financial modeling tools lack accuracy and practicality in valuing financial assets, particularly in determining absolute valuations, due to limitations in existing asset pricing theories and models.
Innovation Solution
A numerical modeling apparatus and method that utilizes Risk Relation Matrices to decompose risk vectors into eigenvectors and eigenvalues, deriving components of risk vectors in a basis of unit independent risks, and calculates the term structure of the price of risk to provide more detailed and accurate financial modeling information.
Engineering Contradictions & Design Principles
Engineering Contradiction Analysis
1Ease of manufacture
If Mean-Variance approach is used for asset pricing, then portfolio construction becomes mathematically tractable, but accuracy of valuation deteriorates due to neglecting higher order moments like skew and kurtosis
Solution Approach 1:
The patent changes the parameters used in asset pricing from only mean and variance to include higher order moments (skewness, kurtosis) and the full probability distribution function. This allows the model to capture more nuanced risk characteristics while maintaining mathematical tractability through the use of characteristic functions and Fourier transforms.
2Stability of the object's composition
If CAPM equilibrium theory is applied, then market clearing condition is satisfied, but practical applicability deteriorates due to unrealistic assumptions about market participant views
Solution Approach 1:
The patent segments the aggregate market equilibrium into individual asset-level pricing relationships. Instead of assuming all market participants share identical views, the model allows each asset to be priced based on its own characteristic function and risk parameters, while still satisfying overall market clearing conditions through the equilibrium constraint on the pricing kernel.
Solution Approach 2:
The patent introduces dynamic elements by allowing the pricing kernel and risk parameters to vary over time and across different states of the world. The model uses time-varying characteristic functions and state-dependent pricing, enabling adaptation to changing market conditions while maintaining theoretical rigor.
3Adaptability or versatility
If dynamic asset pricing models in general equilibrium setting are used, then derivative pricing becomes feasible, but effectiveness in determining valuations of underlying non-derivative assets deteriorates
Solution Approach 1:
The patent inverts the traditional approach by first establishing accurate pricing for underlying assets using their characteristic functions and direct risk parameters, then using these calibrated parameters to price derivatives. This ensures that underlying asset valuations remain the primary focus and are not distorted by derivative market assumptions.
Data Source
AI summary
A numerical modelling apparatus and method of performing numerical modelling are described. An input unit receives signals giving information relating to a set of assets. A processor unit is arranged to provide a set of Risk Relation Matrices Vτ for set of investment horizons indicated by τ. Each of the Risk Relation Matrices Vτ comprises a plurality of elements, wherein each of the elements represents a relationship of risk related to a respective pair of the assets and each element is given by a scalar product of two risk vectors, such that each of the assets has an associated risk vector according to the elements of the risk relation matrix. The processor unit is arranged to decompose each of the Risk Relation Matrices Vτ into eigenvectors and eigenvalues according to Vτ=Eτ·Λτ·E′τ, where, at each tenor τ, Eτ is a set of eigenvectors of the risk matrix Vτ in columns, Λτ is the corresponding diagonal eigenvalue matrix, and E′τ is the transpose of Eτ. Components of each of the risk vectors are derived at each tenor τ in the basis of unit independent risks by the corresponding row of the matrix product Eτ·Λτ1/2 relating to each of the assets. An output unit is arranged to output the components of each of the risk vectors as a risk vector data set.