Option pricing method and system for stochastic volatility models

Abstract

The application relates to the technical field of financial data processing and machine learning, and provides a random volatility model option pricing method and system. The method comprises the following steps: acquiring asset price data, interest rate parameters and contract parameters; determining a drawdown process, an upswing process and a random volatility process of asset prices according to the asset price data and the interest rate parameters; performing viscosity processing on the drawdown process and the upswing process to obtain a viscous drawdown process and a viscous upswing process; establishing a state equation with asset prices and volatility as state variables based on the viscous drawdown process, the viscous upswing process and the random volatility process; and inputting state variables and contract parameters of an option to be priced into a target neural network to obtain a theoretical price of the option. The application describes the continuous stay of prices at historical running extreme values and the random change of volatility in the pricing model, and outputs the theoretical price of the option by using the target neural network, so that market feature description and pricing calculation efficiency are considered.

Description

Technical Field

[0001] This invention relates to the fields of financial data processing and machine learning technology, and in particular to a method and system for pricing options using a stochastic volatility model. Background Technology

[0002] Option pricing is widely used in market making, portfolio valuation, hedging, and risk management. The accuracy and computational efficiency of option pricing directly affect the execution quality of these processes. Existing technologies for option pricing typically employ: constant volatility pricing models, which assume asset prices follow geometric Brownian motion and use a fixed volatility parameter to describe price uncertainty; stochastic volatility pricing models, which characterize volatility clustering, mean reversion, and leverage effects related to asset returns through stochastic variance processes, allowing volatility to dynamically change with market conditions; and numerical solution-based pricing methods, which use Monte Carlo simulations, finite difference methods, or finite element methods to discretely solve models containing features such as stochastic volatility to obtain option prices.

[0003] The aforementioned existing technologies exhibit significant shortcomings when the underlying asset experiences a one-sided market trend. In the real market, asset prices exhibit path dependence: when prices approach or reach historical highs, chasing the rally and trend trading strategies cause prices to continue moving at high levels and continuously reach new highs; when prices approach or reach historical lows, selling pressure and liquidity contraction cause prices to continue moving at low levels and continuously reach new lows. That is, asset prices often linger at historical extremes, and the random changes in volatility coexist with the above phenomena. Under these operating conditions: constant volatility models assume constant volatility and cannot characterize the changes in volatility with market conditions; stochastic volatility models, while characterizing the random changes in volatility, do not include information about the position of prices relative to historical extremes, and the ordinary diffusion process leaves immediately after reaching the boundary, with zero time spent at the boundary, failing to express the sustained lingering of prices at extremes; option prices obtained based on such models deviate from the actual market conditions, and the pricing error is particularly significant in market phases where continuous new highs and new lows occur frequently. Summary of the Invention

[0004] Based on this, it is necessary to propose a stochastic volatility model option pricing method and system to address the following technical problems: Under the operating conditions of one-sided market trends in the underlying asset, prices lingering at historical extreme values, and random volatility changes, neither constant volatility pricing models nor stochastic volatility pricing models can express the continuous lingering of prices at operating extreme values; When the number of state dimensions increases, Monte Carlo simulation and numerical solution are computationally intensive and time-consuming, making it difficult to quickly obtain stable and convergent option pricing results.

[0005] Firstly, a stochastic volatility model option pricing method is provided, including: Obtain asset price data, interest rate parameters, and contract parameters; Based on asset price data and interest rate parameters, determine the retracement, rise, and stochastic volatility processes of asset prices; By applying a sticky processing method to the pullback and upward processes, we obtain sticky pullback and sticky upward processes. Based on the sticky retracement process, the sticky rise process, and the stochastic volatility process, a state equation is established with asset price and volatility as state variables. The state variables and contract parameters of the option to be priced are input into the target neural network to obtain the theoretical price of the option. The target neural network is trained based on the state equation and contract parameters.

[0006] Secondly, a stochastic volatility model option pricing system is provided, including a processor and a memory. The memory stores a computer program, and the processor executes the computer program to implement the aforementioned stochastic volatility model option pricing method.

[0007] Thirdly, a computer device is provided, including a memory, a processor, and a computer program stored in the memory and executable on the processor, wherein the processor executes the computer program to implement the steps of the above-described stochastic volatility model option pricing method.

[0008] Fourthly, a computer-readable storage medium is provided, which stores a computer program that, when executed by a processor, implements the steps of the above-described stochastic volatility model option pricing method.

[0009] The stochastic volatility model option pricing method and system provided in this application obtains asset price data, interest rate parameters, and contract parameters to provide basic inputs for option pricing, such as underlying asset market conditions, risk-free interest rates, and option contract terms, enabling the pricing process to be executed under unified data conditions.

[0010] By using asset price data and interest rate parameters, the model determines the retracement, rise, and random volatility processes of asset prices. It separates the retracement state of prices relative to historical highs, the rise state of prices relative to historical lows, and the random change state of volatility itself from asset price data, and incorporates the continuous rise, continuous fall, and volatility clustering phenomena in market price movements into the same model framework.

[0011] By applying a sticky processing method to the pullback and upward processes, sticky pullback and sticky upward processes are obtained, enabling asset prices to remain at historical highs and lows. This compensates for the limitation of ordinary stochastic processes where the dwell time at extreme values ​​is zero, allowing the model to express the actual phenomenon of continuous price increases and decreases in the market.

[0012] By establishing a state equation with asset price and volatility as state variables based on the sticky retracement process, sticky uptrend process, and stochastic volatility process, stochastic volatility and sticky dwell behavior at historical extreme values ​​are integrated into the same dynamic system. The state equation describes the evolution of the state on which option price depends over time, providing a self-consistent model basis for option pricing.

[0013] By inputting the state variables and contract parameters of the option to be priced into a target neural network, the theoretical price of the option is obtained. The target neural network is trained based on the state equation and contract parameters. During the training process, the financial constraints satisfied by option pricing are encoded into the network parameters. In the pricing stage, only forward computation on the input is needed to obtain the theoretical price of the option, avoiding the repeated large-scale numerical calculations for each pricing scenario. The above process enables option pricing to take into account both the market characteristic of prices lingering at extreme values ​​and the execution efficiency of pricing calculation. The more complete the market characteristics described by the state equation, the closer the neural network output is to the option value under real market conditions.

[0014] The stochastic volatility model option pricing method and system provided by the above embodiments combine the sticky dwell of prices at historical extreme values, the random changes in volatility, and the fast solution of neural networks. The technical effect can be measured and verified by pricing accuracy and computation time: In the experimental verification of specific embodiments, the average percentage error of neural network pricing relative to benchmark simulated pricing is less than 0.4%, the pricing time of a single option is on the order of milliseconds, and the time of a single parameter calibration is on the order of sub-seconds. Attached Figure Description

[0015] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0016] in: Figure 1 This is a flowchart illustrating the option pricing method based on the stochastic volatility model in an embodiment of the present invention. Figure 2 This is a schematic diagram of the structure of the stochastic volatility model option pricing system in an embodiment of the present invention; Figure 3 This is a structural block diagram of the computer device server in an embodiment of the present invention; Figure 4 This is a structural block diagram of the computer device client in an embodiment of the present invention. Detailed Implementation

[0017] 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, not all, of the embodiments of the present invention. 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.

[0018] Please see Figure 1 As shown, Figure 1 A flowchart illustrating the stochastic volatility model option pricing method provided in this embodiment of the invention includes the following steps: Step 01: Obtain asset price data, interest rate parameters, and contract parameters; Step 02: Based on asset price data and interest rate parameters, determine the asset price retracement process, upward process, and stochastic volatility process; Step 03: Apply viscosity processing to the pullback and upward processes to obtain viscous pullback and viscous upward processes; Step 04: Based on the sticky retracement process, the sticky uptrend process, and the stochastic volatility process, establish a state equation with asset price and volatility as state variables; Step 05: Input the state variables and contract parameters of the option to be priced into the target neural network to obtain the theoretical price of the option. The target neural network is trained based on the state equation and contract parameters.

[0019] Specifically, in step 01, the asset price data can be the historical price series or real-time price data of the underlying asset. The underlying asset can be a stock, an index, or other financial asset with options trading. The interest rate parameter can be the risk-free interest rate used for pricing, and the contract parameters can be the terms of the option contract to be priced, including information such as the strike price, expiration date, and option type. The asset price data can be read from a market data system, trading system, or database. The interest rate parameter and contract parameter are determined according to the pricing benchmark. The obtained data is used for subsequent process construction and pricing calculation. The data in step 01 remains unchanged throughout the pricing process, forming the input basis for option pricing.

[0020] In step 02, the pullback process can be a stochastic process describing the position of asset prices relative to historical highs, the upward process can be a stochastic process describing the position of asset prices relative to historical lows, and the stochastic volatility process can be a stochastic process describing the random changes in asset return volatility over time. In real market price movements, prices may continue to rise after approaching or reaching historical highs, forming continuous new highs; conversely, prices may continue to fall after approaching or reaching historical lows, forming continuous new lows. The pullback and upward processes respectively characterize these two types of positional states, while the stochastic volatility process characterizes the dynamic characteristics of volatility such as clustering and mean reversion. Step 02 identifies these three types of processes from asset price data and interest rate parameters, providing a process basis for establishing the state equation.

[0021] In step 03, the "stickiness treatment" can be a method of causing a stochastic process to pause at a specific boundary. A sticky retracement process can be a stochastic process obtained by applying the stickiness treatment to a retracement process, and a sticky upward process can be a stochastic process obtained by applying the stickiness treatment to an upward process. A normal diffusion process leaves immediately after reaching the boundary, with zero dwell time at the boundary, and cannot express a market trend where prices continue to move along historical highs or lows. The stickiness treatment allows the process to accumulate a positive time length when approaching the boundary, thus allowing the price to remain near the boundary in the model description. The sticky retracement process expresses a state of continuously making new highs, and the sticky upward process expresses a state of continuously making new lows.

[0022] In step 04, the state variables can be variables describing the state of the pricing system at any given time. These state variables include at least asset prices and volatility. The state equation can be a dynamic equation describing the evolution of the state variables over time. The states described by sticky retracement processes, sticky uptrend processes, and stochastic volatility processes change over time. The state equation provides the evolutionary relationship of the state variables over time. The option price function becomes a function of the state variables and time. Once the set of state variables is determined, the state space required for option pricing is determined accordingly. The state equation forms the model basis for calculating the theoretical price of options.

[0023] In step 05, the target neural network can be a pre-trained neural network. The target neural network is trained based on the state equation and contract parameters, and the training process learns the pricing constraints described by the state equation into the network parameters. The state variables of the option to be priced and the contract parameters constitute the input of the target neural network. The target neural network performs forward computation on the input and outputs the theoretical option price. During the training phase, the target neural network learns the mapping relationship between the state variables, contract parameters, and option price. During the pricing phase, it directly outputs the theoretical option price for new input combinations. The pricing process does not re-perform numerical solutions for each pricing scenario. The theoretical option price output by the target neural network is consistent with the model described by the state equation and is used for transaction pricing, valuation, or risk calculation.

[0024] In some embodiments, step 02 above includes: The risk-adjusted price is determined based on interest rate parameters and asset price data; Based on risk-adjusted pricing, determine the maximum and minimum operating values; The drawdown process is determined by the ratio of the risk-adjusted price to the maximum operating value; The upward trend is determined by the ratio of the risk-adjusted price to the minimum operating value; The random volatility process is determined through a square root diffusion process based on the preset mean volatility recovery rate, the preset long-term variance mean, and the preset variance volatility.

[0025] Specifically, the original asset price in the asset price data is denoted as S_t, where t represents the current time, and the risk-free interest rate in the interest rate parameter is denoted as r. Determining the risk-adjusted price involves removing the risk-free growth trend from the original asset price to obtain the risk-adjusted asset price. The risk-adjusted price can be expressed as: _t=e^{ rt}·S_t (1) In equation (1), S_t represents the original asset price at time t, r represents the risk-free interest rate, e represents the natural constant, and e^{ rt} represents the discount factor corresponding to the risk-free rate. _t represents the risk-adjusted price at time t. After applying risk adjustment to the asset price, the running maximum and running minimum reflect the asset's position relative to the market state, and are not affected by long-term discount or risk-free growth trends.

[0026] Based on the risk-adjusted price, the maximum and minimum operating values ​​are determined. The maximum operating value can be expressed as: _t=max_{0≤u≤t} _u, _t=min_{0≤u≤t} _u (2) In equation (2), u represents the time variable between 0 and t, max_{0≤u≤t} represents finding the maximum value for the time interval between 0 and t, and min_{0≤u≤t} represents finding the minimum value for the time interval between 0 and t. _t represents the maximum operating value at time t, which is the maximum value reached by the risk-adjusted price within the time period from 0 to t. _t represents the minimum operating value at time t, which is the minimum value reached by the risk-adjusted price within the period from 0 to t. The maximum operating value is updated as the price reaches a new high, and the minimum operating value is updated as the price reaches a new low.

[0027] The drawdown process is determined by the ratio of the risk-adjusted price to the maximum operating value. The drawdown process can be expressed as: D_t= _t / _t (3) In equation (3), _t represents the risk-adjusted price at time t. _t represents the maximum value at time t, and D_t represents the retracement process at time t. The retracement process ranges from 0 to 1. When the retracement process takes the boundary value of 1, the risk-adjusted price equals the maximum value, the asset price reaches a historical high and is in a state of continuous new highs. The closer the retracement process is to 1, the smaller the retracement of the price relative to the maximum value.

[0028] The upward trend is determined by the ratio of the risk-adjusted price to the minimum operating value. The upward trend can be represented as: U_t= _t / _t (4) In equation (4), _t represents the risk-adjusted price at time t. _t represents the minimum value at time t, and U_t represents the upward process at time t. When the upward process takes the boundary value of 1, the risk-adjusted price equals the minimum value, the asset price reaches a historical low and is in a state of continuous new lows. The closer the upward process is to 1, the smaller the distance between the price and the minimum value.

[0029] Based on preset volatility mean reversion rate, preset long-term variance mean, and preset variance volatility, a stochastic volatility process is determined through a square root diffusion process. The volatility mean reversion rate is denoted as κ, the long-term variance mean as θ, the variance volatility as σ, and the instantaneous variance in the stochastic volatility process as V_t. The stochastic volatility process can be represented as: dV_t=κ·(θ V_t)·dt+σ·√V_t·dW_t (5) In equation (5), V_t represents the instantaneous variance at time t, κ represents the mean recovery rate of volatility, θ represents the long-term variance mean, σ represents the variance volatility, W_t represents the Brownian motion corresponding to the variance process, dW_t represents the random increment of the Brownian motion, dt represents the time infinitesimal, and dV_t represents the random derivative of the instantaneous variance. The drift term κ·(θ) of the instantaneous variance V_t... V_t)·dt causes the instantaneous variance to revert to the long-term variance mean θ at a reversion rate κ. The diffusion term σ·√V_t·dW_t causes the instantaneous variance to fluctuate randomly around the mean-reverting path. The instantaneous variance is always non-negative, and the volatility exhibits both clustering and mean-reverting characteristics. Once the random volatility process of asset prices is determined, the diffusion intensity of asset prices changes with the instantaneous variance.

[0030] In one implementation, the instantaneous variance remains non-negative when the variance process parameters satisfy 0 < σ ≤ √(2·κ·θ), where σ represents variance volatility, κ represents the mean recovery rate of volatility, θ represents the long-term variance mean, and √ represents the square root operation. The above inequality is the parameter consistency condition of the CIR variance process, which avoids the instantaneous variance from becoming negative and causing the pricing equation to lose its meaning.

[0031] Under the physical measure, the dynamics of asset prices after introducing stochastic volatility can be expressed as dS_t = μ·S_t·dt + √V_t·S_t·dB_t, where μ represents the asset return drift rate under the physical measure, S_t represents the asset price, B_t represents the Brownian motion corresponding to the asset price, and dS_t represents the stochastic derivative of the asset price. The diffusion intensity of asset prices varies with the square root of the instantaneous variance, which is determined by equation (5). The two constitute the dynamics of the stochastic volatility model under the physical measure. Option pricing is conducted under the risk-neutral measure, and the drift rate in the asset price dynamics under the risk-neutral measure is taken as the risk-free interest rate r.

[0032] In some embodiments, step 03 above includes: Obtain the retracement viscosity coefficient and the upward viscosity coefficient; Based on the retracement viscosity coefficient, the duration of the retracement process at the retracement boundary is adjusted to obtain the viscous retracement process; By adjusting the duration of the upward movement at the upward boundary based on the upward viscosity coefficient, a viscous upward movement process can be obtained.

[0033] Specifically, the drawdown stickiness coefficient is denoted as ξ, which controls the strength of the asset price's stay at the boundary of its maximum operating value. The uptrend stickiness coefficient is denoted as η, which controls the strength of the asset price's stay at the boundary of its minimum operating value. The drawdown boundary of the drawdown process is located at the boundary value 1 of the drawdown process, that is, the position where the risk-adjusted price equals the maximum operating value. The uptrend boundary of the uptrend process is located at the boundary value 1 of the uptrend process, that is, the position where the risk-adjusted price equals the minimum operating value.

[0034] An ordinary stochastic process has zero dwell time at the boundary and leaves immediately after reaching the boundary. To achieve sustained dwell at the boundary, a time transformation is performed on the pullback and rise processes using a random clock, converting the accumulated dwell time at the boundary into sustained dwell time in real time. The accumulated dwell time of the pullback process at boundary 1 is described by the local time L^1_t(D) of the pullback process at boundary 1, and the accumulated dwell time of the rise process at boundary 1 is described by the local time L^1_t(U) of the rise process at boundary 1. The local time is used to measure the accumulated dwell time of the stochastic process at the boundary position. The random clock can be represented as: R(t)=t+ξ·L^1_t(D)+η·L^1_t(U) (6) In equation (6), R(t) represents the random clock, t represents the real time, ξ represents the retracement viscosity coefficient, η represents the upward viscosity coefficient, L^1_t(D) represents the local time of the retracement process D at boundary 1, and L^1_t(U) represents the local time of the upward process U at boundary 1. The random clock is superimposed with the local time at the boundary on the real time. The larger ξ is, the greater the contribution of the local time accumulated at boundary 1 in the retracement process to the random clock, and the longer the price stays at the maximum value boundary. The larger η is, the greater the contribution of the local time accumulated at boundary 1 in the upward process to the random clock, and the longer the price stays at the minimum value boundary. When ξ is 0, the retracement process does not exhibit viscous dwell. When η is 0, the upward process does not exhibit viscous dwell. When η is positive, the corresponding boundary produces a viscous effect.

[0035] By applying an inverse transformation of the random clock to the retracement and upward movements, we obtain sticky retracement and sticky upward movements. In sticky movements, clock time is consumed more than physical time at the boundaries, thus lengthening the time the price spends along the boundaries in physical time. The accumulated dwell time at the boundaries is proportional to the local time, with the proportionality coefficients being the retracement stickiness coefficient ξ and the upward stickiness coefficient η, respectively. The sustained dwell at the retracement boundary in a sticky retracement process maintains the state of continuously making new price highs in the model, while the sustained dwell at the upward boundary in a sticky upward process maintains the state of continuously making new price lows in the model. The retracement stickiness coefficient and the upward stickiness coefficient can be set by the model user based on the market characteristics of the underlying asset, or they can be determined during the calibration phase based on market option prices.

[0036] In some implementations, the target neural network is trained based on state equations and contract parameters, including: Obtain training samples; The training samples are input into the initial neural network to obtain approximate option prices; Determine the pricing residual based on the state equation and the approximate option price; The parameters of the initial neural network are updated based on the pricing residuals to obtain the target neural network.

[0037] Specifically, training samples are used to train an initial version of the target neural network, denoted as the initial neural network. During the training phase, sampling rules are determined in the state and parameter domains, covering the possible states and parameter ranges of the option to be priced. The system extracts samples according to the sampling rules in the internal region, the maximum operating value boundary, the minimum operating value boundary, and the expiration terminal region. The maximum operating value boundary corresponds to the position where the asset price equals the maximum operating value, the minimum operating value boundary corresponds to the position where the asset price equals the minimum operating value, and the expiration terminal region corresponds to the position at the option's expiration time. Training samples do not rely on manually labeled option prices; the option price at the sample point is determined by the pricing constraints corresponding to the state equation.

[0038] Training samples are input into an initial neural network to obtain approximate option prices. The overall option price function is approximated by the neural network. The input of the neural network includes time, state variables, and contract parameters. The neural network outputs an approximate option price, denoted as P^. , This represents all trainable parameters of the neural network. The parameters of the initial neural network are initialized before training.

[0039] Based on the state equation and the approximate option price, the pricing residual is determined. The pricing residual measures the degree of deviation of the approximate option price from the pricing constraints corresponding to the state equation. The state equation determines the partial differential equation for pricing that the option price function satisfies in its internal region, the boundary conditions satisfied at the maximum and minimum operating boundaries, and the terminal conditions satisfied at expiration. Substituting the approximate option price and its corresponding partial derivatives into these equations and conditions yields the difference, which is the pricing residual. The smaller the pricing residual, the closer the approximate option price is to the option price function corresponding to the state equation.

[0040] The target neural network is obtained by updating the parameters of the initial neural network based on the pricing residuals. This means that the pricing residuals are combined into a training objective, and the parameters of the initial neural network are updated along the descent direction of the objective. After the parameter update, the option price output by the network is closer to the pricing constraint corresponding to the state equation. The target neural network is obtained when the training stopping condition is met.

[0041] In some implementations, the pricing residual is determined based on the state equation and the option price approximation, including: Based on the state equation, determine the partial differential equation for pricing that the option price function satisfies; Based on the dwell time of the sticky retracement process at the retracement boundary, determine the first boundary condition of the option price function at the retracement boundary; Based on the dwell time of the sticky upward process at the upward boundary, determine the second boundary condition of the option price function at the upward boundary; Based on the payoff of the unpriced option at expiration, determine the terminal condition of the option price function at expiration; The partial derivatives of the approximate option price with respect to the remaining term of the option and the state variables are calculated using automatic differentiation. Substitute the approximate option price and its partial derivative into the pricing partial differential equation, the first boundary condition, the second boundary condition, and the terminal condition to calculate the pricing residual.

[0042] Specifically, the state variables include the current asset price, the maximum operating value, the minimum operating value, and the current variance, denoted as x, y, z, and v, respectively. Here, x represents the current asset price, y represents the maximum operating value, z represents the minimum operating value, and v represents the current instantaneous variance. Time is denoted as t. The option price function is denoted as P(t,x,y,z,v;Φ), where Φ represents the set of parameters consisting of model parameters and contract parameters. Taking a European call option as an example, the strike price is denoted as K, the expiration date is T, and the payoff of the European call option at expiration is (x... K)^+, the symbol (·)^+ means taking the larger value between the value inside the parentheses and 0.

[0043] Under a risk-neutral measure, the theoretical price of a European call option is equal to the conditional expectation of the discounted option payoff at expiration, where E^Q represents the conditional expectation under the risk-neutral measure Q. The option price function satisfies the pricing partial differential equation in the internal region, satisfies the first boundary condition at the maximum boundary, satisfies the second boundary condition at the minimum boundary, and satisfies the terminal condition at expiration. The option price function is determined by the above equations and conditions.

[0044] The state equation upon which the option price function depends, under a risk-neutral measure, uses the current asset price, the maximum operating value, the minimum operating value, and the instantaneous variance as state variables. The drift rate in the asset price dynamics is taken as the risk-free interest rate r. The diffusion intensity of the asset price varies with the square root of the instantaneous variance. The maximum operating value is updated when the asset price reaches and exceeds the current record, and the minimum operating value is updated when the asset price reaches and falls below the current record. The constraints of the sticky retracement process and the sticky rise process on the dwell at the extreme value are entered into the state equation through local time terms. The state equation gives the evolution relationship of the state variables upon which the option price function depends.

[0045] The option price function satisfies the pricing partial differential equation in the internal region 0<z<x<y and v>0, where the internal region is the region where the asset price is between the running minimum value and the running maximum value and the variance is greater than 0.

[0046] According to the state equation, the pricing partial differential equation satisfied by the option price function is determined, and the pricing partial differential equation satisfied by the option price function in the internal region can be expressed as: P_t+r·x·P_x+(1 / 2)·v·x²·P_xx+ρ·σ·v·x·P_xv+(1 / 2)·σ²·v·P_vv+κ·(θ v)·P_v+r·y·P_y+r·z·P_z r·P=0 (7) In formula (7), P represents the option price function P(t,x,y,z,v;Φ), P_t represents the first-order partial derivative of the option price function with respect to time t, r·x·P_x represents the asset price drift term, x represents the current asset price, v represents the current instantaneous variance, P_xx represents the second-order partial derivative of the option price function with respect to the current asset price, (1 / 2)·v·x²·P_xx represents the asset price diffusion term, ρ represents the correlation coefficient between price and volatility, σ represents the variance volatility, P_xv represents the mixed partial derivative of the option price function with respect to the current asset price and the current variance, ρ·σ·v·x·P_xv represents the correlation term between price and variance, P_vv represents the second-order partial derivative of the option price function with respect to the current variance, (1 / 2)·σ²·v·P_vv represents the variance diffusion term, κ represents the mean reversion speed of volatility, θ represents the long-term variance mean, P_v represents the first-order partial derivative of the option price function with respect to the current variance, κ·(θ v)·P_v represents the variance mean reversion term, y represents the running maximum value, P_y represents the first-order partial derivative of the option price function with respect to the running maximum value, r·y·P_y represents the growth term of the running maximum value, z represents the running minimum value, P_z represents the first-order partial derivative of the option price function with respect to the running minimum value, r·z·P_z represents the growth term of the running minimum value, r represents the risk-free interest rate, and r·P represents the discount term. The pricing partial differential equation integrates asset price diffusion, price-variance correlation, stochastic variance volatility and mean reversion, and the growth of running extreme value state variables into the same equation, and the solution of the equation corresponds to the option price function under the state equation.

[0047] The first boundary condition of the option price function at the retracement boundary is determined by the dwell time of the sticky retracement process at the retracement boundary. The retracement boundary corresponds to the position where the asset price equals the maximum value, i.e., x=y. The first boundary condition satisfied by the option price function at x=y is determined by the dwell time of the sticky retracement process at the maximum value boundary. The larger the retracement stickiness coefficient ξ, the stronger the influence of the maximum value boundary on the option price function. The first boundary condition characterizes the effect of the price remaining at the maximum value on the option price.

[0048] Based on the dwell time of the sticky upward process at the upward boundary, the second boundary condition of the option price function at the upward boundary is determined. The upward boundary corresponds to the position where the asset price equals the operating minimum, i.e., x=z. The second boundary condition satisfied by the option price function at x=z is determined by the dwell time of the sticky upward process at the operating minimum boundary. The larger the upward stickiness coefficient η, the stronger the influence of the operating minimum boundary on the option price function. The second boundary condition characterizes the effect of the price remaining at the operating minimum on the option price.

[0049] Based on the payoff of the unpriced option at expiration, determine the terminal condition of the option price function at expiration. The terminal condition can be expressed as: P(T,x,y,z,v;Φ)=(x K)^+ (8) In equation (8), T represents the maturity date, x represents the asset price at maturity, K represents the exercise price, (x K)^+ represents the payoff of a European call option at expiration, and P(T,x,y,z,v;Φ) represents the value of the option price function at expiration. The terminal condition stipulates that the option price function equals the option payoff at expiration, the option price function internally satisfies the pricing partial differential equation, satisfies the boundary conditions at the boundaries, and satisfies the terminal condition at expiration. These three conditions collectively constrain the option price function.

[0050] The partial derivatives of the approximate option price P^ are calculated using automatic differentiation. Automatic differentiation involves taking the derivative layer by layer with respect to the function constructed by the neural network according to the computational graph. Automatic differentiation can calculate the approximate option price P^. The first-order partial derivative with respect to the remaining time of the option, as well as the first-order partial derivatives with respect to the current asset price, the maximum value, the minimum value, and the current variance, are calculated. The second-order partial derivatives and mixed partial derivatives of the approximate option price with respect to the current asset price and the current variance are also calculated. The remaining time of the option is the difference between the expiration date and the current time. The partial derivative of the approximate option price with respect to the remaining time of the option corresponds to the time partial derivative term in the pricing partial differential equation. The existence of each order of partial derivative is guaranteed by the differentiability of the neural network.

[0051] The pricing residuals are calculated by substituting the approximate option price and its partial derivatives into the pricing partial differential equation, the first boundary condition, the second boundary condition, and the terminal condition. The internal residuals, which measure the deviation of the approximate option price from the pricing partial differential equation within the internal region, are then substituting the approximate option price into the first boundary condition, which measures the deviation of the approximate option price from the first boundary condition at the drawdown boundary. The second boundary residuals, substituting the approximate option price into the second boundary condition, and finally, the terminal residuals, are also substituting the internal, first, second, and terminal residuals. These internal, first, and second boundary residuals, along with the terminal residuals, constitute the pricing residuals. During training, the pricing residuals are used to constrain the neural network output to approximate the internal pricing partial differential equation, the two sticky boundary conditions, and the payoff condition.

[0052] In some implementations, the state variables and contract parameters of the option to be priced are input into a target neural network to obtain the theoretical price of the option, including: Determine the remaining term of the option to be priced based on the contract parameters; The remaining time limit, state variables, model parameters of the state equation, and contract parameters are normalized to a preset numerical range to obtain normalized data. The normalized data is input into the target neural network, and gating processing is performed layer by layer to obtain the output layer result. Based on the output layer results, determine the theoretical price of the option.

[0053] Specifically, the remaining term of the option to be priced is determined based on the contract parameters, including the expiration date T and the remaining term denoted as τ, where τ = T. t and τ represent the time remaining from the option's expiration date, with the remaining time determining the approximate time position of the option price. State variables include the current asset price x, the maximum running value y, the minimum running value z, and the current instantaneous variance v. Model parameters for the state equation include the price-volatility correlation coefficient ρ, the mean volatility reversion rate κ, the long-term variance mean θ, the variance volatility σ, the drawdown stickiness coefficient ξ, and the upside stickiness coefficient η. Contract parameters also include the strike price K and the option type. The risk-free interest rate r is determined by the interest rate parameter. The input vector of the target neural network contains the remaining time, four state variables, model parameters, the strike price, and the risk-free interest rate; the input vector has a dimension of thirteen.

[0054] The remaining maturity, state variables, model parameters of the state equation, and contract parameters are normalized to a preset numerical range to obtain normalized data. Since the remaining maturity, state variables, model parameters, and contract parameters have different value ranges, and price parameters and interest rate parameters have different dimensions, directly inputting them into the neural network would cause an imbalance in the contributions of different dimensions to network training. Normalization linearly maps each dimension to a preset numerical range, which can take any of the following values: The closed interval from 1 to 1, the normalized data remain comparable across different dimensions, and all input variables are sampled and normalized in the parameter domain and state domain during training.

[0055] The normalized data is input into the target neural network, and gating processing is performed layer by layer. The normalized data constitutes the input vector, denoted as . The target neural network consists of multiple gated network layers, denoted as L. The hidden state of the Xi-th layer is denoted as S^Xi, where Xi ranges from 1 to L. After gating the Xi-th layer, the hidden state of the (Xi+1)-th layer is obtained. The gating process can be represented as: S^{Xi+1}=(1 G^Xi)⊙H^Xi+Z^Xi⊙S^Xi (9) In Equation (9), S^Xi represents the hidden state of layer Xi, G^Xi represents the forget gate of layer Xi, H^Xi represents the candidate state of layer Xi, Z^Xi represents the update gate of layer Xi, ⊙ represents element-wise multiplication, and S^{Xi+1} represents the hidden state of layer Xi+1. The update gate controls the retention ratio of the hidden state information of the previous layer, and the forget gate and candidate state determine the newly written candidate information. The gating structure selects to retain state information between layers, so that the deep network remains stable in the approximation of the high-dimensional pricing function.

[0056] The update gate and forget gate of layer Xi are represented as follows: Z^Xi=σ( ·U_z^Xi+S^Xi·W_z^Xi+b_z^Xi), G^Xi=σ( ·U_g^Xi+S^Xi·W_g^Xi+b_g^Xi) (10) In equation (10), Let U_z^Xi represent the input vector, W_z^Xi represent the weight matrix of the Xi-th update gate with respect to the input vector, b_z^Xi represent the weight matrix of the Xi-th update gate with respect to the hidden state, and b_z^Xi represent the bias of the Xi-th update gate. U_g^Xi, W_g^Xi, and b_g^Xi represent the weight matrix of the Xi-th forget gate with respect to the input vector, the weight matrix with respect to the hidden state, and the bias, respectively. σ represents the activation function.

[0057] The update gate and forget gate take the input vector and the hidden state of the previous layer as input, and obtain them after linear transformation and activation function. The input vector participates in the gating calculation repeatedly in each network layer, so that the input information is preserved and passed on in each layer.

[0058] The reset gate and candidate state of layer Xi are represented as follows: R^Xi=σ( ·U_r^Xi+S^Xi·W_r^Xi+b_r^Xi), H^Xi=σ( ·U_h^Xi+(S^Xi⊙R^Xi)·W_h^Xi+b_h^Xi) (11) In Equation (11), U_r^Xi, W_r^Xi, and b_r^Xi represent the weight matrix of the reset gate in layer Xi with respect to the input vector, the weight matrix of the reset gate with respect to the hidden state, and the bias, respectively. R^Xi represents the reset gate in layer Xi. U_h^Xi, W_h^Xi, and b_h^Xi represent the weight matrix of the candidate state in layer Xi with respect to the input vector, the weight matrix of the candidate state with respect to the hidden state, and the bias, respectively. H^Xi represents the candidate state in layer Xi. ⊙ represents element-wise multiplication, and σ represents the activation function. The reset gate controls the part of the hidden state in the previous layer that participates in the calculation of the candidate state. The candidate state is generated with the input vector and the reset hidden state as input. The candidate state carries the new information to be written. The hidden state is updated layer by layer under the control of the gated sublayer.

[0059] The final gating layer provides the input to the output layer. The output layer performs a linear transformation on the last hidden state to obtain the output result, which is a single numerical value, i.e., the approximate option price P^. The theoretical option price is determined based on the output layer results, and the theoretical option price is the price value corresponding to the output layer results. The activation function used in each layer of the network can be the hyperbolic tangent (tanh). The hyperbolic tangent function provides the network with non-linear expressive power and keeps the output of each layer within a finite range, which is beneficial to the stability of network training.

[0060] In some implementations, updating the parameters of the initial neural network based on the pricing residuals to obtain the target neural network includes: Construct a loss function based on the pricing residual; An adaptive moment estimation algorithm is used to update the parameters of the initial neural network according to a piecewise decaying learning rate; When the preset number of iterations is reached or the loss function is lower than the first preset threshold, the parameters of the initial neural network are stopped from being updated, and the target neural network is obtained.

[0061] Specifically, the pricing residuals include internal residuals, first boundary residuals, second boundary residuals, and terminal residuals. Internal residuals correspond to the pricing partial differential equation; first boundary residuals correspond to the first boundary condition at the retracement boundary; second boundary residuals correspond to the second boundary condition at the upward boundary; and terminal residuals correspond to the terminal condition at maturity. Constructing a loss function based on the pricing residuals involves combining the four types of residuals according to preset weights into a unified loss function, which can be expressed as: J=w_1·J_in+w_2·J_bcy+w_3·J_bcz+w_4·J_te (12) In equation (12), J_in represents the internal residual term, which is determined by the residual of the option price approximation relative to the pricing partial differential equation at the internal region sample points; J_bcy represents the first boundary residual term, which is determined by the residual of the maximum value boundary sample points relative to the first boundary condition; J_bcz represents the second boundary residual term, which is determined by the residual of the minimum value boundary sample points relative to the second boundary condition; J_te represents the terminal residual term, which is determined by the residual of the terminal sample points relative to the terminal condition; w_1, w_2, w_3, and w_4 represent the preset weights of the internal residual term, the first boundary residual term, the second boundary residual term, and the terminal residual term, respectively; and J represents the loss function. In a preferred embodiment, the weights of the internal residual term, the first boundary residual term, the second boundary residual term, and the terminal residual term are all set to one-quarter, and the loss function is the arithmetic mean of the four types of residual terms. The loss function also constrains the approximate option price to satisfy the internal pricing partial differential equation, two sticky boundary conditions, and the payoff condition at maturity, so that training does not rely on a large number of manually labeled prices.

[0062] An adaptive moment estimation (Adam) algorithm is employed to update the initial neural network parameters using a piecewise decaying learning rate. Adam (Adaptive Moment Estimation, an optimization algorithm combining first-order and second-order momentum to adaptively adjust the parameter update step size) estimates momentum based on the gradient of the parameters using the loss function and adaptively adjusts the learning step size for each parameter. The piecewise decaying learning rate involves using a larger learning rate in the early stages of training to quickly reduce the loss, and gradually decreasing the learning rate in the later stages to improve convergence accuracy. The learning rate decays at preset proportions at predetermined training stage nodes. The parameters are updated along the descent direction of the loss function, resulting in an updated option price output by the neural network that more closely approximates the pricing constraints.

[0063] When the preset number of iterations is reached or the loss function falls below the first preset threshold, the parameters of the initial neural network are stopped, and the target neural network is obtained. The preset number of iterations is the maximum number of parameter update rounds allowed during training, and the first preset threshold is the target level of the loss function. Training ends when the preset number of iterations is reached or the loss function falls below the first preset threshold. The updated network parameters are saved, and the target neural network is obtained. The target neural network can perform fast forward computation on the input and output the theoretical price of the option.

[0064] In some embodiments, the above method further includes: Initialize the model parameters to be calibrated, including the mean volatility recovery rate, the long-term variance mean, the variance volatility, the price-volatility correlation coefficient, the drawdown stickiness coefficient, and the upside stickiness coefficient. Initialize the state parameters to be calibrated, which include instantaneous variance, running maximum value, and running minimum value; Obtain market option quotes; The model parameters to be calibrated, the state parameters to be calibrated, and the contract parameters are input into the target neural network. Forward propagation is performed on each option corresponding to the market option quote to obtain the model price of each option. Determine the market price residual based on the model price and market option quotes; Based on the market price residual, the model parameters and state parameters to be calibrated are iteratively updated to obtain the calibrated model parameters and calibrated state parameters.

[0065] Specifically, before the model is used for actual market pricing, its parameters and state parameters need to be determined based on market option quotes to ensure that the model price aligns with the market price. The model parameters to be calibrated include the mean volatility reversion rate κ, the long-term variance mean θ, variance volatility σ, the price-volatility correlation coefficient ρ, the drawdown stickiness coefficient ξ, and the upside stickiness coefficient η. The state parameters to be calibrated include the instantaneous variance v, the running maximum y, and the running minimum z. The running maximum and minimum are included in the calibration to allow for differences between the belief in historical running maximums and minimums under a risk-neutral measure and historical extremes under a physical measure. Each parameter to be calibrated is initialized before iteration, and these parameters constitute the parameter vector φ.

[0066] Obtain market option quotes. Market option quotes come from options that are actually traded or quoted in the market on a specified trading day. Market option quotes correspond to multiple option contracts. Each option contract carries contract parameters such as strike price, expiration time and contract type. Market option quotes can be obtained from the market information system or the quotation system.

[0067] The model parameters, state parameters, and contract parameters to be calibrated are input into the target neural network. Forward propagation is performed on each option corresponding to the market option quote to obtain the model price for each option. The network parameters of the target neural network remain unchanged during forward propagation. The target neural network acts as a fast price maker, directly outputting the price for the input combination. The price of the underlying asset on a specified trading day is denoted as x_s, the strike price of the i-th option is denoted as K_{s,i}, the expiration time of the i-th option is denoted as T_{s,i}, the market quote of the i-th option is denoted as P^MKT_{s,i}, and the set of parameters to be calibrated is denoted as φ.

[0068] Based on the model price and the market option quote, the market price residual is determined. The market price residual of the i-th option can be expressed as: r_i(φ)=P^ _i(φ) P^MKT_{s,i} (13) In equation (13), P^ _i(φ) represents the model price of the i-th option under the parameter set φ to be calibrated. The model price is obtained by performing a forward propagation of the target neural network on the input of the i-th option. P^MKT_{s,i} represents the market price of the i-th option, r_i(φ) represents the market price residual of the i-th option, and s represents the trading day number. The market price residual measures the difference between the model price and the market price. The market price residuals of all options constitute the residual vector r(φ). The calibration objective is to minimize the sum of squares of the elements of the residual vector. Market parameter calibration is transformed into a nonlinear least squares problem. Compared with traditional calibration methods, each candidate parameter evaluation only calls the target neural network to perform a forward propagation, without needing to re-execute Monte Carlo simulations or re-solve partial differential equations.

[0069] Based on the market price residual, the model parameters and state parameters to be calibrated are iteratively updated to obtain the calibrated model parameters and calibrated state parameters. This means that the parameters to be calibrated are repeatedly adjusted under the calibration target to gradually reduce the market price residual. The parameter values ​​corresponding to the iteration stop are the calibrated model parameters and calibrated state parameters.

[0070] In some implementations, the model parameters and state parameters to be calibrated are iteratively updated based on the market price residual, including: Calculate the Jacobian matrix of the market price residual with respect to the model parameters and state parameters to be calibrated; The direction of parameter update is determined based on the Jacobian matrix and the damping factor; Candidate parameters are determined based on the parameter update direction; Adjust the damping factor based on the relationship between the candidate parameters and the changes in market price residuals; When the change in market price residual is less than the second preset threshold or the preset number of iterations is reached, the iteration stops, and the calibrated model parameters and calibrated state parameters are obtained.

[0071] Specifically, the iterative update employs a damped nonlinear least squares algorithm, specifically the Levenberg-Marquardt algorithm (a nonlinear least squares optimization algorithm that switches between Gauss-Newton's method and gradient descent by using a damping factor). The Jacobian matrix of the market price residuals with respect to the model parameters and state parameters to be calibrated is calculated. Each element of the Jacobian matrix is ​​the partial derivative of an element in the market price residual vector with respect to a parameter in the parameter set φ to be calibrated. The target neural network is differentiable with respect to its parameters, and the Jacobian matrix can be obtained through automatic differentiation or numerical differencing. The Jacobian matrix is ​​denoted as J.

[0072] Based on the Jacobian matrix and damping factor, the parameter update direction is determined, and the damped normal equation can be expressed as: (J^T·J+λ·I)·δ= J^T·r (14) In equation (14), J represents the Jacobian matrix of the market price residual with respect to the parameter to be calibrated, J^T represents the transpose of the Jacobian matrix, λ represents the damping factor (a real number greater than 0), I represents the identity matrix, r represents the market price residual vector, δ represents the parameter update direction, and J^T·r represents the gradient direction. When the damping factor is small, the normal equation is close to the Gauss-Newton method, and the convergence speed is fast. When the damping factor is large, the normal equation is close to the gradient descent method, the update step size is limited, and the algorithm stability is enhanced. The parameter update direction δ is obtained by solving equation (14).

[0073] Based on the parameter update direction, candidate parameters are determined. These candidate parameters are obtained by adjusting the current parameter to be calibrated along the parameter update direction, and are denoted as φ_new = φ + δ. Based on the relationship between the candidate parameters and the changes in the market price residuals, the damping factor is adjusted. The candidate parameters are then substituted into the target neural network to calculate the market price residuals under the candidate parameters. The change in the market price residuals is the difference between the sum of squares of the residuals under the candidate parameters and the sum of squares of the residuals under the current parameters. When a candidate parameter decreases the market price residuals, it is accepted as a new parameter to be calibrated, and the damping factor is reduced, making the next update closer to the Gauss-Newton method. When a candidate parameter increases the market price residuals, it is rejected, and the damping factor is increased, making the next update closer to the gradient descent method and enhancing iterative stability.

[0074] When the change in market price residuals is less than the second preset threshold or the preset number of iterations is reached, iteration stops, and the calibrated model parameters and calibrated state parameters are obtained. The second preset threshold is the target level of change in market price residuals, and the preset number of iterations is the maximum number of update rounds allowed for calibration. Iteration also stops when the update magnitude of the parameters to be calibrated is lower than the preset update magnitude threshold. When iteration stops, the current parameters to be calibrated are retained as the calibration result. The calibrated parameter set is matched with market option quotes, and the calibrated model parameters and calibrated state parameters are used for subsequent option pricing and risk management.

[0075] This application also provides a stochastic volatility model option pricing system, which includes a processor and a memory. The memory stores a computer program, and the processor executes the computer program to implement the steps of the above-mentioned stochastic volatility model option pricing method.

[0076] To verify the effectiveness of the above technical solution, numerical experiments and empirical verification were conducted on the method of this embodiment in one example. The numerical experiment used a European call option as the pricing object. The target neural network adopted a 4-layer gated network, with each layer having a hidden state dimension of 110. The activation function was the hyperbolic tangent function, and the output layer output a one-dimensional option price. Training samples were uniformly sampled in the state domain and parameter domain: the strike price was sampled between 50 and 131, the risk-free interest rate was sampled between 0.01 and 0.3, and the correlation coefficient between price and volatility was... Sampling is performed between 1 and 1; the mean volatility recovery rate is sampled between 0.01 and 5; the mean long-term variance is sampled between 0.01 and 1; variance volatility, drawdown viscosity, and upside viscosity are sampled between 0.01 and 10, respectively; and the time to maturity is sampled between 7 / 365 and 1.1 years. The sampling satisfies the Feller condition (2·κ·θ>σ²) to ensure non-negativity of instantaneous variance. All input variables are linearly normalized to... The training uses an adaptive moment estimation algorithm, with the learning rate decreasing in segments from 0.001 to 0.000001 according to preset stage nodes, and the number of iterations is up to 6000.

[0077] Pricing accuracy and efficiency verification: Monte Carlo simulation based on continuous-time Markov chain approximation was used as the pricing benchmark, with a step size of 0.01 and 60,000 sample paths. Under 1219 different parameter combinations, the average percentage error of the theoretical option price output by the target neural network relative to the benchmark pricing was 0.3596%, and the average absolute error was 0.02734. Regarding pricing time, on a workstation equipped with a graphics processor, the target neural network took approximately 0.018 seconds to perform one forward calculation for a single option, while the Monte Carlo simulation, with 60,000 sample paths, took approximately 1241 seconds to price an option with a remaining maturity of 0.3 years and approximately 469 seconds to price an option with a remaining maturity of 0.1 years. The method in this embodiment reduces the pricing time of a single option by more than four orders of magnitude, providing the theoretical option price within milliseconds.

[0078] Parameter calibration and verification: For multiple pre-set sets of model parameters, option price samples were generated using continuous-time Markov chains for approximate simulation and calibration was performed. The average absolute errors between the calibrated parameters and the actual parameters were as follows: price-volatility correlation coefficient 0.0156, mean volatility recovery rate 0.0651, long-term variance mean 0.00834, variance volatility 0.0199, upward stickiness coefficient 0.0407, instantaneous variance 0.00166, drawdown stickiness coefficient 0.0320, drawdown ratio 0.0110, and upward ratio 0.0781. All parameters to be calibrated could be accurately recovered. The average calibration time was approximately 164.2 milliseconds, meeting the timeliness requirements for online calibration.

[0079] Empirical Verification: Using European call option market data for the S&P 500 index in 2021 and 2022 (approximately 2.97 million quote records after liquidity screening and no-arbitrage constraint filtering), the model in this embodiment was compared with the Heston model, a comparative model including only sticky retracement processes, and a comparative model including only sticky upward processes. The average percentage error of in-sample pricing was as follows: In 2021, the model in this embodiment was 0.5380%, the sticky retracement-only model was 0.6610%, the sticky upward-only model was 0.6960%, and the Heston model was 0.7410%; In 2022, the model in this embodiment was 0.6970%, the sticky upward-only model was 0.8250%, the sticky retracement-only model was 0.8710%, and the Heston model was 1.1140%. The model in this embodiment achieved the lowest pricing error in both years, and the improvement in error compared to the Heston model was most significant during market phases where consecutive new highs and lows frequently occurred, verifying the necessity of simultaneously introducing sticky retracement and sticky upward processes.

[0080] The above experimental verification shows that, while depicting the price's sustained stay at historical extremes and the random changes in volatility, the pricing error of this embodiment is measurable and controllable within 0.4% in simulation verification and within 1.2% in empirical verification. The pricing time for a single option is on the order of milliseconds, and the time for a single parameter calibration is on the order of sub-seconds. The technical effect is clear and verifiable, and it can meet the accuracy and timeliness requirements of online pricing, batch valuation, and market making scenarios.

[0081] Please see Figure 2 As shown, Figure 2 This is a schematic diagram of the structure of a stochastic volatility model option pricing system provided in an embodiment of the present invention. In one embodiment, the system includes a market data access module, a state variable construction module, a model calculation module, a neural network pricing module, and a pricing output module connected in sequence. A constraint generation module is connected between the model calculation module and the neural network pricing module. A training and optimization module is connected to both the neural network pricing module and the constraint generation module. A parameter calibration module is connected to both the neural network pricing module and the pricing output module. A model validation module is connected to the pricing output module. A model management module is connected to both the market data access module and the training and optimization module.

[0082] The market data access module reads underlying asset price data, option quotes, strike prices, expiration dates, and risk-free interest rates from market data systems, trading systems, or databases. It performs standardized formatting, handles missing values, and filters out abnormal quotes, eliminating option samples with erroneous quotes, insufficient liquidity, or violations of no-arbitrage constraints. The state variable construction module calculates the risk-adjusted price, maximum running value, minimum running value, drawdown process, upward process, and initial variance estimate, organizing the calculation results into a model state vector.

[0083] The model computation module maintains the model state based on the viscous retracement process, viscous upward process, and stochastic volatility process, and approximates the sample path during the validation phase using a continuous-time Markov chain (a stochastic process with no aftereffect). The constraint generation module generates pricing partial differential equations, viscous boundary conditions, and maturity conditions based on the state equation, and provides a residual calculation interface to the training and optimization module.

[0084] The neural network pricing module constructs a deep neural network with thirteen input dimensions, a gated hidden layer, and a single-dimensional price as the output. It uses automatic differentiation to calculate the required partial derivatives, performs forward propagation on the input combination, and outputs the theoretical option price. The training and optimization module samples in the state and parameter domains, calculates a uniform loss function, updates the network parameters using the Adam algorithm, and saves the trained pricing network.

[0085] The parameter calibration module reads market option quotes, calls the trained pricing network to perform fast pricing, and uses a nonlinear least squares method to obtain model parameters and state parameters that match the market quotes. The pricing output module provides online pricing, batch valuation, error reporting, and risk sensitivity calculation interfaces for the business system. Batch valuation can utilize the parallel computing capabilities of a graphics processing unit (GPU) to complete a large number of option valuations in a short time, meeting the needs of portfolio valuation, market making, and risk management scenarios.

[0086] The target neural network is differentiable. The risk sensitivity calculation interface uses automatic differentiation to calculate the partial derivatives of the theoretical option price with respect to the underlying asset price, volatility, and other input quantities, thereby obtaining risk sensitivity indicators such as Delta, Gamma, and Vega. Delta represents the first-order partial derivative of the theoretical option price with respect to the underlying asset price, Gamma represents the second-order partial derivative of the theoretical option price with respect to the underlying asset price, and Vega represents the first-order partial derivative of the theoretical option price with respect to volatility. The risk sensitivity indicators are used for hedging transactions and risk limit management.

[0087] The model validation module is used to evaluate the pricing accuracy of the neural network and compare the pricing performance of the model with the Heston model (a pricing model that describes stochastic volatility using the CIR variance process), a comparative model that only includes a sticky retracement process, and a comparative model that only includes a sticky uptrend process. The model validation module is also used to output visualized data such as implied volatility (IV) curves, pricing errors, calibration parameter time series, and comparison results with different models, to help business personnel understand the model's performance under different market conditions.

[0088] The model management module is used to save network parameters at different training stages and supports model version updates and rollbacks in the production environment.

[0089] In one implementation, this method is applied to the theoretical pricing of option contracts. The underlying asset price data can be taken from the historical price series of the underlying asset. The interest rate parameter is the risk-free rate used for pricing. The contract parameters are the option contract's strike price, expiration date, and option type. The method outputs the theoretical option price, which is then used in business processes such as trading, quantitative trading, market making, valuation, and risk management. Model building and network training are offline processes, while option pricing and parameter calibration are online or quasi-online processes. Offline training first obtains a target neural network that approximates the state equation corresponding to the option price function. Online calibration then uses the target neural network as a fast pricer to deduce the model parameters and state parameters for the current day or a specified time based on market option quotes.

[0090] In one embodiment, a computer device is provided, which may be a server, and its internal structure diagram may be as follows: Figure 3 As shown, the computer device includes a processor, memory, network interface, and database connected via a system bus. The processor provides computational and control capabilities. The memory includes non-volatile and / or volatile storage media and internal memory. The non-volatile storage media stores the operating system, computer programs, and database. The internal memory provides an environment for the operation of the operating system and computer programs in the non-volatile storage media. The network interface is used to communicate with external clients via a network connection. When the computer program is executed by the processor, it implements the functions or steps of a server-side method for pricing options using a stochastic volatility model.

[0091] In one embodiment, a computer device is provided, which may be a client, and its internal structure diagram may be as follows: Figure 4 As shown, the computer device includes a processor, memory, network interface, display screen, and input devices connected via a system bus. The processor provides computational and control capabilities. The memory includes non-volatile storage media and internal memory. The non-volatile storage media stores the operating system and computer programs. The internal memory provides an environment for the operation of the operating system and computer programs in the non-volatile storage media. The network interface is used to communicate with an external server via a network connection. When the computer program is executed by the processor, it implements the client-side functions or steps of a stochastic volatility model option pricing method.

[0092] It should be noted that the functions or steps that can be implemented by the computer-readable storage medium or computer device described above can be referred to the relevant descriptions on the server side and client side in the foregoing method embodiments. To avoid repetition, they will not be described one by one here.

[0093] Those skilled in the art will understand that all or part of the processes in the methods of the above embodiments can be implemented by a computer program instructing related hardware. The computer program can be stored in a non-volatile computer-readable storage medium. When executed, the computer program can include the processes of the embodiments of the above methods. Any references to memory, storage, databases, or other media used in the embodiments provided in this application can include non-volatile and / or volatile memory. Non-volatile memory may include read-only memory (ROM), programmable ROM (PROM), electrically programmable ROM (EPROM), electrically erasable programmable ROM (EEPROM), or flash memory. Volatile memory may include random access memory (RAM) or external cache memory. By way of illustration and not limitation, RAM is available in a variety of forms, such as static RAM (SRAM), dynamic RAM (DRAM), synchronous DRAM (SDRAM), dual data rate SDRAM (DDRSDRAM), enhanced SDRAM (ESDRAM), synchronous link DRAM (SLDRAM), RAMbus direct RAM (RDRAM), direct memory bus dynamic RAM (DRDRAM), and memory bus dynamic RAM (RDRAM), etc.

[0094] Those skilled in the art will clearly understand that, for the sake of convenience and brevity, the above-described division of functional units and modules is used as an example. In practical applications, the above functions can be assigned to different functional units and modules as needed, that is, the internal structure of the device can be divided into different functional units or modules to complete all or part of the functions described above.

[0095] The above embodiments are only used to illustrate the technical solutions of the present invention, and are not intended to limit it. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some of the technical features. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention, and should all be included within the protection scope of the present invention.

Claims

1. A method for pricing options using a stochastic volatility model, characterized in that, Includes the following steps: Obtain asset price data, interest rate parameters, and contract parameters; Based on the asset price data and the interest rate parameters, determine the asset price retracement process, upward process, and stochastic volatility process; The pullback process and the upward process are subjected to viscosity processing to obtain a viscous pullback process and a viscous upward process; Based on the sticky retracement process, the sticky upward process, and the stochastic volatility process, a state equation is established with asset price and volatility as state variables. The state variables of the option to be priced and the contract parameters are input into the target neural network to obtain the theoretical price of the option. The target neural network is trained based on the state equation and the contract parameters. The step of determining the asset price retracement process, upward process, and stochastic volatility process based on the asset price data and the interest rate parameter includes: The risk-adjusted price is determined based on the interest rate parameters and the asset price data. Based on the risk-adjusted price, determine the maximum and minimum operating values; The drawdown process is determined based on the ratio of the risk-adjusted price to the maximum operating value; The upward trend is determined based on the ratio of the risk-adjusted price to the minimum operating value. The random volatility process is determined through a square root diffusion process based on the preset mean volatility recovery rate, the preset long-term variance mean, and the preset variance volatility.

2. The option pricing method based on a stochastic volatility model according to claim 1, characterized in that, The process of determining stochastic volatility through a square root diffusion process based on a preset mean volatility recovery rate, a preset long-term variance mean, and a preset variance volatility includes: Based on the mean recovery rate of volatility, the preset long-term variance mean, and the preset variance volatility, the stochastic volatility process is determined according to the square root of the instantaneous variance and the Brownian motion corresponding to the variance process. Among them, for the mean recovery rate of volatility κ, the mean long-term variance θ, and the variance volatility σ, σ≤√(2·κ·θ).

3. The option pricing method based on a stochastic volatility model according to claim 1, characterized in that, The step of applying a viscous processing method to the pullback process and the upward process to obtain a viscous pullback process and a viscous upward process includes: Obtain the retracement viscosity coefficient and the upward viscosity coefficient; Based on the retraction viscosity coefficient, the duration of the retraction process at the retraction boundary is adjusted to obtain a viscous retraction process; Based on the rising viscosity coefficient, the duration of the stay at the rising boundary during the rising process is adjusted to obtain a viscous rising process.

4. The option pricing method based on a stochastic volatility model according to claim 1, characterized in that, The target neural network is trained based on the state equation and the contract parameters, including: Obtain training samples; The training samples are input into the initial neural network to obtain approximate option prices; Based on the state equation and the approximate option price, determine the pricing residual; The parameters of the initial neural network are updated based on the pricing residual to obtain the target neural network.

5. The option pricing method based on a stochastic volatility model according to claim 4, characterized in that, The step of determining the pricing residual based on the state equation and the approximate option price includes: Based on the state equation, determine the partial differential equation for pricing that the option price function satisfies; Based on the sticky retracement process's dwell time at the retracement boundary, the first boundary condition of the option price function at the retracement boundary is determined; Based on the dwell time of the sticky upward process at the upward boundary, determine the second boundary condition of the option price function at the upward boundary; Based on the payoff of the option to be priced at expiration, determine the terminal condition of the option price function at expiration; The partial derivatives of the approximate option price with respect to the remaining term of the option and the state variable are calculated by automatic differentiation. The pricing residual is calculated by substituting the approximate option price and the partial derivative into the pricing partial differential equation, the first boundary condition, the second boundary condition, and the terminal condition.

6. The option pricing method based on a stochastic volatility model according to claim 4, characterized in that, The step of inputting the state variables of the option to be priced and the contract parameters into the target neural network to obtain the theoretical price of the option includes: Based on the contract parameters, determine the remaining term of the option to be priced; The remaining term, the state variables, the model parameters of the state equation, and the contract parameters are normalized to a preset numerical range to obtain normalized data. The preset numerical range is a closed interval from negative one to one. The normalized data is input into the target neural network, and gating processing is performed layer by layer to obtain the output layer result. The theoretical price of the option is determined based on the output layer results.

7. The option pricing method based on a stochastic volatility model according to claim 4, characterized in that, The acquisition of training samples includes: Training samples are obtained by sampling in the state domain and parameter domain. The state domain includes an internal region, a maximum operating value boundary, a minimum operating value boundary, and an expiration terminal region. The internal region is the region where the asset price is between the minimum and maximum operating values ​​and the variance is greater than zero. The maximum operating value boundary is the position where the asset price equals the maximum operating value. The minimum operating value boundary is the position where the asset price equals the minimum operating value. The expiration terminal region is the position at the time of option expiration.

8. The option pricing method based on a stochastic volatility model according to claim 1, characterized in that, The method further includes: Initialize the model parameters to be calibrated, which include the mean volatility recovery rate, the long-term variance mean, the variance volatility, the price-volatility correlation coefficient, the drawdown stickiness coefficient, and the upside stickiness coefficient. Initialize the state parameters to be calibrated, which include instantaneous variance, operating maximum value, and operating minimum value; Obtain market option quotes; The model parameters to be calibrated, the state parameters to be calibrated, and the contract parameters are input into the target neural network, and forward propagation is performed on each option corresponding to the market option quote to obtain the model price of each option; The market price residual is determined based on the model price and the market option quote. Based on the market price residual, the model parameters and state parameters to be calibrated are iteratively updated to obtain calibrated model parameters and calibrated state parameters. Specifically, the Jacobian matrix of the market price residual with respect to the model parameters and state parameters to be calibrated is calculated. Based on the Jacobian matrix and the damping factor, the parameter update direction is determined. Based on the parameter update direction, candidate parameters are determined. Based on the relationship between the candidate parameters and the change value of the market price residual, the damping factor is adjusted. When the change value of the market price residual is less than a second preset threshold or reaches a preset number of iterations, the iteration stops, resulting in calibrated model parameters and calibrated state parameters.

9. A stochastic volatility model option pricing system, characterized in that, It includes a processor and a memory, the memory storing a computer program, and the processor executing the computer program to implement the stochastic volatility model option pricing method as described in any one of claims 1 to 8.

10. A computer-readable storage medium, characterized in that, The computer-readable storage medium stores a computer program that, when executed by a processor, implements the steps of the stochastic volatility model option pricing method as described in any one of claims 1 to 8.

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