Jump diffusions

Jump-diffusions models are a class of stochastic processes that combine a diffusion process with a jump process. The jump process is a Poisson process that generates jumps in the value of the underlying asset. The jump-diffusion model is a generalization of the Black-Scholes model that allows for the possibility of large, discontinuous jumps in the value of the underlying asset.

The most famous jump-diffusion model is the Merton model, which was introduced by Robert Merton in 1976. The Merton model assumes that the underlying asset follows a geometric Brownian motion with jumps that are normally distributed.

class quantflow.sp.jump_diffusion.JumpDiffusion(*, diffusion: ~quantflow.sp.weiner.WeinerProcess = <factory>, jumps: ~quantflow.sp.poisson.CompoundPoissonProcess)

A generic jump-diffusion model

\[dx_t = \sigma d w_t + d N_t\]

where \(w_t\) is a Weiner process with standard deviation \(\sigma\) and \(N_t\) is a CompoundPoissonProcess with intensity \(\lambda\) and generic jump distribution D

Methods:

analytical_mean	Analytical mean of the process at time t
analytical_variance	Analytical variance of the process at time t
characteristic_exponent	Characteristic exponent at time t for a given input parameter
create	Create a jump-diffusion model with a given jump distribution, volatility and jump fraction.
sample	Generate random Paths from the process.
sample_from_draws	Sample Paths from the process given a set of draws

Attributes:

diffusion	The diffusion process is a standard WeinerProcess
jumps	The jump process is a generic CompoundPoissonProcess
model_config	Configuration for the model, should be a dictionary conforming to [ConfigDict][pydantic.config.ConfigDict].

analytical_mean(t: ndarray[tuple[int, ...], dtype[floating[Any]]] | float) → ndarray[tuple[int, ...], dtype[floating[Any]]] | float

Analytical mean of the process at time t

Implement if available

analytical_variance(t: ndarray[tuple[int, ...], dtype[floating[Any]]] | float) → ndarray[tuple[int, ...], dtype[floating[Any]]] | float

Analytical variance of the process at time t

Implement if available

characteristic_exponent(t: ndarray[tuple[int, ...], dtype[floating[Any]]] | float, u: int | float | complex | ndarray | Series) → int | float | complex | ndarray | Series

Characteristic exponent at time t for a given input parameter

classmethod create(jump_distribution: type[D], vol: float = 0.5, jump_intensity: float = 100, jump_fraction: float = 0.5, jump_asymmetry: float = 0.0) → JumpDiffusion

Create a jump-diffusion model with a given jump distribution, volatility and jump fraction.

Parameters:

• jump_distribution – The distribution of jump sizes (currently only Normal and DoubleExponential are supported)

• vol – total annualized standard deviation

• jump_intensity – The average number of jumps per year

• jump_fraction – The fraction of variance due to jumps (between 0 and 1)

• jump_asymmetry – The asymmetry of the jump distribution (0 for symmetric, only used by distributions with asymmetry)

If the jump distribution is set to the Normal distribution, the model reduces to a Merton jump-diffusion model.

sample(n: int, time_horizon: float = 1, time_steps: int = 100) → Paths

Generate random Paths from the process.

Parameters:

• n – number of paths

• time_horizon – time horizon

• time_steps – number of time steps to arrive at horizon

sample_from_draws(path_w: Paths, *args: Paths) → Paths

Sample Paths from the process given a set of draws

diffusion: WeinerProcess

The diffusion process is a standard WeinerProcess

jumps: CompoundPoissonProcess[D]

The jump process is a generic CompoundPoissonProcess

model_config: ClassVar[ConfigDict] = {'extra': 'forbid'}

Configuration for the model, should be a dictionary conforming to [ConfigDict][pydantic.config.ConfigDict].