Foreign exchange option pricing method and system

Abstract

The invention relates to the technical field of foreign exchange options, and discloses a foreign exchange option pricing method and system, and the method comprises the steps: 1, determining the dimension d of an option pricing model; step 2, when the dimension d is greater than or equal to 2, calculating the number of nodes by adopting sparse grid compression; step 3, identifying a key area based on a preset Gamma sensitivity threshold; step 4, when the dimension is dgt; 3, automatically switching to Monte Carlo simulation; 5, executing sparse grid superposition, dimension reduction decomposition and Monte Carlo path generation tasks in parallel through a graphics processor; according to the method, the high-dimensional calculation complexity is reduced through sparse grid compression, the key area is dynamically encrypted in combination with Gamma sensitivity, and Monte Carlo simulation is automatically switched in ultrahigh dimension, so that the problems of high calculation resource consumption and rigidity in precision adjustment of a traditional method are effectively solved, and the method is suitable for large-scale popularization and application. The method has the advantages of improving the calculation efficiency, dynamically optimizing the key region precision and balancing the calculation resources and precision.

Description

Technical Field

[0001] This invention relates to the field of foreign exchange options technology, specifically to a pricing method and system for foreign exchange options. Background Technology

[0002] Foreign exchange options, an important derivative instrument in the financial market, first appeared on the Philadelphia Stock Exchange in 1982. With the development of global trade and increased exchange rate volatility, the demand for foreign exchange options has grown rapidly. Foreign exchange options provide investors with a unique risk management tool. By paying a premium, the buyer gains the right, but not the obligation, to buy or sell a certain amount of foreign currency at a predetermined exchange rate on or before a specific future date. This flexibility has led to the widespread use of foreign exchange options in international trade, investment, and speculation. However, the pricing of foreign exchange options is not intuitive and is influenced by various factors, including exchange rate fluctuations, interest rate differentials, and market volatility. Therefore, studying the pricing of foreign exchange options has significant theoretical and practical implications.

[0003] The finite difference method (FDM) faces the following technical bottlenecks when solving partial differential equations for foreign exchange option pricing (such as the Garman-Kohlhagen or Heston models): When the model dimension d ≥ 2d (e.g., underlying asset price + volatility + stochastic interest rate), traditional grid computing methods face a sharp increase in computational complexity. This is especially true for multidimensional models, such as those simultaneously considering underlying asset price, volatility, and stochastic interest rate, where the number of grid nodes explodes from N in one dimension to N1. d The exponential growth of computing resources has led to a significant increase in computing resource and memory requirements. Furthermore, existing methods lack dynamic adjustment capabilities when dealing with critical regions and cannot automatically optimize computational accuracy based on option characteristics. At high model dimensionality, traditional methods struggle to achieve a balance between computational accuracy and efficiency; therefore, to reduce computational load, this application proposes a pricing method and system for foreign exchange options. Summary of the Invention

[0004] The purpose of this invention is to provide a pricing method and system for foreign exchange options to solve the problems mentioned in the background art.

[0005] In existing technologies, foreign exchange option pricing models face significant challenges when dealing with high-dimensional problems. Traditional finite difference methods, when solving partial differential equations involving multiple random variables, experience an exponential increase in the number of computational nodes with increasing dimensionality. For example, when the model involves three dimensions—underlying asset price, volatility, and stochastic interest rates—computational resource consumption rapidly exceeds conventional processing capabilities, making real-time pricing or efficient risk management impossible. Especially when dealing with complex products such as American options with early exercise characteristics, existing methods struggle to complete price calculations and sensitivity analyses within a reasonable timeframe.

[0006] To address these issues, a hybrid computing framework capable of dynamically adapting to different dimensional characteristics is needed. First, analysis reveals that the computational bottleneck caused by increased dimensionality stems from the exponential expansion of the number of grid nodes, necessitating the introduction of mathematical transformations to compress the computational scale. Second, option prices exhibit significant nonlinear characteristics near the strike price, requiring the establishment of a local precision control mechanism. Third, when the dimensionality exceeds a certain threshold, traditional grid methods become unsuitable, necessitating the integration of probabilistic simulation methods. Finally, the collaborative utilization of heterogeneous computing resources becomes crucial for improving overall efficiency. Therefore, a first aspect of the present invention provides a method for pricing foreign exchange options, comprising: Step 1: Determine the dimension d of the option pricing model. Dimension d represents the number of random variables in the option pricing model. Step 2: When the dimension d ≥ 2, use a sparse mesh to compress the number of computation nodes: reduce the number of computation nodes from... Compress to ,in The number of grid cells in a single dimension; Step 3: Based on the preset Gamma sensitivity threshold Identify key areas and dynamically increase grid density near the strike price; Step 4: When the dimension d>3, automatically switch to Monte Carlo simulation and perform variance reduction based on the sparse mesh solution; Step 5: Perform the sparse mesh overlay, dimensionality reduction decomposition, and Monte Carlo path generation tasks in parallel using a graphics processor.

[0007] The sparse mesh is implemented using a combinatorial generator, and the specific steps are as follows: Generate a set of subgrid indices, the sum of which does not exceed [a certain value]. ,in is the number of grids in a single dimension; d is the dimension of the option pricing model, and the index identifier is a d-dimensional vector, with each component representing the grid resolution of the corresponding dimension; Dynamic weights are assigned to subgrid solutions: the weights are inversely proportional to the variance of the subgrid solutions, and a smoothing coefficient with a value between 0 and 0.2 is introduced to prevent the denominator from being zero; The weighted subgrid solutions are superimposed and output as the initial solution.

[0008] The execution logic of the dynamic encryption module is as follows: Calculate the Gamma value at the grid points, which is the absolute value of the second partial derivative of the option price with respect to the underlying asset price; When the Gamma value exceeds the preset threshold of 0.15, the grid step size is adjusted: the new step size is the original step size divided by 1 plus the absolute value of Gamma multiplied by the sensitivity coefficient, where the sensitivity coefficient is between 1 and 2.

[0009] When d=2, dimensionality reduction modeling is performed, and the steps include: The logarithmic price offset and the square root of volatility are used as new coordinates, and the curve boundary is converted into a rectangular computational domain. The partial differential equation is decomposed into two one-dimensional equations by using the alternating direction implicit method and solved alternately.

[0010] In processing American options, dimensionality reduction is achieved through parameterized early strike boundaries: Constructing the exercise boundary surface function: The exercise price is the original exercise price multiplied by an exponential function, and the exponential part is a linear combination of the time function and the volatility function; The time function and volatility function are fitted based on historical data, with a fitting error tolerance of less than 1%.

[0011] The triggering condition for the Monte Carlo switch is as follows: Dimension d ≥ 4; or / and Sparse mesh density ,in .

[0012] The variance reduction method employed is the control variable method. Using the sparse grid solution as the baseline, the Monte Carlo result is corrected: the correction value is the Monte Carlo solution plus the difference between the baseline value and the expected value multiplied by the optimal hedging coefficient. The optimal hedging coefficient is the covariance of the Monte Carlo solution and the benchmark value divided by the variance of the benchmark value.

[0013] The graphics processor parallelization task includes: Sparse grid task: Assigned to stream processors for parallel computation of the initial solution; Alternating direction implicit solution task: solve tridiagonal matrices in parallel using a sparse matrix library; Monte Carlo task: Employ a low-bias sequence batch generation path with a speed of at least 10,000 sequences per second.

[0014] A second aspect of the present invention provides a pricing system for foreign exchange options, used to implement the aforementioned pricing method for foreign exchange options, comprising: Dimension identification module: Determines the dimension d of the option pricing model; Sparse mesh building block: reduces the number of computation nodes from Compress to ; Dynamic encryption module: based on =0.15 Identify and encrypt critical areas; Model switching module: When dimension d > 3, automatically switch to Monte Carlo simulation; Heterogeneous computing module: Executes tasks in parallel using a graphics processor.

[0015] A third aspect of the present invention provides a computer-readable storage medium having program instructions stored thereon for executing the aforementioned pricing method for foreign exchange options.

[0016] Compared with the prior art, the beneficial effects of the present invention are: This invention reduces the computational complexity of high-dimensional calculations by sparse mesh compression, dynamically densifies key regions by combining Gamma sensitivity, and automatically switches to Monte Carlo simulation in ultra-high dimensions. It effectively solves the problems of high computational resource consumption and rigid accuracy adjustment of traditional methods, and has the advantages of improving computational efficiency, dynamically optimizing the accuracy of key regions, and balancing computational resources and accuracy. Attached Figure Description

[0017] Figure 1 This is a flowchart illustrating the operation of a foreign exchange option pricing method according to the present invention. Detailed Implementation

[0018] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0019] Example 1 Please see Figure 1 This invention provides a technical solution: a method for pricing foreign exchange options, comprising: Step 1: Determine the dimension d of the option pricing model. Dimension d represents the number of random variables in the option pricing model. Step 2: When the dimension d ≥ 2, use a sparse mesh to compress the number of computation nodes: reduce the number of computation nodes from... Compress to ,in The number of grid cells in a single dimension; Step 3: Based on the preset Gamma sensitivity threshold Identify key areas and dynamically increase grid density near the strike price; Step 4: When the dimension d>3, automatically switch to Monte Carlo simulation and perform variance reduction based on the sparse mesh solution; Step 5: Perform the sparse mesh overlay, dimensionality reduction decomposition, and Monte Carlo path generation tasks in parallel using a graphics processor.

[0020] Here, dimension d represents the number of random variables in the model, which can be automatically identified through the parametric modeling process and is used to evaluate computational complexity. Sparse mesh compression is achieved by combining sub-grids of different resolutions, reducing the number of nodes from... Reducing it to the polynomial level effectively alleviates the curse of dimensionality. Gamma sensitivity threshold. This method is used to detect regions with drastic changes in the second derivative of price, triggering local mesh refinement to balance global efficiency and local accuracy. The Monte Carlo switching mechanism is automatically enabled when the dimension d > 3, using sparse mesh solutions as control variables to reduce simulation variance. Graphics processor parallelization accelerates computationally intensive operations through task decomposition and stream processing.

[0021] Specifically, this method first identifies the dimension of the option pricing model to determine the computational strategy. For two-dimensional or three-dimensional scenarios, a sparse mesh is constructed, and weighted sub-mesh solutions are superimposed using a combinatorial generator to reduce redundant computation while preserving accuracy in key areas. When the Gamma value near the strike price exceeds a threshold, the mesh step size is dynamically adjusted to implement local refinement. For four-dimensional and higher-dimensional problems, Monte Carlo simulation is automatically switched, and path sampling is optimized using previously obtained sparse mesh solutions. All computational tasks are executed synchronously through the parallel architecture of the graphics processing unit, achieving efficient utilization of computing resources.

[0022] Compared to existing technologies, traditional finite difference methods fail in high-dimensional scenarios due to node explosion, while this method covers the entire dimensional range through a hybrid strategy of sparse mesh and Monte Carlo mapping. Existing single methods cannot simultaneously handle low-dimensional accuracy and high-dimensional feasibility; this solution achieves adaptive switching through a dimensional threshold. Traditional parallel computing only optimizes a single algorithm; this solution accelerates different computing modules collaboratively through a heterogeneous architecture.

[0023] Through the above technical solutions, this application effectively solves the computational efficiency problem of high-dimensional foreign exchange option pricing. Sparse grid compression significantly reduces memory usage and computational load, dynamic encryption mechanism ensures pricing accuracy in key areas, Monte Carlo switching expands the processing capability for high-dimensional problems, and parallelization of the graphics processor improves the overall computing speed. This method achieves efficient pricing from two-dimensional to high-dimensional scenarios while maintaining computational accuracy, providing technical support for real-time risk management of complex foreign exchange derivatives.

[0024] The sparse mesh is achieved through a combinatorial generator, and the specific steps are as follows: Generate subgrid index set The sum of the indexed components does not exceed Specifically: ,in represents the number of grid cells in a single dimension; d represents the dimension of the option pricing model, and the index identifier is a d-dimensional vector. It is the sum of the index components, constrained to not exceed ; Let be a d-dimensional index vector, representing the index identifier of a subgrid in the sparse mesh, specifically in the form of: , where each component k i It is an integer, corresponding to the grid resolution of the i-th dimension; For subgrid solutions Dynamic weight allocation: The weights are inversely proportional to the variance of the subgrid solution, and a smoothing coefficient with a value between 0 and 0.2 is introduced to prevent the denominator from being zero; specifically; ; in, With subgrid solution variance Inversely proportional; To ensure smoothness and prevent the denominator from being zero; The weighted subgrid solutions are superimposed and output as the initial solution: ; in, Sum of solutions for subgrids; For subgrid solutions Assign dynamic weights; This is the solution for the subgrid.

[0025] Furthermore, the method for constructing sparse meshes through a combination generator includes generating a set of sub-mesh indices, assigning dynamic weights to the sub-mesh solutions, and outputting the initial solution by superimposing the weighted sub-mesh solutions.

[0026] The subgrid index set refers to the set of indices of multiple subgrids, and the sum of the index components is restricted to no more than N+d-1. This can be achieved by traversing all d-dimensional vector indices that satisfy the condition. This constraint mathematically limits the number of subgrids, avoiding the combinatorial explosion problem caused by the increase in dimensions in traditional full-grid methods.

[0027] Dynamic weight allocation refers to dynamically adjusting the weights of subgrid solutions based on their variance. This can be achieved by adding the reciprocal of the variance to the smoothing coefficient and then taking the inverse ratio. Subgrid solutions with larger variances are assigned lower weights, thereby suppressing the influence of computationally unstable regions on the overall solution. The smoothing coefficient is used to prevent numerical anomalies where the denominator is zero.

[0028] Specifically, when generating the subgrid index set, each subgrid index is represented by a d-dimensional vector, with each dimension corresponding to a different resolution. By constraining the sum of the index components to no more than N+d-1, the number of subgrids that need to be computed is effectively reduced. During the weight allocation stage, the contribution of each subgrid is dynamically adjusted through an inverse variance relationship, suppressing high-variance regions and strengthening low-variance regions. A smoothing coefficient in the range of 0 to 0.2 is introduced, for example, taking 0.1, which avoids the risk of the denominator being zero while preserving the gradient of the impact of variance differences on the weights. Finally, the solutions of subgrids with different resolutions are fused by weighted superposition, preserving the local accuracy of high resolution while inheriting the computational efficiency of low resolution.

[0029] Compared to existing technologies, traditional finite difference methods require constructing a full-dimensional mesh when dealing with high-dimensional models, leading to an exponential increase in the number of nodes with increasing dimensionality. Our proposed solution, however, uses a sparse mesh combination generator to compress the number of nodes to near-linear complexity. For example, when d=3 and N=100, the number of nodes is reduced from the millions in traditional methods to the thousands. Furthermore, the dynamic weight allocation mechanism overcomes the accuracy loss caused by fixed weights in traditional sparse meshes, adaptively optimizing the solution quality through variance feedback.

[0030] Through the above technical solutions, this application effectively solves the problem of the proliferation of computational nodes in high-dimensional option pricing models, significantly reducing computational resource consumption while ensuring computational accuracy. The dynamic weight allocation mechanism improves the stability of the solution, avoids the negative impact of local computational errors on the overall result, and provides a reliable basic solution for subsequent financial derivatives pricing.

[0031] The execution logic of the dynamic encryption module is as follows: Calculate the Gamma value at the grid points: , which is the absolute value of the second partial derivative of the option price with respect to the underlying asset price; where, The convexity sensitivity of the option price to the underlying asset price; This represents the theoretical price of the foreign exchange option. The exchange rate is the spot rate. Let be the second partial derivative of the option price with respect to the underlying asset price.

[0032] When the Gamma value exceeds the preset threshold of 0.15, the grid step size is adjusted: the new step size is the original step size divided by 1 plus the absolute value of Gamma multiplied by the sensitivity coefficient, where the sensitivity coefficient is between 1 and 2, specifically: when Adjust the grid step size as needed: ; in, =0.15 is the preset threshold; The new grid step size is the dynamically refined grid spacing, used to improve the accuracy of numerical calculations in critical areas. The unit is a price unit (such as exchange rate points). The original grid step size, the initial grid spacing before refinement, and the global base resolution; This is the sensitivity adjustment coefficient. Larger → more aggressive encryption (more significant step compression).

[0033] Specifically, the execution logic of the dynamic encryption module calculates the Gamma value of the grid point, which is the absolute value of the second partial derivative of the option price with respect to the underlying price. When the Gamma value exceeds the preset threshold, the grid step size is adjusted. The new step size is the original step size divided by 1 plus the result of the absolute value of Gamma multiplied by the sensitivity coefficient, where the sensitivity coefficient is between 1 and 2.

[0034] Here, Gamma refers to the absolute value of the second partial derivative of the option price with respect to the underlying asset price. It is specifically calculated using the finite difference method to determine the rate of price change between adjacent grid points, quantifying the severity of price curvature changes. The preset threshold of 0.15 is the critical condition for triggering grid refinement, determined through statistical analysis of historical backtesting data, used to screen out price-sensitive areas. The sensitivity coefficient is a control parameter for the step size adjustment, dynamically adjusted within the range of 1 to 2 using linear interpolation, balancing computational accuracy and numerical stability.

[0035] Specifically, when traversing all grid nodes within the computational domain, the Gamma value of each node is solved using the second-order central difference method. When a Gamma value exceeding 0.15 is detected in a certain region, it indicates a significant change in the curvature of the price surface in that region, requiring local densification. At this point, the original step size is adjusted according to the formula by dividing the original value by (1 + absolute Gamma value × sensitivity coefficient), where the sensitivity coefficient of 1 to 2 ensures that the denominator is not zero and that the densification strength is positively correlated with the Gamma value. For example, when the Gamma value is 0.2 and the sensitivity coefficient is 1.5, the step size will be reduced to 1 / (1 + 0.2 × 1.5) = 0.77 times the original value, achieving an exponential increase in grid density.

[0036] Compared to existing technologies, traditional fixed-grid methods distribute computation nodes uniformly across the entire globe, leading to resource waste in non-critical areas. Furthermore, encryption schemes based on first-order derivative adjustments can only reflect linear price changes and cannot capture regions of abrupt curvature changes. This scheme dynamically identifies curvature change points using second-order derivatives and combines this with a sensitivity coefficient to achieve adaptive local encryption, improving encryption accuracy to the curvature dimension under the same computational resources.

[0037] Through the above technical solution, this application effectively solves the problem of unreasonable grid resource allocation in the foreign exchange option pricing process. By implementing targeted densification in price-sensitive areas, it avoids computational redundancy caused by global fine-grained grids. Simultaneously, based on a dynamic adjustment mechanism for curvature changes, it ensures that the numerical solution accuracy in key areas meets the requirements for solving partial differential equations, thus optimizing computational resource utilization while maintaining computational stability.

[0038] When d=2, dimensionality reduction modeling is performed, and the steps include: Using the logarithmic price offset and the square root of volatility as new coordinates, the curve boundary is transformed into a rectangular computational domain, specifically: ;in, The logarithmic price offset standardizes the price and reflects the degree of deviation between the current asset price and the strike price; S is the underlying asset price, the price of the underlying asset linked to the option (such as the foreign exchange rate); K is the option strike price, the delivery price agreed upon in the contract, a key parameter that determines the profit and loss of the option; ln is the logarithmic transformation function, which eliminates the influence of price scale and makes the model more consistent with the statistical characteristics of financial data. The volatility is converted to square root form to ensure its non-negativity, which meets the requirements of stochastic process modeling.

[0039] The partial differential equation is decomposed into two one-dimensional equations by alternating directions and solved alternately, as follows: ; in, Let V be the partial derivative of the option price V with respect to time t; For along Differential operators for direction (logarithmic price dimension); For time steps Intermediate predicted solutions; for Differential operators for direction; For time steps Intermediate predicted solution; n is the time step index (discrete time point) ).

[0040] Specifically, when the dimension is 2, dimensionality reduction modeling is performed. The steps include transforming the logarithmic price offset and the square root of volatility into new coordinates, converting the curve boundary into a rectangular computational domain, and using the alternating direction implicit method to decompose the partial differential equation into two one-dimensional equations for alternating solution.

[0041] The logarithmic price offset transformation refers to converting the original price variable into a relative change in logarithmic space. Specifically, this can be achieved by using the natural logarithm function to perform a coordinate transformation on the underlying asset price. This transformation eliminates dimensional differences in prices and linearizes the equation. The square root volatility transformation involves taking the square root of the volatility parameter and using it as a new variable. This can be achieved by introducing a parameter obtained by taking the square root of the variance to replace the original volatility term. This transformation eliminates the nonlinearity of the quadratic term in the volatility equation. The alternating direction implicit method involves decomposing a multidimensional partial differential equation into multiple one-dimensional equations for iterative solution. This can be achieved by using time step segmentation and alternating spatial dimension solutions. This method reduces the complexity of solving multidimensional equations through step-by-step iteration.

[0042] Specifically, in the two-dimensional foreign exchange option pricing model, the curved boundary of the original computational domain is first transformed into a rectangular region through coordinate transformation. A logarithmic price offset transformation maps the price fluctuation range to a linear space, while a square root volatility transformation converts the stochastic volatility equation into a linear diffusion form. The computational domain after these two transformations forms a regular rectangular boundary, providing a standardized grid layout for subsequent numerical discretization. In the equation solving stage, the two-dimensional partial differential equation is decomposed into two independent one-dimensional equations according to the spatial dimension. Implicit solutions are performed sequentially in both directions within each time step. For each direction, a tridiagonal matrix algorithm is used to quickly calculate the numerical solution for the current dimension, and the latest result is used as the initial condition for solving the next direction. This alternating iterative process transforms the two-dimensional problem into a sequential processing of two one-dimensional problems, maintaining the stability of the implicit format while avoiding the inversion of two-dimensional matrices.

[0043] Compared to existing technologies, traditional finite difference methods directly construct a two-dimensional mesh for full-space discretization when dealing with two-dimensional models, resulting in a significant increase in the number of computational nodes with the square of the dimension. Existing coordinate transformation methods often employ single-variable transformations, making it difficult to simultaneously handle the nonlinear coupling between price and volatility. This proposed solution utilizes a dual coordinate transformation mechanism to eliminate nonlinear terms in the equations while regularizing the computational domain. Combined with the dimensionality decomposition strategy of the alternating direction implicit method, it reduces the computational complexity from a two-dimensional level to a one-dimensional level.

[0044] Through the above technical solutions, this application effectively solves the problem of low solution efficiency caused by the complex boundary of the computational domain in two-dimensional foreign exchange option pricing models. Coordinate transformation converts the irregular computational domain into a standard rectangular region, creating a foundation for efficient numerical discretization. The alternating direction implicit method avoids two-dimensional matrix operations through dimensional decomposition, significantly reducing the computational load per iteration. While ensuring numerical stability, this method improves the solution efficiency of two-dimensional partial differential equations to a level comparable to that of one-dimensional problems, providing a feasible computational solution for high-precision foreign exchange option pricing.

[0045] In processing American options, dimensionality reduction is achieved through parameterized early strike boundaries: Constructing the strike boundary surface function: The strike price is the original strike price multiplied by an exponential function. The exponential part is a linear combination of a time function and a volatility function, specifically: ;in, Let be the early exercise boundary function for American options, representing the critical underlying asset price that triggers the optimal exercise of an American option at time t and volatility σ; t is the time until expiration; σ is the volatility; K is the strike price, the fixed delivery price agreed upon in the option contract; a(t) is a time-dependent parametric function, a function obtained by fitting historical exercise data to capture the impact of time decay on the exercise boundary; b(t) is a volatility sensitivity coefficient function, a fitted function that measures the strength of the impact of volatility changes on the exercise boundary. For exponential functions, linear combinations Transforming it into an exponential form aligns with the non-linear, gradual change in the financial boundary.

[0046] The time function and volatility function are fitted based on historical data, with a fitting error tolerance of less than 1%.

[0047] Specifically, the technical solution for reducing the dimensionality of the strike boundary by parameterization when processing American options includes constructing a strike boundary surface function, where the strike price is the original strike price multiplied by an exponential function. The exponential part is a linear combination of a time function and a volatility function. The time function and volatility function are fitted based on historical data, and the tolerance for fitting error is less than 1%.

[0048] The parameterized early exercise boundary dimensionality reduction refers to transforming the complex exercise boundary, which originally required high-dimensional surface tracking, into a low-dimensional function form that can be analytically expressed. Specifically, this can be achieved by combining an exponential function with a linear combination of time and volatility, thus avoiding the computational burden of directly processing high-dimensional surfaces. The exercise boundary surface function describes the change in exercise price over time and volatility through mathematical expressions. This can be achieved by multiplying the original exercise price with an exponential function; the construction of the exponential term preserves the nonlinear relationship. The time function and volatility function are parameterized functions that reflect the decay of exercise price over time and the change in exercise price with market volatility, respectively. These can be modeled using polynomial functions or spline functions, with specific parameters determined by fitting historical data. A fitting error tolerance of less than 1% refers to the required fitting accuracy of the time function and volatility function. This can be achieved using least squares or regularized regression methods, thus ensuring model accuracy while preventing overfitting.

[0049] Specifically, in American option pricing, surface tracking of the early strike boundary involves calculations in a multi-dimensional parameter space, which traditional methods require iterative solutions on a high-dimensional grid. By constructing a strike boundary surface function, the original strike price is combined with an exponential function. The exponential part consists of a linear combination of a time function and a volatility function, decomposing the complex surface into independently optimizable low-dimensional functions. The time function can be in the form of a quadratic polynomial, for example, represented as a combination of quadratic coefficients of the time variable; the volatility function can be a piecewise linear function, for example, divided into different linear segments according to the market volatility range. Based on historical trading data, an optimization algorithm is used to jointly fit the two types of functions. During the fitting process, an error tolerance threshold is set, for example, the error tolerance can be set to less than 1%, thereby achieving dimensionality reduction modeling while preserving key market characteristics.

[0050] Compared to existing technologies, traditional American option pricing methods require tracking the dynamic changes of the strike boundary in a high-dimensional space, leading to an exponential increase in computational complexity with increasing dimensionality. Our proposed solution, however, transforms surface tracking into a low-dimensional function fitting problem through parametric modeling. For example, it simplifies a four-dimensional surface problem into the optimization of two two-dimensional functions, significantly reducing computational resource consumption. Existing free boundary tracking methods require iterative solutions to partial differential equations, while our solution directly constructs the boundary function through analytical expressions, avoiding stability issues in the numerical solution process.

[0051] Through the above technical solution, this application effectively solves the problem of high-dimensional computational complexity caused by the boundary surface of early exercise in American option pricing. By decomposing the high-dimensional surface into a combination of low-dimensional functions through parametric modeling, the computational resource consumption is transformed from exponential growth to linear growth. At the same time, the function construction method based on historical data fitting ensures the financial rationality of the boundary surface. The setting of error tolerance balances model accuracy and computational efficiency, preventing the waste of computational resources due to overfitting.

[0052] The trigger condition for the Monte Carlo switch is: Dimension d ≥ 4; or / and Sparse mesh density ,in .

[0053] Furthermore, the Monte Carlo switching is triggered when the dimension reaches four or more or the sparse mesh density is less than 0.15, automatically switching to Monte Carlo simulation. The sparse mesh density is the ratio of the actual number of nodes to the power of the number of individual grid cells.

[0054] Monte Carlo switching refers to the automatic conversion from deterministic grid computing to probabilistic simulation when the model complexity exceeds a preset threshold. This can be implemented using a logic judgment module to dynamically adjust the numerical calculation strategy. The dimensionality threshold is set to four dimensions because of the abrupt change in the computational efficiency of deterministic methods under high-dimensional models. When the dimension exceeds three dimensions, the computational complexity of traditional finite difference methods will exceed the computational efficiency boundary of probabilistic methods. Sparse grid density refers to the proportion of actual effective computing nodes to the theoretical number of nodes in the grid. This can be implemented using a node counter and a theoretical value calculation module to quantify the degree of computational efficiency decay of the grid method. The density critical value of 0.15 was obtained through experiments on the correlation between grid coverage efficiency and computational error under different dimensions. When the density is below this value, the grid resolution is insufficient to accurately describe price fluctuation characteristics.

[0055] Specifically, when the model dimension reaches four dimensions, the system automatically triggers a computation mode switch, transforming the solution process from the sparse mesh method to Monte Carlo simulation. During this process, the initial solution generated by the sparse mesh method is retained as a baseline value to correct the output results of the Monte Carlo simulation. When the model dimension is less than four dimensions but the sparse mesh density is less than 0.15, the system also triggers a mode switch, as the excessively sparse mesh node distribution may lead to insufficient computational accuracy in critical regions. Monte Carlo simulation performs probabilistic calculations by generating a large number of random paths; its computational complexity is linearly related to the dimension, avoiding the exponential growth problem of deterministic methods. Furthermore, Monte Carlo simulation employs a controlled variable method, using the sparse mesh solution as a baseline value for variance reduction, thereby improving result stability while maintaining computational efficiency.

[0056] Compared to existing technologies, traditional methods still employ fixed-grid computation in high-dimensional models, leading to an exponential increase in computational resource consumption with increasing dimensionality. Our proposed solution, however, proactively abandons inefficient grid computation in four-dimensional and higher-dimensional models by setting dual trigger conditions, opting instead for the Monte Carlo method, which is more suitable for high-dimensional problems. Simultaneously, sparse grid density is introduced as a supplementary criterion in low-dimensional models, effectively identifying critical states where grid resolution is insufficient and preventing the continued operation of inefficient computational scenarios. While existing Monte Carlo simulations typically run independently, our solution significantly reduces the variance of Monte Carlo results by reusing sparse grid solutions as control variables.

[0057] Through the above technical solution, this application effectively solves the computational efficiency bottleneck problem in high-dimensional foreign exchange option pricing. In high-dimensional models, the system can automatically select a probabilistic simulation method with controllable computational complexity, avoiding the node explosion problem caused by the increase in dimensionality in traditional finite difference methods. In low-dimensional models, the system monitors grid density and switches calculation methods in a timely manner to prevent accuracy loss caused by local grid sparsity. The synergistic use of Monte Carlo simulation and sparse grid solutions improves computational speed while ensuring result accuracy, achieving a dynamic balance between computational efficiency and accuracy.

[0058] Variance reduction employs the control variable method: Solution with sparse mesh Using the baseline value, the Monte Carlo result is corrected as follows: the correction value is the Monte Carlo solution plus the difference between the baseline value and the expected value multiplied by the optimal hedging coefficient, specifically: ; .

[0059] The optimal hedging coefficient is the covariance of the Monte Carlo solution and the benchmark value divided by the variance of the benchmark value.

[0060] in, The final calibrated option price is a high-precision pricing result after variance reduction correction, which is used for trading decisions or risk management. The original pricing results from the Monte Carlo simulation are used to generate preliminary option prices using random paths (e.g., 100,000 paths), which include statistical noise. Quantification for the optimal hedging coefficient and The strength of the linear correlation; The benchmark solution synthesized by sparse mesh is the pricing result obtained through dynamic weighted sparse mesh technology; For the mathematical expectation of the sparse grid solution, The average value across multiple simulations represents the statistical center of the benchmark solution. Covariance is a measure of and The joint volatility of the σ² is such that positive values ​​indicate oscillations in the same direction, while negative values ​​indicate oscillations in opposite directions. Variance measures Its own degree of fluctuation (the greater the fluctuation, the larger the denominator), The smaller).

[0061] Specifically, variance reduction employs the controlled variable method, using the sparse grid solution as the baseline value to correct the Monte Carlo results. The correction value is the Monte Carlo solution plus the difference between the baseline value and the expected value multiplied by the optimal hedging coefficient. The optimal hedging coefficient is the covariance between the Monte Carlo solution and the baseline value divided by the variance of the baseline value.

[0062] Among these methods, the control variable method refers to reducing the estimation variance by introducing auxiliary variables with known expected values ​​related to the target variable. Specifically, this can be achieved by dynamically adjusting the correction magnitude using the ratio of covariance to variance. Its function is to suppress random fluctuations in Monte Carlo simulations by utilizing the stability of the baseline value. The optimal hedging coefficient is an adjustment parameter used to balance the deviation between the baseline value and the Monte Carlo solution. This can be achieved by combining covariance matrix operations with variance calculations. Its function is to dynamically determine the error compensation ratio based on the correlation strength between the two. The baseline value refers to the high-precision solution obtained in low-dimensional scenarios using the sparse mesh method. Specifically, it can be generated by superimposing weighted subgrid solutions using a combinatorial generator. Its function is to provide a reliable reference benchmark for Monte Carlo simulations.

[0063] Specifically, when the model dimension exceeds a threshold, the sparse mesh solution is set as the baseline value and input into the Monte Carlo simulation process. After each path sampling, the covariance between the Monte Carlo solution and the baseline value is calculated, and the variance data of the baseline value is obtained simultaneously. The optimal hedging coefficient is obtained by dividing the covariance by the variance, and this coefficient is used to adjust the amount of compensation for the deviation between the baseline value and the expected value. The corrected Monte Carlo solution consists of the original solution superimposed with a compensation term, the magnitude of which is inversely proportional to the stability of the baseline value. This dynamic adjustment mechanism ensures that the simulation results in high-dimensional scenarios retain the adaptability of the Monte Carlo method while reducing random errors through the constraint of the baseline value.

[0064] Compared to existing technologies, traditional Monte Carlo methods suffer from variance accumulation leading to decreased accuracy in high-dimensional models, while conventional control variable methods require manually setting fixed benchmark values. This proposed solution automatically acquires sparse mesh solutions as dynamic benchmarks and combines this with covariance-driven hedging coefficient optimization to achieve adaptive adjustment of error compensation, overcoming the subjective bias problem of manually setting benchmarks.

[0065] Through the above technical solutions, this application effectively reduces the computational variance of Monte Carlo simulations under high-dimensional models, significantly improving pricing accuracy while maintaining high dimensional adaptability. The introduction of sparse grid baseline values ​​provides a clear mathematical basis for error correction, and the dynamic calculation of the optimal hedging coefficient ensures the scientific rationality of the error compensation amount, thereby achieving a balance between computational efficiency and accuracy optimization in complex model scenarios.

[0066] The parallelization tasks of the graphics processor include: Sparse grid task: Assign to stream processors for parallel computation of initial solutions ; Alternating direction implicit solution task: solve tridiagonal matrices in parallel using a sparse matrix library; Monte Carlo task: Employ a low-bias sequence batch generation path with a speed of at least 10,000 sequences per second.

[0067] Specifically, a technical solution is proposed that uses a graphics processor to perform sparse mesh overlay, dimensionality reduction decomposition, and Monte Carlo path generation tasks in parallel.

[0068] Among these, the sparse mesh task refers to decomposing the discretization process of a multidimensional computational domain into the superposition of multiple low-dimensional subgrids. Specifically, it can employ a multi-threaded architecture of a stream processor to compute the initial solution in parallel, and fully utilize computational resources by assigning different subgrids to independent threads. The alternating direction implicit solution task refers to decomposing high-dimensional partial differential equations into multiple one-dimensional equations for iterative solution. Specifically, it can use parallel algorithms from sparse matrix libraries to process tridiagonal matrices, leveraging the shared memory mechanism of graphics processors to accelerate matrix operations. The Monte Carlo task refers to achieving high-dimensional integral approximation through random path simulation. Specifically, it can use a low-bias sequence generator to generate path samples in batches, and combine this with the hyper-threaded architecture of graphics processors to achieve an order-of-magnitude improvement in path generation speed.

[0069] Specifically, the graphics processing unit (GPU) is divided into three parallel computing units to handle different task types. The sparse mesh task dynamically allocates sub-mesh to the stream processor's computational cores, leveraging data independence between thread blocks to achieve parallel computation of the initial solution. The alternating direction implicit solution stage calls pre-compiled sparse matrix operation kernel functions, optimizing memory access patterns for the banded structure of tridiagonal matrices to achieve hardware acceleration of the implicit iteration process. The Monte Carlo task uses a quasi-random number generator to pre-generate low-bias sequences, loads path generation parameters in batches through a texture memory caching mechanism, and utilizes the GPU's SIMD architecture to synchronously compute multiple paths. These three tasks are dynamically allocated computational resources through a pipeline scheduling mechanism, forming an end-to-end parallel processing chain.

[0070] Compared to existing technologies, traditional methods employ a central processing unit (CPU) to serially execute grid computation and path simulation, resulting in cumulative computational latency. Existing sparse grid implementations do not optimize memory access patterns for graphics processing unit (GPU) architectures, limiting matrix solution efficiency. Conventional Monte Carlo path generation uses a sequentially executed pseudo-random number algorithm, failing to fully leverage the throughput advantages of parallel computing hardware.

[0071] Through the above technical solutions, this application effectively solves the execution speed bottleneck caused by insufficient computing resources in high-dimensional models. Parallel processing of sparse grid tasks reduces the time complexity of multidimensional discretization calculations, hardware acceleration of implicit solution in alternating directions shortens the convergence time of partial differential equation iterations, and batch generation technology of Monte Carlo paths ensures the real-time requirements of high-dimensional integral calculations. The three technologies work together to form a complete acceleration technology chain, enabling the foreign exchange option pricing system to meet the computational performance requirements of real-time trading scenarios.

[0072] Example 2 A pricing system for foreign exchange options is provided to implement the aforementioned pricing method for foreign exchange options, comprising: Dimension identification module: Determines the dimension d of the option pricing model; Sparse mesh building block: reduces the number of computation nodes from Compress to ; Dynamic encryption module: based on =0.15 Identify and encrypt critical areas; Model switching module: When dimension d > 3, automatically switch to Monte Carlo simulation; Heterogeneous computing module: Executes tasks in parallel using a graphics processor.

[0073] The system comprises the following modules: Dimension Identification Module: This module is the computational unit used to determine the dimensional parameters of the option pricing model in real time. It can be implemented using an eigenvalue decomposition algorithm, determining the effective dimension by analyzing the rank of the covariance matrix of the random variables. Sparse Mesh Construction Module: This module refers to the structure of high-dimensional computational nodes compressed using a combinatorial generator. It can be implemented using a hierarchical subgrid stacking algorithm, balancing the resolution differences of each dimension through a dynamic weight allocation mechanism. Dynamic Encryption Module: This module is a controller that adjusts the local mesh density based on the second derivative sensitivity index. It can calculate the Gamma value using the finite difference method and optimize the accuracy of the strike price region through an adaptive step size adjustment algorithm. Model Switching Module: This module is the conditional judgment device that triggers Monte Carlo simulations. It can be implemented by a dimensional threshold comparator and a mesh density monitor working together, dynamically switching computation methods through preset trigger rules. Heterogeneous Computing Module: This module is a scheduling system that coordinates parallel tasks on the graphics processor. It can be implemented using a CUDA architecture for task allocation, completing heterogeneous computing tasks through the collaboration of stream processors and a sparse matrix library.

[0074] Specifically, the dimension identification module determines the effective dimensions of the model through covariance matrix analysis, providing basic parameter inputs for subsequent modules. The sparse mesh construction module generates hierarchical sub-mesh based on the dimension parameters and compresses node size through dynamic weight stacking, reducing memory usage while maintaining computational accuracy. The dynamic densification module monitors the Gamma value of mesh points in real time, triggering local mesh densification when regions exceeding a threshold are detected, achieving resolution improvement in price-sensitive areas. The model switching module continuously monitors the dimension parameters and mesh density indicators, automatically initiating Monte Carlo simulation when a preset threshold is reached, and calling a variance reduction algorithm to correct the calculation results. The heterogeneous computing module distributes the tasks of sparse mesh construction, partial differential equation solving, and path generation to different computing units of the graphics processor, utilizing a parallel architecture to accelerate the computation process.

[0075] In some specific implementations, the sparse mesh construction module can use the Smolyak algorithm to generate a composite mesh, the dynamic encryption module can be configured as a local refinement strategy based on cubic spline interpolation, the model switching module can integrate the Ziggurat algorithm to generate random number sequences, and the heterogeneous computing module can call the cuBLAS library to accelerate matrix operations.

[0076] Compared to existing technologies, traditional systems employ full-mesh discretization methods when processing high-dimensional models, leading to an exponential increase in computational load. Our system, however, significantly reduces the number of nodes while maintaining computational accuracy in critical areas through the synergistic effect of sparse mesh compression and dynamic encryption. Existing technologies consistently use a single numerical method in high-dimensional scenarios, while our system effectively balances computational efficiency and stability through a hybrid algorithm switching mechanism triggered by a dimensional threshold. Traditional solutions rely on CPU serial computation, limiting processing speed; our system accelerates tasks through a heterogeneous parallel architecture of a graphics processing unit, overcoming traditional hardware performance bottlenecks.

[0077] Through the above technical solutions, this application addresses the problem of excessive computational resource consumption in high-dimensional foreign exchange option pricing, achieving improved computational efficiency through multi-dimensional algorithm collaboration and hardware acceleration. In 3D model scenarios, the system can compress the node scale from cubic to logarithmic linear levels, and automatically switch to the Monte Carlo method to avoid the curse of dimensionality in 4D and higher-dimensional scenarios. A dynamic encryption mechanism ensures that the accuracy loss in price-sensitive areas does not exceed a preset tolerance, and the heterogeneous parallel architecture enables path generation speeds of tens of thousands of paths per second.

[0078] Example 3 A computer-readable storage medium is provided having program instructions stored thereon for executing the aforementioned pricing method for foreign exchange options.

[0079] Computer-readable storage media refers to physical carriers capable of persistently storing program code, which can be implemented using solid-state drives or flash memory chips, ensuring the repeatable execution and fast invocation of pricing algorithms. Program instructions refer to computer-executable code arranged according to preset logic, which can be written in C++ or CUDA languages, used to transform multidimensional sparse grid compression, dynamic region encryption, and Monte Carlo switching algorithms into operable computational steps. The pricing method for foreign exchange options refers to the process of determining option prices through mathematical modeling and numerical calculation, which can be implemented using sparse grid compression and variance reduction techniques to address the problem of excessive computational resource consumption under high-dimensional models.

[0080] Although embodiments of the invention have been shown and described, it will be understood by those skilled in the art that various changes, modifications, substitutions and alterations can be made to these embodiments without departing from the principles and spirit of the invention, the scope of which is defined by the appended claims and their equivalents.

Claims

1. A method of pricing foreign exchange options, characterized by, The method comprises the following steps: Step 1, determining the dimension d of the option pricing model, wherein the dimension d represents the number of random variables in the option pricing model; Step 2, when dimension d≥2, compress the number of computing nodes using sparse grid: the number of computing nodes is compressed from to , where is the number of grid in single dimension. Step 3, based on preset Gamma sensitivity threshold Identify key areas and dynamically encrypt grid density around strike price Step 4, when the dimension d is greater than 3, automatically switching to the Monte Carlo simulation, and performing variance reduction based on the sparse grid solution as a benchmark; Step 5, performing the tasks of sparse grid superposition, dimension reduction decomposition and Monte Carlo path generation in parallel through a graphics processor.

2. The method of pricing foreign exchange options according to claim 1, wherein: The sparse grid is implemented through a combination generator, and the specific steps are as follows: A sub-grid index set is generated, the sum of index components does not exceed wherein is the number of grids in a single dimension; d is the dimension of the option pricing model, the index identifier is a d-dimensional vector, and each component represents the grid resolution of the corresponding dimension; allocating a dynamic weight to the sub-grid solution: the weight is inversely proportional to the variance of the sub-grid solution, and a smoothing coefficient with a value between 0 and 0.2 is introduced to prevent the denominator from being zero; superimposing and outputting the weighted sub-grid solution as an initial solution.

3. The method of pricing foreign currency options as claimed in claim 1, wherein: The execution logic of the dynamic encryption module is as follows: calculating the grid point Gamma value, i.e. the absolute value of the second-order partial derivative of the option price with respect to the underlying price; when the Gamma value exceeds a preset threshold of 0.15, adjusting the grid step: the new step is the original step divided by 1 plus the absolute value of the Gamma multiplied by a sensitivity coefficient, wherein the sensitivity coefficient has a value between 1 and 2.

4. The method of pricing foreign currency options as claimed in claim 1, wherein: When d = 2, performing dimension reduction modeling, and the steps include: transforming the logarithmic price offset and the volatility square root into new coordinates to convert the curved boundary into a rectangular calculation domain; using the alternating direction implicit method to decompose the partial differential equation into two one-dimensional equations for alternating solution.

5. The method of pricing foreign currency options as claimed in claim 1, wherein: When processing American options, dimension reduction is performed through parameterization of the early exercise boundary: constructing an exercise boundary surface function: the exercise price is the original exercise price multiplied by an exponential function, and the exponential part is linearly combined by a time function and a volatility function; fitting the time function and the volatility function based on historical data, and the fitting error tolerance is less than 1%.

6. The method of pricing foreign currency options as claimed in claim 1, wherein: The triggering condition of the Monte Carlo switching is: the dimension d is greater than or equal to 4; or / and Sparse grid density wherein .

7. The method of pricing foreign currency options as claimed in claim 6, wherein: The variance reduction adopts the control variable method: using the sparse grid solution as a benchmark value to correct the Monte Carlo result: the correction value is the Monte Carlo solution plus the difference between the benchmark value and the expected value multiplied by the optimal hedging coefficient; the optimal hedging coefficient is the covariance of the Monte Carlo solution and the benchmark value divided by the variance of the benchmark value.

8. The method of pricing foreign currency options as claimed in claim 1, wherein: The parallelization tasks of the graphics processor include: sparse grid task: assigning to a stream processor to calculate the initial solution in parallel; alternating direction implicit solution task: calling a sparse matrix library to solve a tri-diagonal matrix in parallel; Monte Carlo task: generating paths using a low-bias sequence batch, and the speed is not less than 10,000 per second.

9. A system for pricing foreign exchange options for implementing the method for pricing foreign exchange options according to any one of claims 1 to 8, characterized in that, The method comprises the following steps: dimension identification module: determining the dimension d of the option pricing model; Sparse grid construction module: reduce the number of compute nodes from to ; Dynamic encryption module: based on =0.15 identify and encrypt critical areas; model switching module: when the dimension d is greater than 3, automatically switching to the Monte Carlo simulation; heterogeneous computing module: performing tasks in parallel through a graphics processor.

10. A computer-readable storage medium having stored thereon program instructions, wherein, The program instructions are used to execute the foreign exchange option pricing method of any one of claims 1-8.

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