tf_quant_finance.models.ItoProcess

tf_quant_finance.models.ItoProcess#

Interface for specifying Ito processes.

Interface for defining stochastic process defined by the Ito SDE:

  dX_i = a_i(t, X) dt + Sum(S_{ij}(t, X) dW_j for 1 <= j <= n), 1 <= i <= n

The vector coefficient a_i is referred to as the drift of the process and the matrix S_{ij} as the volatility of the process. For the process to be well defined, these coefficients need to satisfy certain technical conditions which may be found in Ref. [1]. The vector dW_j represents independent Brownian increments.

For a simple and instructive example of the implementation of this interface, see models.GenericItoProcess.

References#

[1]: Brent Oksendal. Stochastic Differential Equations: An Introduction with Applications. Springer. 2010.

Methods#

`dim`

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dim()

The dimension of the process. A positive python integer.

`drift_fn`

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drift_fn()

Python callable calculating instantaneous drift.

The callable should accept two real Tensor arguments of the same dtype. The first argument is the scalar time t, the second argument is the value of Ito process X - Tensor of shape batch_shape + [dim]. Here batch_shape is an arbitrary shape. The result is the value of drift a(t, X). The return value of the callable is a real Tensor of the same dtype as the input arguments and of shape batch_shape + [dim].

Returns:#

The instantaneous drift rate callable.

`dtype`

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dtype()

The data type of process realizations.

`fd_solver_backward`

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fd_solver_backward(
    start_time, end_time, coord_grid, values_grid, discounting=None,
    one_step_fn=None, boundary_conditions=None, start_step_count=0, num_steps=None,
    time_step=None, values_transform_fn=None, dtype=None, **kwargs
)

Returns a solver for Feynman-Kac PDE associated to the process.

This method applies a finite difference method to solve the final value problem as it appears in the Feynman-Kac formula associated to this Ito process. The Feynman-Kac PDE is closely related to the backward Kolomogorov equation associated to the stochastic process and allows for the inclusion of a discounting function.

For more details of the Feynman-Kac theorem see [1]. The PDE solved by this method is:

  V_t + Sum[mu_i(t, x) V_i, 1<=i<=n] +
    (1/2) Sum[ D_{ij} V_{ij}, 1 <= i,j <= n] - r(t, x) V = 0

In the above, V_t is the derivative of V with respect to t, V_i is the partial derivative with respect to x_i and V_{ij} the (mixed) partial derivative with respect to x_i and x_j. mu_i is the drift of this process and D_{ij} are the components of the diffusion tensor:

  D_{ij}(t,x) = (Sigma(t,x) . Transpose[Sigma(t,x)])_{ij}

This method evolves a spatially discretized solution of the above PDE from time t0 to time t1 < t0 (i.e. backwards in time). The solution V(t,x) is assumed to be discretized on an n-dimensional rectangular grid. A rectangular grid, G, in n-dimensions may be described by specifying the coordinates of the points along each axis. For example, a 2 x 4 grid in two dimensions can be specified by taking the cartesian product of [1, 3] and [5, 6, 7, 8] to yield the grid points with coordinates: [(1, 5), (1, 6), (1, 7), (1, 8), (3, 5) ... (3, 8)].

This method allows batching of solutions. In this context, batching means the ability to represent and evolve multiple independent functions V (e.g. V1, V2 …) simultaneously. A single discretized solution is specified by stating its values at each grid point. This can be represented as a Tensor of shape [d1, d2, … dn] where di is the grid size along the ith axis. A batch of such solutions is represented by a Tensor of shape: [K, d1, d2, … dn] where K is the batch size. This method only requires that the input parameter values_grid be broadcastable with shape [K, d1, … dn].

The evolution of the solution from t0 to t1 is often done by discretizing the differential equation to a difference equation along the spatial and temporal axes. The temporal discretization is given by a (sequence of) time steps [dt_1, dt_2, … dt_k] such that the sum of the time steps is equal to the total time step t0 - t1. If a uniform time step is used, it may equivalently be specified by stating the number of steps (n_steps) to take. This method provides both options via the time_step and num_steps parameters. However, not all methods need discretization along time direction (e.g. method of lines) so this argument may not be applicable to some implementations.

The workhorse of this method is the one_step_fn. For the commonly used methods, see functions in math.pde.steppers module.

The mapping between the arguments of this method and the above equation are described in the Args section below.

For a simple instructive example of implementation of this method, see models.GenericItoProcess.fd_solver_backward.

TODO(b/142309558): Complete documentation.

Args:#

start_time: Real positive scalar Tensor. The start time of the grid. Corresponds to time t0 above.
end_time: Real scalar Tensor smaller than the start_time and greater than zero. The time to step back to. Corresponds to time t1 above.
coord_grid: List of n rank 1 real Tensors. n is the dimension of the domain. The i-th Tensor has shape, [d_i] where d_i is the size of the grid along axis i. The coordinates of the grid points. Corresponds to the spatial grid G above.
values_grid: Real Tensor containing the function values at time start_time which have to be stepped back to time end_time. The shape of the Tensor must broadcast with [K, d_1, d_2, ..., d_n]. The first axis of size K is the values batch dimension and allows multiple functions (with potentially different boundary/final conditions) to be stepped back simultaneously.
discounting: Callable corresponding to r(t,x) above. If not supplied, zero discounting is assumed.
one_step_fn: The transition kernel. A callable that consumes the following arguments by keyword:
1. ‘time’: Current time
2. ‘next_time’: The next time to step to. For the backwards in time evolution, this time will be smaller than the current time.
3. ‘coord_grid’: The coordinate grid.
4. ‘values_grid’: The values grid.
5. ‘boundary_conditions’: The boundary conditions.
6. ‘quadratic_coeff’: A callable returning the quadratic coefficients of the PDE (i.e. (1/2)D_{ij}(t, x) above). The callable accepts the time and coordinate grid as keyword arguments and returns a Tensor with shape that broadcasts with [dim, dim].
7. ‘linear_coeff’: A callable returning the linear coefficients of the PDE (i.e. mu_i(t, x) above). Accepts time and coordinate grid as keyword arguments and returns a Tensor with shape that broadcasts with [dim].
8. ‘constant_coeff’: A callable returning the coefficient of the linear homogenous term (i.e. r(t,x) above). Same spec as above. The one_step_fn callable returns a 2-tuple containing the next coordinate grid, next values grid.
boundary_conditions: A list of size dim containing boundary conditions. The i’th element of the list is a 2-tuple containing the lower and upper boundary condition for the boundary along the i`th axis.
start_step_count: Scalar integer Tensor. Initial value for the number of time steps performed. Default value: 0 (i.e. no previous steps performed).
num_steps: Positive int scalar Tensor. The number of time steps to take when moving from start_time to end_time. Either this argument or the time_step argument must be supplied (but not both). If num steps is k>=1, uniform time steps of size (t0 - t1)/k are taken to evolve the solution from t0 to t1. Corresponds to the n_steps parameter above.
time_step: The time step to take. Either this argument or the num_steps argument must be supplied (but not both). The type of this argument may be one of the following (in order of generality): (a) None in which case num_steps must be supplied. (b) A positive real scalar Tensor. The maximum time step to take. If the value of this argument is dt, then the total number of steps taken is N = (t0 - t1) / dt rounded up to the nearest integer. The first N-1 steps are of size dt and the last step is of size t0 - t1 - (N-1) * dt. (c) A callable accepting the current time and returning the size of the step to take. The input and the output are real scalar Tensors.
values_transform_fn: An optional callable applied to transform the solution values at each time step. The callable is invoked after the time step has been performed. The callable should accept the time of the grid, the coordinate grid and the values grid and should return the values grid. All input arguments to be passed by keyword.
dtype: The dtype to use.
**kwargs: Any other keyword args needed by an implementation.

Returns:#

A tuple object containing at least the following attributes: final_values_grid: A Tensor of same shape and dtype as values_grid. Contains the final state of the values grid at time end_time. final_coord_grid: A list of Tensors of the same specification as the input coord_grid. Final state of the coordinate grid at time end_time. step_count: The total step count (i.e. the sum of the start_step_count and the number of steps performed in this call.). final_time: The final time at which the evolution stopped. This value is given by max(min(end_time, start_time), 0).

`fd_solver_forward`

View source

fd_solver_forward(
    start_time, end_time, coord_grid, values_grid, one_step_fn=None,
    boundary_conditions=None, start_step_count=0, num_steps=None, time_step=None,
    values_transform_fn=None, dtype=None, **kwargs
)

Returns a solver for the Fokker Plank equation of this process.

The Fokker Plank equation (also known as the Kolmogorov Forward equation) associated to this Ito process is given by:

  V_t + Sum[(mu_i(t, x) V)_i, 1<=i<=n]
    - (1/2) Sum[ (D_{ij} V)_{ij}, 1 <= i,j <= n] = 0

with the initial value condition $$V(0, x) = u(x)$$.

This method evolves a spatially discretized solution of the above PDE from time t0 to time t1 > t0 (i.e. forwards in time). The solution V(t,x) is assumed to be discretized on an n-dimensional rectangular grid. A rectangular grid, G, in n-dimensions may be described by specifying the coordinates of the points along each axis. For example, a 2 x 4 grid in two dimensions can be specified by taking the cartesian product of [1, 3] and [5, 6, 7, 8] to yield the grid points with coordinates: [(1, 5), (1, 6), (1, 7), (1, 8), (3, 5) ... (3, 8)].

Batching of solutions is supported. In this context, batching means the ability to represent and evolve multiple independent functions V (e.g. V1, V2 …) simultaneously. A single discretized solution is specified by stating its values at each grid point. This can be represented as a Tensor of shape [d1, d2, … dn] where di is the grid size along the ith axis. A batch of such solutions is represented by a Tensor of shape: [K, d1, d2, … dn] where K is the batch size. This method only requires that the input parameter values_grid be broadcastable with shape [K, d1, … dn].

The evolution of the solution from t0 to t1 is often done by discretizing the differential equation to a difference equation along the spatial and temporal axes. The temporal discretization is given by a (sequence of) time steps [dt_1, dt_2, … dt_k] such that the sum of the time steps is equal to the total time step t1 - t0. If a uniform time step is used, it may equivalently be specified by stating the number of steps (n_steps) to take. This method provides both options via the time_step and num_steps parameters. However, not all methods need discretization along time direction (e.g. method of lines) so this argument may not be applicable to some implementations.

The workhorse of this method is the one_step_fn. For the commonly used methods, see functions in math.pde.steppers module.

The mapping between the arguments of this method and the above equation are described in the Args section below.

For a simple instructive example of implementation of this method, see models.GenericItoProcess.fd_solver_forward.

TODO(b/142309558): Complete documentation.

Args:#

start_time: Real positive scalar Tensor. The start time of the grid. Corresponds to time t0 above.
end_time: Real scalar Tensor smaller than the start_time and greater than zero. The time to step back to. Corresponds to time t1 above.
coord_grid: List of n rank 1 real Tensors. n is the dimension of the domain. The i-th Tensor has shape, [d_i] where d_i is the size of the grid along axis i. The coordinates of the grid points. Corresponds to the spatial grid G above.
values_grid: Real Tensor containing the function values at time start_time which have to be stepped back to time end_time. The shape of the Tensor must broadcast with [K, d_1, d_2, ..., d_n]. The first axis of size K is the values batch dimension and allows multiple functions (with potentially different boundary/final conditions) to be stepped back simultaneously.
one_step_fn: The transition kernel. A callable that consumes the following arguments by keyword:
1. ‘time’: Current time
2. ‘next_time’: The next time to step to. For the backwards in time evolution, this time will be smaller than the current time.
3. ‘coord_grid’: The coordinate grid.
4. ‘values_grid’: The values grid.
5. ‘quadratic_coeff’: A callable returning the quadratic coefficients of the PDE (i.e. (1/2)D_{ij}(t, x) above). The callable accepts the time and coordinate grid as keyword arguments and returns a Tensor with shape that broadcasts with [dim, dim].
6. ‘linear_coeff’: A callable returning the linear coefficients of the PDE (i.e. mu_i(t, x) above). Accepts time and coordinate grid as keyword arguments and returns a Tensor with shape that broadcasts with [dim].
7. ‘constant_coeff’: A callable returning the coefficient of the linear homogenous term (i.e. r(t,x) above). Same spec as above. The one_step_fn callable returns a 2-tuple containing the next coordinate grid, next values grid.
boundary_conditions: A list of size dim containing boundary conditions. The i’th element of the list is a 2-tuple containing the lower and upper boundary condition for the boundary along the i`th axis.
start_step_count: Scalar integer Tensor. Initial value for the number of time steps performed. Default value: 0 (i.e. no previous steps performed).
num_steps: Positive int scalar Tensor. The number of time steps to take when moving from start_time to end_time. Either this argument or the time_step argument must be supplied (but not both). If num steps is k>=1, uniform time steps of size (t0 - t1)/k are taken to evolve the solution from t0 to t1. Corresponds to the n_steps parameter above.
time_step: The time step to take. Either this argument or the num_steps argument must be supplied (but not both). The type of this argument may be one of the following (in order of generality): (a) None in which case num_steps must be supplied. (b) A positive real scalar Tensor. The maximum time step to take. If the value of this argument is dt, then the total number of steps taken is N = (t1 - t0) / dt rounded up to the nearest integer. The first N-1 steps are of size dt and the last step is of size t1 - t0 - (N-1) * dt. (c) A callable accepting the current time and returning the size of the step to take. The input and the output are real scalar Tensors.
values_transform_fn: An optional callable applied to transform the solution values at each time step. The callable is invoked after the time step has been performed. The callable should accept the time of the grid, the coordinate grid and the values grid and should return the values grid. All input arguments to be passed by keyword.
dtype: The dtype to use.
**kwargs: Any other keyword args needed by an implementation.

Returns:#

A tuple object containing at least the following attributes: final_values_grid: A Tensor of same shape and dtype as values_grid. Contains the final state of the values grid at time end_time. final_coord_grid: A list of Tensors of the same specification as the input coord_grid. Final state of the coordinate grid at time end_time. step_count: The total step count (i.e. the sum of the start_step_count and the number of steps performed in this call.). final_time: The final time at which the evolution stopped. This value is given by max(min(end_time, start_time), 0).

`name`

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name()

The name to give to ops created by this class.

`sample_paths`

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sample_paths(
    times, num_samples=1, initial_state=None, random_type=None, seed=None, **kwargs
)

Returns a sample of paths from the process.

Args:#

times: A Tensor of positive real values of a shape [T, k], where T is either empty or a shape which is broadcastable to batch_shape (as defined when this ItoProcess was initialised) and k is the number of time points. The definition of batch_shape and how it is set is dictated by the individual child classes. Typically batch_shape is set based on the shapes of the input parameters that define the ItoProcess, see GeometricBrownianMotion for an example. The times at which the path points are to be evaluated.
num_samples: Positive scalar int. The number of paths to draw.
initial_state: Tensor of shape [dim]. The initial state of the process. Default value: None which maps to a zero initial state.
random_type: Enum value of RandomType. The type of (quasi)-random number generator to use to generate the paths. Default value: None which maps to the standard pseudo-random numbers.
seed: Seed for the random number generator. The seed is only relevant if random_type is one of [STATELESS, PSEUDO, HALTON_RANDOMIZED, PSEUDO_ANTITHETIC, STATELESS_ANTITHETIC]. For PSEUDO, PSEUDO_ANTITHETIC and HALTON_RANDOMIZED the seed should be an Python integer. For STATELESS and STATELESS_ANTITHETIC must be supplied as an integer Tensor of shape [2]. Default value: None which means no seed is set.
**kwargs: Any other keyword args needed by an implementation.

Returns:#

A real Tensor of shape [batch_shape, num_samples, k, n] where k is the size of times, n is the dimension of the process and batch_shape is the batch dimensions of the ItoProcess.

`volatility_fn`

View source

volatility_fn()

Python callable calculating the instantaneous volatility matrix.

The callable should accept two real Tensor arguments of the same dtype and shape times_shape. The first argument is the scalar time t, the second argument is the value of Ito process X - Tensor of shape batch_shape + [dim]. Here batch_shape is an arbitrary shape. The result is value of volatility S_ij(t, X). The return value of the callable is a real Tensor of the same dtype as the input arguments and of shape batch_shape + [dim, dim].

Returns:#

The instantaneous volatility callable.

tf_quant_finance.models.ItoProcess

Contents

tf_quant_finance.models.ItoProcess#

References#

Methods#

dim

drift_fn

Returns:#

dtype

fd_solver_backward

Args:#

Returns:#

fd_solver_forward

Args:#

Returns:#

name

sample_paths

Args:#

Returns:#

volatility_fn

Returns:#

`dim`

`drift_fn`

`dtype`

`fd_solver_backward`

`fd_solver_forward`

`name`

`sample_paths`

`volatility_fn`