CN102609879A - Option pricing method and apparatus based on random backward stochastic differential equation - Google Patents

Abstract

The invention discloses an option pricing method and apparatus based on the random backward stochastic differential equation. The method comprises: achieving discreteness of BSDE (backward stochastic differential equation) in a time layer, constructing a time-space discrete grid, and calculating a termination condition for option pricing according to the time-space discrete grid; according to the termination condition, performing Monte Carlo simulation for each space point in each time layer among neighbor time layers, and calculating a final option pricing mathematical desired value by deductive calculation layer by layer according to an isosceles triangle model. According to the option pricing method and apparatus, by using the BSDE mode and based on the isosceles triangle model and powerful parallel computation capabilities of the GPU (graphics processing unit), option pricing operations can be speeded up by hundreds of times, and meanwhile, since the cost of the GPU is low, the option pricing method and the option pricing apparatus have the advantages of low cost, high degree of parallelism and high operation performance.

Description

Option valuation method and device based on BSDE

Technical field

The present invention relates to the method for option valuation, relate in particular to option valuation method and device based on BSDE (BSDE, Backward Stochastic Differential Equation).

Background technology

Monte Carlo method (Monte Carlo Method) is an algorithm through the statistic sampling solve problem.In early days, this algorithm is applied in the development of atomic bomb during the World War II.To the integral and calculating greater than the arbitrary function of 6 DOF, Monte Carlo method is the method for unique practicality.It also has many other application, for example separates PDE, sharpening satellite photo, cell groups volume modeling, and the approximate solution that in polynomial time circle, finds np problem (can verify in polynomial time that whether correct separated a problem).

Monte Carlo method; Be to waiting to ask problem and according to the statistical law of physical phenomenon itself or construct the probability model of a suitable dependence stochastic variable artificially; The statistic that makes some stochastic variable is to wait to ask separating of problem, and carries out statistical experiment method or the Computer Random Simulation method of big statistic N → ∞.

BSDE (BSDE), i.e. " Bach shut out (Pardoux)-Peng Fangcheng " obtained very high international popularity in stochastic analysis, STOCHASTIC CONTROL and financial mathematic circle, and it can solve non-linear option valuation problem.

The general type of BSDE numerical solution is:

-dY _t＝f(Y _t，Z _t，t)dt-Z _tdW _t，t∈[O，T]

Y _T＝ξ

Wherein, W _tBe the Brownian movement that is defined on the complete probability, Y _TBe the terminal condition of BSDE; Y in option valuation _tFor at t option premium constantly, Z _tValue is with helping risk measurement, and generating function f (*) is Y _tAnd Z _tAnd the function of time t; ξ is the value of last one deck terminal condition, and being worth with this is the Yt that benchmark can be calculated next layer.

" BSDE " theory has been set up the bridge between " at random " and " confirming ", makes people go to solve uncertain problem at random with strategy, the method confirmed, or carries out optimal treatment to uncertain thing at random.The approach that it is opened up can be widely used in many aspects of social and economic activities, goes to solve to relate to a lot of major issues that international academic community generally is concerned about in the fields such as computer science, finance, economics and engineering science.

Option (Option) is a kind of right to choose, refers to it is a kind of right that can buy in or sell certain particular commodity of some at following certain special time with certain price.It is a kind of financial instrument that on the basis of futures, produces, and gives the right that the buyer (or holder) bought or sold target assets (underlying asset).The holder of option can select to buy or do not buy, sell or right not for sale in this option official hour, and he can implement this right, also can abandon this right, and the seller that goes out of option then only bears the obligation that option contract is stipulated.

Right by option is divided, and two types of call option (Call Options) and put options (Put Options) are arranged.Wherein, call option is meant the buyer of option after the premium of the certain number of seller payments, just has in the term of validity of option contract the right of commodity of buying in the contract regulation of some by the price of prior agreement, but does not bear the obligation that must buy.And the commodity of contract regulation are sold in the obligated term of validity planted agent option buyer's in option regulation of option seller requirement with the price of option contract regulation.Put option is meant the buyer of option after the premium of the certain number of seller payments of option, just has in the term of validity of option contract the right of commodity of selling the contract regulation of some by the price of prior agreement, but does not bear the obligation that must sell.And the commodity of contract regulation are bought in the obligated term of validity planted agent option buyer's in option regulation of option seller requirement with the price of option contract regulation.

Cloth Rec-Si Keersi Black-Scholes Option Pricing Model Black-Scholes (B-S; Black Scholes Option Pricing Model), for various derivative financial instruments are laid a good foundation based on the arm's length pricing of fluctuation in comprising the emerging financial market of stock, bond, currency and commodity.

But carrying out option valuation with the B-S model must have following 5 important hypothesis:

(1) financial assets' incoming rate obeys logarithm normal distribution;

(2) in option expiration date, risk free rate and financial asset income variable are constant;

(3) market does not have friction, does not promptly have tax revenue and transaction cost;

(4) financial asset dividend off and other gained (this hypothesis back is abandoned) in option expiration date;

(5) this option is an European style option, promptly before option expires, can not implement.

Analysis and research show; When adopting traditional Black-Scholes equation method to carry out option valuation; Because this equation is to be based upon on the bases of five big hypothesis and to utilize the fluctuation of stock price to follow geometric Brownian motion to obtain, and can not all satisfy these hypothesis in the reality.For European style option, utilize this equation can obtain the formula of separating, but its mathematical derivation and solution procedure financial world difficulty be widely accepted and grasp.Especially for American style option and since this equation to decide the problem of separating more complicated, can not obtain the expression formula of separating, so it's a pity, this equation can not solve non-linear option valuation problem.

When the option valuation serial approach that adopts the BSDE method solves the option valuation problem; Most of the time all consumes in this step: promptly according to terminal (promptly final possible final result) condition; Between adjacent layer, use Monte Carlo (Monte-Carlo) method that each time horizon is retrodicted, find the solution the computational mathematics expectation.Owing to the Monte Carlo simulation of each net point often, cause this step not only to be carried out often, and the data volume that will calculate is big; Simultaneously, these data computing can only satisfy the characteristics of SISD single instruction single data, and are difficult to satisfy the characteristics of single instruction multiple data (SIMD, Single Instruction Multiple Data).Therefore, adopt the option valuation serial approach of BSDE to solve the option valuation problem, have certain performance bottleneck.

The unified calculation equipment framework (CUDA of tall and handsome reaching (NVIDIA) company; Compute Unified Device Architecture); Be a kind of development environment that graphic process unit (GPU, Graphics Processing Unit) is calculated that is used for, it is a brand-new software and hardware architecture; Can GPU be regarded as the equipment that parallel data is calculated, the calculating of being carried out is distributed and manages.In the framework of CUDA; These calculate no longer as general graphical processor (GPGPU) framework in past, must calculating be mapped in the graphics application program interface (API, such as OpenGL and Direct 3D); Therefore for the developer, the exploitation threshold of CUDA greatly reduces.The measured C language of the GPU programming language of CUDA, therefore the user of any C of having language basis develops the application program of CUDA at an easy rate.

Because the characteristics of GPU are to handle intensive data and parallel data calculating, so CUDA is fit to the field that the needs large-scale parallel calculates very much.CUDA also provides the application programming interfaces of formula translation except using the C language development at present, can estimate that following CUDA will support each speech like sound such as C++, Java, Python.CUDA can be widely used in every field such as graphic animations, science calculating, geology, biology and physical simulation.

At present, still do not have a kind of method of option valuation fast, can improve user's operating experience, so that the maximization of client's interests.

Summary of the invention

Technical matters to be solved by this invention provides a kind of option valuation method and device based on BSDE, can quicken option valuation greatly and calculate, and improves user experience.

In order to solve the problems of the technologies described above, the invention provides a kind of option valuation method based on BSDE, comprising:

On time horizon, BSDE is dispersed, make up the space-time discrete grid block, calculate the terminal condition that realizes option valuation according to the space-time discrete grid block;

According to terminal condition, each spatial point to each time horizon between adjacent time horizon is carried out Monte Carlo simulation, and adopts the isosceles triangle model to the calculating of successively retrodicting of each time horizon, solves final option valuation mathematical expectation.

Further, on time horizon, BSDE is dispersed, makes up the space-time discrete grid block, calculate the terminal condition that realizes option valuation, specifically carry out following steps and realize through CPU according to said space-time discrete grid block:

On time horizon, stochastic differential equation is dispersed with the theta-form;

Time period [0, T] is divided into N time step, and N is a positive integer; If each time step is dt=T/N, be divided into N+1 time horizon with this; Space step-length dh set equal the dt size, simultaneously, be provided with and deposit the strike price array variable X of respective value of each space networks lattice point, and each spatial point is carried out the times N of Monte Carlo simulation _E, make up the space-time discrete grid block with this;

According to the assets price S on the N layer of time horizon _TObtain option premium Y possible when time T arrives with the X that strikes price _T, be help risk measurement Z in time T _TInitial value is set; Option premium Y _TWith help risk measurement Z _TJust as terminal condition.

Further, according to the assets price S on the N layer of time horizon _TObtain option premium Y possible when time T arrives with the X that strikes price _T, specifically comprise:

Obtain the assets price S on the N layer through formula _T:

S _T=initial price * exp (stability bandwidth * X [i]+(expected revenus rate-0.5* stability bandwidth square) * is constantly maximum);

Corresponding to call option, with assets price S _TGive option premium Y with difference and the 0 maximal value assignment relatively of the X that strikes price _TFor seeing option, X and assets price S will strike price _TDifference and 0 maximal value assignment relatively give option premium Y _T

Further; According to terminal condition; Each spatial point to each time horizon between adjacent time horizon is carried out Monte Carlo simulation; And based on the isosceles triangle model to the calculating of successively retrodicting of each time horizon, solve final option valuation mathematical expectation, specifically carry out following steps and realize through GPU:

Black-Scholes Option Pricing Model Black-Scholes is optimized for an isosceles triangle model, obtains the space subdivision through the base of finding the solution this isoceles triangle shape and count M=2*ps*N; Wherein, ps=c/dh+1 is the maximum space grid number up or down of Brownian movement on each time step, and c=5.0*sqrtf (dt) is the bound of Brownian movement on each time step;

Between adjacent layer, each spatial point of each time horizon is carried out Monte Carlo simulation according to terminal condition, based on the isoceles triangle model since the N-1 layer, in each circulation according to the calculating of successively retrodicting of following formula:

$y_t^n=E_{h,t_n}^{t_n,x_i}\left[y^{n+1}\right]+\mathrm{\Δ}{t}_n\left\{(1-{\mathrm{\θ}}_1^n)E_{h,t_n}^{t_n,x_i}\right[f(t_{n+1},y^{n+1},z^{n+1})]+{\mathrm{\θ}}_1^{n}f(t_n,y_i^n,z_i^n)\}$

Calculate at the 0th layer of temporal option mathematical expectation of time horizon according to following formula:

$E_{h,t_n}^{t_n,x_i}\left[y^{n+1}\right]=\frac{\underset{k=1}{\overset{N_E}{\mathrm{\Σ}}}{I}_h{y}^{n+1}\left({\hat{X}}_i^k\right)}{N_E};$

In the formula, I _hGo out through the usage space interpolation calculation.

Further, GPU through parallel mode to the calculating of successively retrodicting of each time horizon.

In order to solve the problems of the technologies described above, the invention provides a kind of option valuation device based on BSDE, comprise that host computer module and association handle computing module, wherein:

The host computer module is used on time horizon, BSDE being dispersed, and makes up the space-time discrete grid block, calculates the terminal condition that realizes option valuation according to the space-time discrete grid block;

Association handles computing module; Be used for carrying out Monte Carlo simulation according to terminal condition each spatial point between adjacent time horizon to each time horizon; And adopt the isosceles triangle model to the calculating of retrodicting of each time horizon, solve final option valuation mathematical expectation.

Further, the host computer module realizes that through CPU association handles computing module and realizes through GPU.

Further,

The host computer module is carried out the discrete of theta-form to stochastic differential equation on time horizon, the time period [0, T] is divided into N time step, and N is a positive integer; If each time step is dt=T/N, be divided into N+1 time horizon with this; Space step-length dh set equal the dt size, simultaneously, be provided with and deposit the strike price array variable X of respective value of each space networks lattice point, and each spatial point is carried out the times N of Monte Carlo simulation _EAccording to strike price X and the assets price S on time horizon N layer _TObtain at the possible option premium Y of time T _T, with Y _TWith said Z _TExport to association as terminal condition and handle computing module;

The base of association's this isoceles triangle shape of processing computing module calculating is tried to achieve the space subdivision and is counted M; Carry out Monte Carlo simulation according to terminal condition each spatial point between adjacent layer to each time horizon; And adopt the isosceles triangle model since the N-1 layer, in each circulation according to the calculating of successively retrodicting of following formula:

$y_t^n=E_{h,t_n}^{t_n,x_i}\left[y^{n+1}\right]+\mathrm{\Δ}{t}_n\left\{(1-{\mathrm{\θ}}_1^n)E_{h,t_n}^{t_n,x_i}\right[f(t_{n+1},y^{n+1},z^{n+1})]+{\mathrm{\θ}}_1^{n}f(t_n,y_i^n,z_i^n)\}$

Calculate Y, the Z value of current time i layer, and preserve two pairs of arrays of Y and Z; And solve the mathematical expectation of final option valuation through following formula:

$E_{h,t_n}^{t_n,x_i}\left[y^{n+1}\right]=\frac{\underset{k=1}{\overset{N_E}{\mathrm{\Σ}}}{I}_h{y}^{n+1}\left({\hat{X}}_i^k\right)}{N_E};$

In the formula, I _hGo out through the usage space interpolation calculation.

Further,

The host computer module is corresponding to call option, with the assets price X and the S that strikes price _TDifference and 0 maximal value assignment relatively give option premium Y _TFor seeing option, S will strike price _TGive option premium Y with difference and the 0 maximal value assignment relatively of assets price X _T

Further, GPU through parallel mode to the calculating of successively retrodicting of each time horizon.

The present invention utilizes the BSDE mode; And, can hundreds of times quicken the computing of option valuations, simultaneously based on isosceles triangle model and the powerful computation capability of GPU; Because the cost of GPU is lower, so the present invention has low cost, high degree of parallelism and operational performance at a high speed.

Description of drawings

Fig. 1 option valuation method embodiment synoptic diagram based on BSDE of the present invention;

Fig. 2 is that the isosceles triangle Black-Scholes Option Pricing Model Black-Scholes makes up synoptic diagram among the method embodiment shown in Figure 1;

Fig. 3 is the tactful synoptic diagram that uses each time horizon option mathematical expectation of GPU CUDA parallel computation among the method embodiment shown in Figure 1.

Embodiment

Below in conjunction with accompanying drawing and preferred embodiment technical scheme of the present invention is at length set forth.Should be appreciated that following listed examples only is used for explanation and explains the present invention, and does not constitute the restriction to technical scheme of the present invention.

Option valuation method embodiment based on BSDE of the present invention comprises:

On time horizon, BSDE is dispersed, make up the space-time discrete grid block, calculate the terminal condition that realizes option valuation based on this space-time discrete grid block;

According to terminal condition, between adjacent time horizon, each spatial point of each time horizon is carried out Monte Carlo simulation, and based on the isosceles triangle model to the calculating of successively retrodicting of each time horizon, solve final option valuation mathematical expectation.

In said method embodiment, on time horizon, BSDE is dispersed, make up the space-time discrete grid block, calculate the terminal condition that realizes option valuation based on this space-time discrete grid block, specifically carry out following steps and realize through CPU:

Step 1 disperses with the theta-form to BSDE on time horizon;

Step 2 is divided into N time step with the time period [0, T], and N is a positive integer; Set each time step and be Δ t=T/N, as shown in Figure 1, be divided into N+1 time horizon with this; The accuracy of calculating for guaranteeing is set at Δ t size with space step delta h, i.e. Δ h=Δ t simultaneously, is provided with and deposits each space networks lattice point strike price the array variable X of respective value and the times N of Monte-Carlo simulation on each spatial point _E, make up the space-time discrete grid block with this;

If X is the float type, be used to deposit the respective value of each space networks lattice point of geometric Brownian motion, promptly depositing the position of space-time discrete grid block point.In fact X is an array X [i]; Each number is deposited the value of grid on the corresponding time horizon in this array, and the size of this array also successively reduces according to the net point number of each time horizon, distributes so this array size is a dynamic size; And being subdivision number (being maximum space step number M) and space step delta h according to space grid on the terminal time layer, calculates together each numerical value in the array; Because mesh space is the separatrix with 0, be divided into two boundaries up and down, so the X array is all from top to bottom values; Comprise negative value, 0 value and on the occasion of, specifically be embodied as:

N _EBe the number of times (SIM_TIMES) of Monte-Carlo simulation on each spatial point, promptly all will carry out the Monte-Carlo simulation each point on each time horizon.

Step 3 is according to the assets price S on the N layer of time horizon _TObtain the possible option premium Y of T at the appointed time with the X that strikes price _T

In this step, assets price S _TObtain through formula:

S _T=initial price * exp (stability bandwidth * X [i]+(expected revenus rate-0.5* stability bandwidth square) * is constantly maximum); (1)

According to assets price S _TObtain at the possible option premium Y of time T _T, corresponding to call option with see that option realizes through following two formula respectively:

For call option (call option):

Be option price Y _TEqual assets price S _TDifference and 0 maximal value relatively with the X that strikes price.

For seeing option (put option):

Be option price Y _TX and assets price S equal to strike price _TDifference and 0 maximal value relatively.

Step 4 is the help risk measurement Z in time T _TInitial value is set.

Z _TInitial value for example be set to 0.

Above step does not need a large amount of calculating, gets N _E=40000, N=64, the used time of these steps is 0.3ms, compares the working time of whole procedure, this part time is undoubtedly less important, goes up operation so it is arranged in CPU.

In said method embodiment; Between adjacent time horizon; Each spatial point to each time horizon is carried out Monte Carlo simulation; And based on the isosceles triangle model each time horizon is successively retrodicted, find the solution and calculate final option valuation mathematical expectation, specifically carry out following steps and realize through GPU:

Step 1 is optimized for an isosceles triangle model with Black-Scholes Option Pricing Model Black-Scholes, the base of calculating this isoceles triangle shape, and promptly the space subdivision is counted M;

As shown in Figure 2, the height of isosceles triangle is known in this model, i.e. time T, and the tangent value at this isoceles triangle shape base angle can calculate through formula (4):

tanθ＝dt/ps (4)

In the formula:

Dt=T/N is a time step, and dh=dt is space step-length (seeing also Fig. 1);

Ps=c/dh+1 is a Brownian movement maximum space grid number up or down on each time step, and wherein: c=5.0*sqrtf (dt) is the bound of Blang (brown) motion on each time step;

N is a time subdivision number (TIME_GRID), i.e. what time horizons expression has divided, and the time number of plies is N+1 altogether, and time horizon is since 0 label: 0,1 ..., N, N layer contain that layer of terminal condition.

Obtain the base M of isosceles triangles by these two elements of T and tan θ, i.e. space subdivision number, it is the maximum space step-length of T constantly, reflection isosceles triangle relation capable of using is calculated and is obtained:

M＝2*T/tanθ＝2*(T/(dt/ps))＝2*ps*T/dt；

By dt=T/N, obtain:

M＝2*ps*N； (5)

The time horizon that is parallel to the isosceles triangle base is that unit is divided into the T+1 row with dt with the height of this isoceles triangle shape, and the net point of every row is several to be obtained through geometrical calculation according to the isosceles triangle model.

Step 2 is according to terminal condition (Y _T, Z _T), between adjacent layer, each spatial point of each time horizon is carried out Monte Carlo simulation, based on the isoceles triangle model each time horizon is used formula (6) calculating of successively retrodicting, solve the mathematical expectation of option valuation.

$y_t^n=E_{h,t_n}^{t_n,x_i}\left[y^{n+1}\right]+\mathrm{\Δ}{t}_n\left\{(1-{\mathrm{\θ}}_1^n)E_{h,t_n}^{t_n,x_i}\right[f(t_{n+1},y^{n+1},z^{n+1})]+{\mathrm{\θ}}_1^{n}f(t_n,y_i^n,z_i^n)\}---\left(6\right)$

Computation cycles is since the N-1 layer, i.e. that one deck in N layer front.Y, Z value through calling current separating (current_solution) function calculation current time i layer in each circulation, preservation Y and Z have two couples of array: Y1 and Z1, Y2 and a Z2; So each circulation need be called the current_solution function twice;

Up to calculating, obtain formula (8) through formula (7) differentiation and realize this calculating at the 0th layer of temporal option mathematical expectation.

$0=E_{h,t_n}^{t_n,x_i}\left[y^{n+1}\mathrm{\Δ}{W}_{t_{n+1}}\right]+\mathrm{\Δ}{t}_n(1-{\mathrm{\θ}}_2^n)E_{h,t_n}^{t_n,x_i}\left[f(t_{n+1},y^{n+1},z^{n+1})\mathrm{\Δ}{W}_{t_{n+1}}\right]-\mathrm{\Δ}{t}_n(1-{\mathrm{\θ}}_2^n)E_{h,t_n}^{t_n,x_i}\left[z^{n+1}\right]+{\mathrm{\θ}}_2^n{z}_i^n---\left(7\right)$

$E_{h,t_n}^{t_n,x_i}\left[y^{n+1}\right]=\frac{\underset{k=1}{\overset{N_E}{\mathrm{\Σ}}}{I}_h{y}^{n+1}\left({\hat{X}}_i^k\right)}{N_E};---\left(8\right)$

Through the usage space interpolation, just can calculate

(be the I in the formula 1 _h).Use cubic spline interpolation (interpolation) can improve the degree of accuracy of calculating greatly.

Implementation process is following:

Certain space networks lattice point on the current time layer carries out N with a thread to it _EInferior simulation, each simulation realizes through kernel function (kernel), comprises following step:

The corresponding relation of computational threads and space networks lattice point

At time horizon t _iOn, corresponding to each spatial point X _jOn option premium need N _EInferior Monte Carlo simulation, this step is the most consuming time in whole algorithm; And on each time horizon, the option valuation process of different spaces point can not have the execution respectively of the ground of dependence each other, so this step is to be fit to very much the process that parallelization is carried out.So adopt GPU CUDA technology to realize this algorithm concurrently.

What CUDA adopted is the multiple programming style of single program multiple data (SPMD); Comprise a GPU program that is called the kernel function in the program; Comprise that the execution be made up of thread block calculates all net points in the kernel program, the thread block of the execution kernel program of forming by thread.Thread in a thread block can communicate through shared storage, and coordinates memory access through the execution of each thread synchronously.

In order to use GPU to realize above-mentioned parallel computation; By all net points on time horizon of kernel function calculation; Option premium by a space networks lattice point of each thread computes; Then the number of thread equals the number of the space networks lattice point on time horizon N, uses the detailed paralleling tactic of CUDA as shown in Figure 3.

In Fig. 3, the left side has comprised all the space networks lattice points on time i layer, and the function of kernel function is exactly to calculate the option premium of all net points on the current time i layer.The black round dot is represented needs calculated space networks lattice point on the current time layer, and this calculating is accomplished by the thread of an activity.Oval dotted line comprises is in order to calculate the net point on the time i+1 layer that need read calculating at the option premium of certain net point on the time i layer.When handling calculating from time i+1 layer to the i layer, the space networks lattice point that can make the kernel function handle according to the isosceles triangle model reduces gradually.The representative of white round dot does not need calculated net point on the current time layer, for these points, thread need not to do any work.

Utilize equality (6) successively to calculate, accomplish BSDE option valuation (Y through formula (8) ₀, Z ₀) calculating.

Therefore, on time i layer, for a net point X _jThe calculating of last option premium is the net point that depends on the time i+1 layer.On single time step, to the geometric Brownian motion definition upper bound and lower bound

C is a constant, and the upper bound of geometric Brownian motion and lower bound also can be defined by the number of spatial point (Ps=C/ Δ h), and in this case, the spatial point data on time i layer are passed through M _i=i*Ps*2 tries to achieve, M ₀=1.

The present invention is when the thread block of design GPU, and the corresponding relation through thread number and array index satisfies merging visits, and is assigned to the visit that reduces in the constant storer global storage through the array X with the parking space net point.In addition, according to concrete example, come further to improve performance through adopting shared storage and texture storage device.

Idle thread can influence efficient in the past.And after optimize codes of the present invention, make the idle thread influence become very slight.BLOCK granularity in CUDA, thread is assigned to the stream multiprocessor, stream scheduling of multiprocessor thread execution.So using the most effective way of GPU is all to do same work at all threads of a thread block.When the thread of free time existed, the thread block that comprises idle thread was not done any work.In the thread block that contains idle and active threads, in the time of synchronizing thread, idle thread will wait for that active threads calculates completion.In program, carry out before the kernel, the number of active threads can be recomputated and defined, and does not so just exist idle thread to influence the efficient of GPU computing.

The present invention is based on said method embodiment, the option valuation device embodiment based on BSDE correspondingly also is provided, comprise that host computer module and association handle computing module, wherein:

The host computer module is used on time horizon, BSDE being dispersed, and makes up the space-time discrete grid block, calculates the terminal condition that realizes option valuation according to this space-time discrete grid block;

Association handles computing module; Be used for according to terminal condition; Each spatial point to each time horizon between adjacent time horizon is carried out Monte Carlo simulation, and based on the isosceles triangle model to the calculating of retrodicting of each time horizon, solve final option valuation mathematical expectation.

In said apparatus embodiment, the host computer module realizes that through CPU association handles computing module and realizes through GPU.

In said apparatus embodiment,

The host computer module disperses with the theta-form to BSDE on time horizon; Time period T is divided into N time step; Setting each time step is Δ t=T/N, is divided into N+1 time horizon with this, simultaneously space step delta h is made as Δ t size; And the array variable X that deposits each space networks lattice point respective value is set, make up the space-time discrete grid block with this; According to strike price X and the assets price S on time N layer _TObtain at the possible option premium Y of time T _T, and be the help risk measurement Z in time T _TInitial value is set, with Y _TAnd Z _TExport to association as terminal condition and handle computing module;

Association handles computing module Black-Scholes Option Pricing Model Black-Scholes is optimized for an isosceles triangle model, tries to achieve the space subdivision through the base of calculating this isoceles triangle shape and counts M, according to terminal condition (Y _T, Z _T), between adjacent layer, use Monte Carlo method and, each time horizon is successively retrodicted based on the isosceles triangle model, find the solution the mathematical expectation that calculates final option valuation.

In said apparatus embodiment,

The host computer module is corresponding to call option, with the assets price X and the S that strikes price _TDifference and 0 maximal value assignment relatively give option premium Y _TFor seeing option, S will strike price _TGive option premium Y with difference and the 0 maximal value assignment relatively of assets price X _T

The base of association's this isoceles triangle shape of processing computing module calculating is tried to achieve the space subdivision and is counted M; Carry out Monte Carlo simulation according to said terminal condition each spatial point between adjacent layer to each time horizon; And adopt said isosceles triangle model since the N-1 layer; According to formula (6) calculatings of successively retrodicting, computation cycles is calculated Y, the Z value of current time i layer, and two pairs of arrays of preservation Y and Z in each circulation; And solve the mathematical expectation of final option valuation, wherein I through formula (8) _hGo out through the usage space interpolation calculation.

Association handles computing module according to terminal condition (Y _T, Z _T); Between adjacent layer, use Monte Carlo (Monte-Carlo) method and based on the isoceles triangle model; Each time horizon is successively retrodicted, and computation cycles is calculated Y, the Z value of current time i layer since the N-1 layer in each circulation; And two pairs of arrays of preserving Y and Z, up to calculating at the 0th layer of temporal option mathematical expectation.

The present invention is after providing said method embodiment and system embodiment; Data scale according to option valuation; Choose the computing test that appropriate C PU and GPU carry out the iterative reconstruction image respectively; Wherein CPU has adopted the i7920 of Intel Company, and GPU adopts the C1060 of Tesla company, and the detail parameters of test environment is seen table 1.

Table 1

Through adopting top test environment to test, get N=32, N _E=20000, test result such as table 2.

Table 2

As can be seen from Table 2, the option premium that originally on CPU, needs 185.653s just can calculate only needs cost 1.525s on GPU, and the GPU execution speed is more than 121 times of CPU execution speed.

Can find out by above-mentioned test result, the present invention is based on the BSDE mode, utilize the isosceles triangle model and the powerful computation capability of GPU of option valuation, can hundreds of times quicken the computing of option valuations.Simultaneously, because the cost of GPU is lower, therefore, the present invention has low cost, high degree of parallelism and operational performance at a high speed.

More than explanation is merely the preferable embodiment of the present invention; But protection scope of the present invention is not limited thereto; Any technician who is familiar with the present technique field is in the technical scope that the present invention discloses, and the variation that can expect easily or replacement all should be encompassed within protection scope of the present invention.Therefore, protection scope of the present invention should be as the criterion with the protection domain of claims.

Claims (10)

1. the option valuation method based on BSDE is characterized in that, comprising:

On time horizon, BSDE is dispersed, make up the space-time discrete grid block, calculate the terminal condition that realizes option valuation according to said space-time discrete grid block;

According to said terminal condition, each spatial point to each time horizon between adjacent time horizon is carried out Monte Carlo simulation, and adopts the isosceles triangle model to the calculating of successively retrodicting of each time horizon, solves final option valuation mathematical expectation.

2. according to the described method of claim 1; It is characterized in that, saidly on time horizon, BSDE is dispersed, make up the space-time discrete grid block; Calculate the terminal condition that realizes option valuation according to said space-time discrete grid block, specifically carry out following steps and realize through CPU:

On said time horizon, carry out said discrete with the theta-form to said stochastic differential equation;

Time period [0, T] is divided into N time step, and said N is a positive integer; If each time step is dt=T/N, be divided into N+1 time horizon with this; Space step-length dh set equal said dt size, simultaneously, be provided with and deposit the strike price array variable X of respective value of each space networks lattice point, and said each spatial point is carried out the times N of Monte Carlo simulation _E, make up said space-time discrete grid block with this;

According to the assets price S on the N layer of said time horizon _TObtain option premium Y possible when said time T arrives with the X that strikes price _T, be help risk measurement Z in said time T _TInitial value is set; Said option premium Y _TWith said help risk measurement Z _TJust as said terminal condition.

3. according to the described method of claim 2, it is characterized in that, said according to the assets price S on the N layer of said time horizon _TObtain option premium Y possible when said time T arrives with the X that strikes price _T, specifically comprise:

Obtain the assets price S on the said N layer through formula _T:

S _T=initial price * exp (stability bandwidth * X [i]+(expected revenus rate-0.5* stability bandwidth square) * is constantly maximum);

Corresponding to call option, with assets price S _TGive said option premium Y with difference and the 0 maximal value assignment relatively of the X that strikes price _TFor seeing option, X and assets price S will strike price _TDifference and 0 maximal value assignment relatively give said option premium Y _T

4. according to the described method of claim 2; It is characterized in that; According to said terminal condition, each spatial point to each time horizon between adjacent time horizon is carried out Monte Carlo simulation, and based on the isosceles triangle model to the calculating of successively retrodicting of each time horizon; Solve final option valuation mathematical expectation, specifically carry out following steps and realize through GPU:

Black-Scholes Option Pricing Model Black-Scholes is optimized for an isosceles triangle model, obtains the space subdivision through the base of finding the solution this isoceles triangle shape and count M=2*ps*N; Wherein, said ps=c/dh+1 is the maximum space grid number up or down of Brownian movement on each time step, and said c=5.0*sqrtf (dt) is the bound of Brownian movement on each time step;

Between adjacent layer, each spatial point of each time horizon is carried out Monte Carlo simulation according to said terminal condition, based on said isoceles triangle model since the N-1 layer, in each circulation according to the calculating of successively retrodicting of following formula:

$y_i^n=E_{h,t_n}^{t_n,x_i}\left[y^{n+1}\right]+\mathrm{\Δ}{t}_n\left\{(1-{\mathrm{\θ}}_1^n)E_{h,t_n}^{t_n,x_i}\right[f(t_{n+1},y^{n+1},z^{n+1})]+{\mathrm{\θ}}_i^{n}f(t_n,y_i^n,z_i^n)\}$

Calculate at the 0th layer of temporal said option mathematical expectation of said time horizon according to following formula:

$E_{h,t_n}^{t_n,x_i}\left[y^{n+1}\right]=\frac{\underset{k=1}{\overset{N_E}{\mathrm{\Σ}}}{I}_h{y}^{n+1}\left({\hat{X}}_i^k\right)}{N_E};$

In the formula, said I _hGo out through the usage space interpolation calculation.

5. according to the described method of claim 4, it is characterized in that said GPU carries out the said calculating of successively retrodicting through parallel mode to each time horizon.

6. the option valuation device based on BSDE is characterized in that, comprises that host computer module and association handle computing module, wherein:

The host computer module is used on time horizon, said BSDE being dispersed, and makes up the space-time discrete grid block, calculates the terminal condition that realizes option valuation according to said space-time discrete grid block;

Association handles computing module; Be used for carrying out Monte Carlo simulation according to said terminal condition each spatial point between adjacent time horizon to each time horizon; And adopt the isosceles triangle model to the calculating of retrodicting of each time horizon, solve final option valuation mathematical expectation.

7. according to the described device of claim 6, it is characterized in that said host computer module realizes that through CPU said association handles computing module and realizes through GPU.

8. according to the described device of claim 7, it is characterized in that,

Said host computer module is carried out the discrete of theta-form to said stochastic differential equation on time horizon, the time period [0, T] is divided into N time step, and said N is a positive integer; If each time step is dt=T/N, be divided into N+1 time horizon with this; Space step-length dh set equal said dt size, simultaneously, be provided with and deposit the strike price array variable X of respective value of each space networks lattice point, and said each spatial point is carried out the times N of Monte Carlo simulation _EAccording to strike price X and the assets price S on said time horizon N layer _TObtain at the possible option premium Y of time T _T, with said Y _TWith said Z _TExport to said association as terminal condition and handle computing module;

The base of said association this isoceles triangle shape of processing computing module calculating is tried to achieve the space subdivision and is counted M; Carry out Monte Carlo simulation according to said terminal condition each spatial point between adjacent layer to each time horizon; And adopt said isosceles triangle model since the N-1 layer, in each circulation according to the calculating of successively retrodicting of following formula:

$y_i^n=E_{h,t_n}^{t_n,x_i}\left[y^{n+1}\right]+\mathrm{\Δ}{t}_n\left\{(1-{\mathrm{\θ}}_1^n)E_{h,t_n}^{t_n,x_i}\right[f(t_{n+1},y^{n+1},z^{n+1})]+{\mathrm{\θ}}_i^{n}f(t_n,y_i^n,z_i^n)\}$

Calculate Y, the Z value of current time i layer, and preserve two pairs of arrays of Y and Z; And solve the mathematical expectation of final option valuation through following formula:

In the formula, said I _hGo out through the usage space interpolation calculation.

9. according to the described device of claim 8, it is characterized in that,

Said host computer module is corresponding to call option, with the assets price X and the S that strikes price _TDifference and 0 maximal value assignment relatively give option premium Y _TFor seeing option, S will strike price _TGive option premium Y with difference and the 0 maximal value assignment relatively of assets price X _T

10. according to the described device of claim 7, it is characterized in that said GPU carries out the said calculating of successively retrodicting through parallel mode to each time horizon.

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