of continuous time portfolio with given properties

Abstract

A continuous-time version of portfolio management problem for a market with multiple stocks is under study. Asset allocation policy is determined implicitly as a solution to the system of ordinary diferential equations for asset quantities. Right-hand sides of equations are derived explicitly in closed form from given portfolio properties using techniques of construction of diferential equations by given integral manifold. Related issues of simulation for portfolio with desired properties are also addressed. We present the results of computer experiment for verification of specified portfolio property using constructed portfolio model in the form of the system of stochastic diferential equations.

Keywords

• Markov decision process
• portfolio policy
• stochastic process
• SDE simulation
• OpenCL framework

1. Introduction

Modeling with stochastic diferential equations is used extensively in many areas of modern research [1]. Stochastic diferential equations (SDE) arise in modeling and simulation of various random dynamical systems in physical [2], chemical [3], biological [4], financial [ 4] and other sciences.

Stochastic models based on SDE are an integral part of quantitative analysis of wireless channels in communications [5]. Modeling of signals and interference in communications requires construction of SDE with specified properties, such as given marginal probability density function and autocovariance function. Applications of SDE in other areas usually also imply construction of SDE with desired characteristics.

We study asset allocation problem for a portfolio with specified properties using continuoustime continuous-state Markov decision process, determined implicitly by a system of diferential equations. Our goal is to derive right-hand sides of equations by known portfolio properties and verify the required portfolio property in a computer experiment using simulation techniques. Workshop on information technology and scientific computing in the framework of the XI International Conference CEUR Workshop Proceedings © 2021 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0). 2. Implicit Markov decision process for portfolio management Markov decision process [6] is a widely used approach to decision making, including applications in finance.

Consider a market in which no dividend stocks are traded continuously and transaction cost and consumption are not taken into account. The stock price processes ( ), = 1, generally satisfy the system of SDE [7]: =1 = (, ) +

∑ (, ) , = 1, , ⩽ , where (, )are drift terms, (, )are volatility terms, 1, ..., are independent standard Wiener processes, = ( 1, ..., ).

The initial stock prices at = 0 are assumed to be known:

Let ( )be the quantity (in units) of the -th stock in the portfolio at time . We assume that the quantity ( )can be positive (long position), negative (short position) or zero (no position) and can take fractional values.

The value process of the stock portfolio ( )at time is determined by the formula ( 0)= (0) > 0, = 1, . =1 ( )= ∑ ( ) ( ). = (, s, x), =

1, The share (weight) ( )of the -th stock in the portfolio ( )is equal to ( )= ( ) ( ) ( ) = ( ) ( ) ∑ =1 ( ) ( ) , = 1, .

The weights ( )are not all independent, because of the norming condition ∑

The state of the stock portfolio can be determined either by prices s = ( 1, ..., )and quantities x = ( 1, ..., ), or by prices s = ( 1, ..., ), weights w = ( 1, ..., )and total portfolio value ( ). =1 ( )= 1.

We assume that the portfolio state is given by price-quantity pair (s, x).

The portfolio state (s, x)varies because of changes in stock prices s due to market situation or changes in stock quantities x due to policy (actions) of portfolio manager.

Possible actions for the -th stock in the portfolio are to hold the stock, to buy or to sell some units of the stock.

As a result of actions for the time period △ > 0 change in the quantity of units of the -th stock in the portfolio is equal to △ = ( + △ )− ( ). If △ > 0, then △ units of the stock are purchased, if △ < 0, then |△ | units of the stock are sold, if △ = 0, then ( )units of the stock are being held in the portfolio.

Asset allocation policy x ( )may be explicit with quantities of stocks being functions of time and/or stock prices, i.e. x = x (, s). We assume that asset allocation policy x ( )is determined implicitly as a solution of the system of diferential equations ( 1 ) ( 2 ) ( 3 ) ( 4 ) ( 5 ) with initial conditions (initial stock quantities) [ (, s ( ), x ( ))] = 0, = 1, , ∀ ⩾ 0,

Actually, the system ( 5 ) should be treated as a system of SDE with zero difusion coeficients, because right-hand sides of system ( 5 ) depend on stochastic stock price processes ( ), following ( 1 ) with initial conditions ( 2 ).

When functions (, s, x)in ( 5 ) are fixed, the portfolio management policy is fully determined by equations ( 1 ) and ( 5 ) with initial conditions ( 2 ) and ( 6 ), the portfolio dynamics depends only on the initial state of the portfolio (s(0), x(0)) at time 0 and is independent of the past ( < 0).

Thus, we determine the portfolio policy by choosing various functions (, s, x) on the right-hand side of equations ( 5 ) and receive Markov decision process for portfolio management. 3. Construction of portfolio with given properties Construction of stock portfolio with desired characteristics and evaluation of its risk and performance using simulation and optimization plays extremely important role in modern ifnancial industry [ 8]. The desired properties of the portfolio can be a given rate of return, a given risk metric (variance of return, value at risk, expected shortfall), given structural (country, industry, rating) ratios, etc.

As we will show below, portfolio management policy may be based on construction of ordinary diferential equations (ODE) by a given integral manifold. Originally, construction of ODE from a given integral curve was proposed by Yerugin [9] and later extended to dynamical systems of general nature by Galiullin [10], Mukharlyamov [11] and many other authors [12].

The method is widely used in investigations of controlled dynamical systems and allows to construct equations of motion from given properties of trajectories taking into account additional requirements, such as stability of the given manifold or optimality in some sense. The problem of construction of equations in the class of Ito SDE by known properties of motion was investigated by Tleubergenov [13, 14, 15].

We assume that price dynamics of stocks in the portfolio is described by SDE ( 1 ) with initial conditions ( 2 ) and quantities of stocks follow Markov portfolio policy, implicitly determined by equations ( 5 ) with initial conditions ( 6 ). The functions on the right-hand sides of equations ( 1 ) and ( 5 ) are assumed to be continuous in time and Lipschitz in portfolio state variables s, x.

We consider portfolio properties in the form of equalities

(, s, x)= 0, = 1, , < ) is of maximum rank . where Jacobian × -matrix ( x

Basically, the stochasticity of stock prices driven by SDE ( 1 ) leads to violation of equalities ( 7 ), so our goal is to find a portfolio policy of the form ( 5 ), which ensures that equalities ( 7 ) are satisfied on ensemble average, i.e. ( 6 ) ( 7 ) ( 8 ) where

(, )is the probability density function of ( )at time .

This statement of the problem difers from common approach to construction of stochastic diferential equations by given integral manifold [ 13], when properties ( 7 ) are supposed to be satisfied exactly. But in portfolio management problem this implies restrictions on price equations ( 1 ), which is not relevant for financial market models.

Stochastic diferentials of can be calculated using the multidimensional Ito’s formula [17] = ( + ∑ =1 =1 + ∑ +

1 ∑ 2 =1 =1 ℎ=1 ∑ ∑ ℎ ℎ 2 ) + + ∑ ∑ ℎ

ℎ, = 1, .

Arbitrary (unknown) functions in ( 5 ) can be chosen in such a way that the following equalities hold true + ∑ =1 + ∑ =1 +

1 ∑ 2 =1 =1 ℎ=1 ∑ ∑ ℎ ℎ

2 = − ∑ , = 1, , =1 where

(, s, x)are arbitrary functions. Equations ( 11 ) can be regarded as a system of linear algebraic equations for unknown functions : where ∑ =1 = − , = 1, , =1 = ∑ + + ∑ =1 +

1 ∑ 2 =1 =1 ℎ=1 ∑ ∑ ℎ ℎ 2 .

Applying the methodology of Moore–Penrose pseudoinverse matrices [18] to system ( 12 ), we obtain the solution of ( 12 ) in the following matrix form: where denotes expected value (ensemble average) [16] and for a stochastic process ( ) +∞ −∞ [ ( )] = ∫ (, ), f = − ( x

+ ) + [ − ( x ) + ] , ( x x

+ ) = x ( x x −1 ) , where =

( 1, ..., ) , is the identity × -matrix, = (Φ1, ..., Φ ) is a vector of arbitrary functions. The pseudoinverse matrix ( of ( 12 ) in the following final form highest rank . Substituting the corresponding expression from ( 12 ) for , we obtain the solution ) x + exists, because the Jacobian matrix x has the f = − ( x

+ ) [Λ + +

+ s 2 1 tr ( 2 x2 )] + [ − ( x ) + ] ,

x 33–44

( 9 ) ( 10 ) ( 11 ) ( 12 ) ( 13 ) ( 14 ) where Λ = ( ), tr is matrix trace operation.

We additionally assume that for any , = receive that

1, the random variables and are independent, then, averaging the equalities ( 10 ) along ensemble of trajectories under the policy ( 14 ), we [ ]

=1 = − ∑ [ ] [ ] , = 1, . ( 15 ) 4. Modeling self-financing portfolio with given structure We consider a self-financing portfolio with cash account and two risky assets (stocks), driven by geometric Brownian motion, and assume that the portfolio management policy is given in The system of ODE ( 15 ) admits the trivial solution [ ] = 0, = 1, , which implies that the portfolio with stock dynamics ( 1 ) and portfolio policy ( 5 ) admits portfolio properties ( 8 ).

Thus, we receive the following statement on Markov portfolio policies with given portfolio properties.

Markov asset allocation policy ( 5 ) with right-hand sides given by ( 14 ) admits the specified portfolio properties ( 8 ), provided that Λ = ( (, s, x))is an arbitrary × -matrix such that the random variables (, s ( ), x ( ))and (, s ( ), x ( ))are independent, (, s, x)is an arbitrary column vector.

Portfolio asset allocation policy ( 5 ) with right-hand sides given by ( 14 ) can be applied to portfolios containing stocks, currencies and commodities. By choosing arbitrary functions and Φ it is possible to ensure the stability of portfolio policy or to find an optimal policy for given reward function and discount factor.

Asset portfolio can include cash account, be self-financing, and incorporate other additional features. The value of portfolio with cash account and risky assets is equal to ( )= 0 ( )+ ∑ ( ) ( ), where 0 ( )denotes the balance of cash account at time .

The portfolio is self-financing [ 17], if no external inflow or outflow of cash and stocks takes place and the purchase of a stock must be financed by cash on cash account or by sale of some stocks from the portfolio. The portfolio with cash account is self-financing, if [ 17] where > 0 is the fixed interest rate of cash account. This implies that the balance of cash account 0 satisfies the following equation [ 19] where functions from ( 5 ) determine the portfolio policy. =1 =1 0 = ( 0 − ∑ ) , implicit form ( 5 ). The portfolio dynamics is determined by the following system of SDE: ⎧ 1 = 1 1 + 1 1 1, ⎪⎪ 2 = 2 2 + 2 2 2,

0 = ( 0 − 1 1 (, s, x)− 2 2 (, s, x)), ⎨ ⎪ 1 = 1 (, s, x), ⎪ ⎩ 2 = 2 (, s, x).

Here 1, 2 are the stock prices, 0 is the balance of cash account, 1, 2 are the quantities of stocks in the portfolio, s = ( 1, 2), x = ( 0, 1, 2), 1, 2 are the instantaneous rates of return, 1, 2 are the volatilities of stocks, 1, 2 are independent standard Wiener processes, is the interest rate of cash account.

We assume that the desired portfolio property is the condition that 80% of the portfolio assets is invested in risky assets (stocks).

The value of the portfolio under consideration is equal to = 0 + 1 1 + 2 2, and weights of cash and stocks are equal to 0 =

0 0 + 1 1 + 2 2 , 1 =

1 1 0 + 1 1 + 2 2 , 2 =

2 2 0 + 1 1 + 2 2 .

The desired portfolio property can be expressed as the following relation between the stock weights 1 and 2: 1 + 2 =

1 1 + 2 2 0 + 1 1 + 2 2 = 0.8, which can be transformed into the equality 33–44 ( 16 ) ( 17 ) ( 18 ) ( 19 ) ( 20 ) ≡ −4 0 + 1 1 + 2 2 = 0.

Our goal is to determine the unknown functions 1, 2 in ( 16 ) to guarantee the portfolio property ( 17 ) on ensemble average.

According to Ito’s lemma the stochastic diferential of is equal to = (−4 0 + 1 1 1 + 2 2 2 + 5 1 1 + 5 2 2) + 1 12 1 + 2 22 2.

We set the unknown functions 1, 2 so, that the drift term at is equal to − , ∈ ℝ : − 4 0 + 1 1 1 + 2 2 2 + 5 1 1 + 5 2 2 = −.

Equality ( 19 ) is the linear algebraic equation for the determination of unknown functions 1, 2, and the general solution of equation ( 19 ) is:

1 = − 12 +1 22 + 2 , 2 = − 12 +2 22 − 1 , where = 0.2 − 0.8

0 + 0.2 1 1 1 + 0.2 2 2 2, ∈ ℝ , is an arbitrary function of , s, x.

Thus, SDE system ( 16 ) for modeling prices and quantities of the portfolio with given property ( 17 ) takes the following form with an arbitrary constant ∈ ℝ and an arbitrary function : ⎧ ⎪ ⎪ ⎪ ⎪ ⎪

1 = 1 1 + ⎪ 2 = 2 2 +

0 = ( + 0.2 ⎨ 1 = (− 12 + 22 ⎪

⎪ 2 = (− 12 + 22

⎩ , , 5. Simulation of self-financing portfolio with given structure version of SDE ( 21 ).

To verify that the portfolio model ( 21 ) satisfies the portfolio property ( 17 ) we simulate the trajectories of system ( 21 ) for the time segment [ 0, ].

So we divide the segment [ 0, ] into equal parts of length ℎ = − 0 and simulate a discretized

There exists a number of discretization schemes available, such as Milstein scheme [20] and stochastic version of Runge-Kutta methods [21, 22], but the most intuitive, easy to implement and common is Euler-Maruyama scheme [23]. Euler-Maruyama discretization of ( 21 ) gives us the following equations: ⎪ ⎪ ⎪ ⎪ ⎩ ⎨⎪ 1,+1 = 1, − ⎪ 2,+1 = 2, − ⎧ 1,+1 = 1, + 1 1, ℎ + 1 1, √ℎ 1, , ⎪ 2,+1 = 2, + 2 2, ℎ + 2 2, √ℎ 2, , ⎪ 0,+1 = 0, + ( + 0.2 0, + 0.2 1 1, 1, + 0.2 2 2, 2, )ℎ, 0.2 − 0.8 0, + 0.2 1 1, 1, + 0.2 2 2, 2, 0.2 − 0.8 0, + 0.2 1 1, 1, + 0.2 2 2, 2, 2 2 1, + 2, 1, + 2, 2 2 1, ℎ + 2, ℎ, 2, ℎ − 1, ℎ, where , = ( ), , = (

), = 0 + ℎ , = −4 0, + 1, 1, + 2, 2, , , are i.i.d. random variables with standard normal distribution, i.e. , ∼ ( 0, 1 ).

Basically, simulation of SDE is a resource-intensive application, so to speed up the simulation one has to enable GPU acceleration. PyOpenCL [24] is a programming environment for access to OpenCL parallel computation framework [25] from Python language.

To apply PyOpenCL environment for portfolio simulation, one has to follow the typical sequence of steps: 1. Import the OpenCL API, obtain an OpenCL platform and a GPU device id: import pyopencl as cl plat = cl.get_platforms()[0] dev = plat.get_devices()[2] 33–44 ( 21 ) ( 22 ) 2. Initialize a context for the selected GPU device, create a Queue object (with profiling enabled to track computation time), create memory bufers on GPU for input and output data (here p, q and s_gpu are NumPy arrays): opencl_context = cl.Context(devices=[dev]) command_queue = cl.CommandQueue(opencl_context,

properties=cl.command_queue_properties.PROFILING_ENABLE) p_buffer = cl.Buffer(opencl_context,

cl.mem_flags.READ_ONLY | cl.mem_flags.COPY_HOST_PTR, hostbuf=p) q_buffer = cl.Buffer(opencl_context,

cl.mem_flags.READ_ONLY | cl.mem_flags.COPY_HOST_PTR, hostbuf=q) s_buffer = cl.Buffer(opencl_context,

cl.mem_flags.WRITE_ONLY, s_gpu.nbytes) 3. Store the source code of C functions for GPU execution in Python strings (sde_step_src contains source code of one step of SDE simulation according to ( 22 ), simulate_sde_src contains source code of main kernel function, being called from Python): sde_step_src = """ // One step of SDE simulation static void sde_step(__global double *p, double *x, double W) { double s1n, s2n, x0n, x1n, x2n; x[5] = -0.8*x[2] + 0.2*x[3]*x[0] + 0.2*x[4]*x[1]; // omega s1n = p[P_M1]*x[0]*p[P_DT] + p[P_S1]*x[0]*p[P_SQRDT]*W; s2n = p[P_M2]*x[1]*p[P_DT] + p[P_S2]*x[1]*p[P_SQRDT]*W; x0n = (p[P_A]*x[5] + 0.2*p[P_R]*x[2] + 0.2*p[P_M1]*x[3]*x[0] + 0.2*p[P_M2]*x[4]*x[1])*p[P_DT]; x1n = -x[0]*(p[P_A]*x[5] - 0.8*p[P_R]*x[2] + 0.2*p[P_M1]*x[3]*x[0] + 0.2*p[P_M2]*x[4]*x[1])*p[P_DT]/(x[0]*x[0]+x[1]*x[1]); x2n = -x[1]*(p[P_A]*x[5] - 0.8*p[P_R]*x[2] + 0.2*p[P_M1]*x[3]*x[0] + 0.2*p[P_M2]*x[4]*x[1])*p[P_DT]/(x[0]*x[0]+x[1]*x[1]); x[0] += s1n; x[1] += s2n; x[2] += x0n; x[3] += x1n; x[4] += x2n; }""" simulate_sde_src = """ // GPU kernel function to simulate trajectories of SDE __kernel void simulate_sde(__global double *p, __global int *q, __global double *s) { // Indexing the current element (trajectory) to process int i = get_global_id(0); // Pointer to the i'th row of s (output) __global double *s_i = &s[i*6]; // 6 values in output // Simultaneous calculation of SDE trajectories within OpenCL kernel double x[6]; // placeholders for s1,s2,x0,x1,x2,omega int q_i = (int) q[i], *seed = &q_i; const double r4_pi = 3.141592653589793; double v1,v2; for (int j = 0; j < 5; j++) { x[j] = p[P_S10+j]; } for (int j = 0; j < (int) p[P_N]/2.; j++) { // 2 steps of SDE v1 = r8_uniform_01 ( seed ); v2 = r8_uniform_01 ( seed ); sde_step( p, x, sqrt(-2.0*log(v1))*cos(2.0*r4_pi*v2) ); sde_step( p, x, sqrt(-2.0*log(v1))*sin(2.0*r4_pi*v2) ); } for (int j = 0; j < 6; j++) { s_i[j] = x[j]; } }""" 4. Compile the GPU C program from source: opencl_program = cl.Program(opencl_context,

sde_hdr_src+sde_rng_src+sde_step_src+simulate_sde_src).build() 5. Enqueue the SDE simulation program on the GPU device and wait until the program completion: event = opencl_program.simulate_sde(command_queue, s_gpu.shape, None, p_buffer, q_buffer, s_buffer) event.wait() 6. Get back the output data from GPU memory into NumPy array s_gpu:

cl.enqueue_copy(command_queue, s_gpu, s_buffer).wait()

Function r8_uniform_01 is an implementation of uniform random number generator [26], C++ versions of other random number generators may be found in [27]. Standard normally distributed random variables , are implemented with Box–Muller transform [28].

Simulation of discretized equations ( 22 ) is performed using computer program in Python with the following parameters: 1,0 = 0.9, 2,0 = 1.1, 1 = 0.3, 1 = 0.4, 2 = −0.1, 2 = 0.3, 0 = 0., = 1.

0,0 = 90., 1,0 = 110., 2,0 = 100., = 0.05, = 1., ≡ 0, = 100.

The results of the computer experiment are shown in Fig. 1. Green line on the last plot with stock weights shows that the total share of stocks 1 + 2 tends to the value of 0.8, which is the target property of the portfolio under consideration.

Thus, constructed equations ( 22 ) generate trajectories with empirical portfolio property close to the given property ( 17 ).

Acknowledgments

The research was funded by RFBR, grant No. 19-08-00261.

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