Levy Models

The levy_models module provides a collection of stochastic models for option pricing.

Modules

geometric_brownian_motion.py: Implements the geometric Brownian motion (GBM) model.
constant_jump_diffusion.py: Implements the constant jump-diffusion (CJD) model.
lognormal_jump_diffusion,py: Implements the lognormal jump-diffusion (LJD) model.
double_exponential_jump_diffusion.py: Implements the double-exponential jump-diffusion (DEJD) model.
variance_gamma.py: Implements the Variance Gamma (VG) model.

Classes

Below are the key classes defined in this module:

GeometricBrownianMotion

Implements the GBM stochastic process for modeling asset prices.

class quantmetrics.levy_models.geometric_brownian_motion.GeometricBrownianMotion(S0: float = 50, mu: float = 0.05, sigma: float = 0.2)[source]

Bases: LevyModel

Geometric Brownian motion model.

Parameters

S0float: Initial stock price.
mufloat: Expected return (drift).
sigmafloat: Volatility (annualized). Divide by the square root of the number of days in a year (e.g., 360) to convert to daily.

fit(data: ndarray, method: str = 'Nelder-Mead', init_params: ndarray = None, brute_tuple: tuple = ((-1, 1, 0.5), (0.05, 2, 0.5)))[source]

Fit the Geometric Brownian Motion model to the data using Maximum Likelihood Estimation (MLE).

Parameters

datanp.ndarray: The data points to fit the model.
methodstr: The minimization method, defualt is “Nelder-Mead”.
init_paramsnp.ndarray: A 2x1-dimensional numpy array containing the initial estimates for the drift (mu) and volatility (sigma).
brute_tupletuple: If initial parameters are not specified, the brute function is applied with a 2x3-dimensional tuple for each parameter as (start value, end value, step size).

Returns

minimize: The result of the minimization process containing the estimated parameters.

property model_params_conds_valid

property mu: float

pdf(data: ndarray, est_params: ndarray) → ndarray[source]

Probability density function for the Geometric Brownian Motion model.

Parameters

datanp.ndarray: The data points for which the PDF is calculated.
est_paramsnp.ndarray: Estimated parameters (mu, sigma).

Returns

np.ndarray: The probability density values.

property sigma: float

ConstantJumpDiffusion

Implements the constant jump diffusion model.

class quantmetrics.levy_models.constant_jump_diffusion.ConstantJumpDiffusion(S0: float = 50, mu: float = 0.05, sigma: float = 0.2, lambda_: float = 1, gamma: float = -0.1, N: int = 10)[source]

Bases: LevyModel

Constant jump-diffusion model.

Parameters

S0float: Initial stock price.
mufloat: Expected return (drift).
sigmafloat: Volatility (annualized). Divide by the square root of the number of days in a year (e.g., 360) to convert to daily.
lambda_float: Jump intensity rate is strictly greater than zero.
gammafloat: Mean jump size is strictly greater than -1 and non-zero.
Nint: Number of big jumps (the Poisson jumps).

fit(data: ndarray, method: str = 'Nelder-Mead', init_params: ndarray | None = None, brute_tuple: tuple = ((-1, 1, 0.5), (0.05, 2, 0.5), (0.1, 0.401, 0.1), (-0.5, 1, 0.1)))[source]

Fit the constant jump-diffusion model to the data using Maximum Likelihood Estimation (MLE).

Parameters

datanp.ndarray: The data points to fit the model.
methodstr: The minimization method, defualt is “Nelder-Mead”. Other options are the same as for the minimize function from scipy.optimize.
init_paramsnp.ndarray: A 4x1-dimensional numpy array containing the initial estimates for the drift (mu) and volatility (sigma).
brute_tupletuple: If initial parameters are not specified, the brute function is applied with a 4x3-dimensional tuple for each parameter as (start value, end value, step size).

Returns

minimize: The result of the minimization process containing the estimated parameters.

property gamma: float

property lambda_: float

property model_params_conds_valid

property mu: float

pdf(data: ndarray, est_params: ndarray) → ndarray[source]

Probability density function for the constant jump-diffusion model.

Parameters

datanp.ndarray: The data points for which the PDF is calculated.
est_paramsnp.ndarray: Estimated parameters (mu, sigma, lambda, gamma).

Returns

np.ndarray: The probability density values.

property sigma: float

LognormalJumpDiffusion

Implements the lognormal jump-diffusion model.

class quantmetrics.levy_models.lognormal_jump_diffusion.LognormalJumpDiffusion(S0: float = 50, mu: float = 0.05, sigma: float = 0.2, lambda_: float = 1, muJ: float = -0.1, sigmaJ: float = 0.1, N: int = 10)[source]

Bases: LevyModel

Lognormal jump-diffusion model.

Parameters

S0float: Initial stock price.
mufloat: Expected return (drift).
sigmafloat: Volatility (annualized). Divide by the square root of the number of days in a year (e.g., 360) to convert to daily.
lambda_float: Jump intensity rate is strictly greater than zero.
muJfloat: Mean jump size is strictly greater than -1 and non-zero.
sigmaJfloat: Standard deviation of the jump size. It is a positive number.
Nint: Number of big jumps (the Poisson jumps).

fit(data: ndarray, method: str = 'Nelder-Mead', init_params: ndarray | None = None, brute_tuple: tuple = ((-1, 1, 0.5), (0.05, 2, 0.5), (0.1, 0.401, 0.1), (-0.5, 1, 0.1), (0.05, 5, 0.5)))[source]

Fit the constant jump-diffusion model to the data using Maximum Likelihood Estimation (MLE).

Parameters

datanp.ndarray: The data points to fit the model.
methodstr: The minimization method, defualt is “Nelder-Mead”. Other options are the same as for the minimize function from scipy.optimize.
init_paramsnp.ndarray: A 5x1-dimensional numpy array containing the initial estimates for the drift (mu) and volatility (sigma).
brute_tupletuple: If initial parameters are not specified, the brute function is applied with a 5x3-dimensional tuple for each parameter as (start value, end value, step size).

Returns

minimize: The result of the minimization process containing the estimated parameters.

property lambda_: float

property model_params_conds_valid

property mu: float

property muJ: float

pdf(data: ndarray, est_params: ndarray) → ndarray[source]

Probability density function for the lognormal jump-diffusion model.

Parameters

datanp.ndarray: The data points for which the PDF is calculated.
est_paramsnp.ndarray: Estimated parameters (mu, sigma, lambda, muJ, sigmaJ).

Returns

np.ndarray: The probability density values.

property sigma: float

property sigmaJ: float

DoubleExponentialJumpDiffusion

Implements the double-exponential jump-diffusion model.

class quantmetrics.levy_models.double_exponential_jump_diffusion.DoubleExponentialJumpDiffusion(S0: float = 100, mu: float = 0.05, sigma: float = 0.16, lambda_: float = 1, eta1: float = 10, eta2: float = 5, p: float = 0.4, N: int = 10)[source]

Bases: LevyModel

Double exponential jump-diffusion model.

Parameters

S0float: Initial stock price.
mufloat: Expected return (drift).
sigmafloat: Volatility (annualized). Divide by the square root of the number of days in a year (e.g., 360) to convert to daily.
lambda_float: Jump intensity rate is strictly greater than zero.
eta1float: to be defined
eta2float: to be defined
pfloat: to be defined
Nint: Number of big jumps (the Poisson jumps).

References

Kou, S. G. (2002). A jump-diffusion model for option pricing. Management science, 48(8), 1086-1101. Ramezani, C. A., & Zeng, Y. (2007). Maximum likelihood estimation of the double exponential jump-diffusion process. Annals of Finance, 3, 487-507.

Examples

property eta1: float

property eta2: float

fit(data: ndarray, method: str = 'Nelder-Mead', init_params: ndarray | None = None, brute_tuple: tuple = ((-1, 1, 0.5), (0.05, 2, 0.5), (0.1, 0.401, 0.1), (-0.5, 1, 0.1), (0.05, 5, 0.5)))[source]

Fit the constant jump-diffusion model to the data using Maximum Likelihood Estimation (MLE).

Parameters

datanp.ndarray: The data points to fit the model.
methodstr: The minimization method, defualt is “Nelder-Mead”. Other options are the same as for the minimize function from scipy.optimize.
init_paramsnp.ndarray: A 5x1-dimensional numpy array containing the initial estimates for the drift (mu) and volatility (sigma).
brute_tupletuple: If initial parameters are not specified, the brute function is applied with a 5x3-dimensional tuple for each parameter as (start value, end value, step size).

Returns

minimize: The result of the minimization process containing the estimated parameters.

property lambda_: float

property model_params_conds_valid

property mu: float

property p: float

pdf(data: ndarray, est_params: ndarray) → ndarray[source]

Probability density function for the lognormal jump-diffusion model.

Parameters

datanp.ndarray: The data points for which the PDF is calculated.
est_paramsnp.ndarray: Estimated parameters (mu, sigma, lambda, muJ, sigmaJ).

Returns

np.ndarray: The probability density values.

property sigma: float

VarianceGamma

Implements the Variance Gamma model for stochastic processes.

class quantmetrics.levy_models.variance_gamma.VarianceGamma(S0: float = 50, mu: float = 0.05, m: float = -0.1, delta: float = 0.2, kappa: float = 0.1)[source]

Bases: LevyModel

Variance Gamma model.

Parameters

S0float: Initial stock price.
mufloat: Expected return (drift).
mfloat: Drift of the subordinated Brownian motion at a random time given by a Gamma process. If m = 0, then we have a symmetric variance gamma distribution.
deltafloat: Volatility of the subordinated Brownian motion at a random time given by a Gamma process.
kappafloat: Variance rate of the subordinator Gamma process.

property delta: float

fit(data: ndarray, method: str = 'Nelder-Mead', init_params: ndarray | None = None, brute_tuple: tuple = ((-1, 1, 0.5), (-1, 1, 0.5), (0.05, 2.05, 0.5), (0.3, 0.5, 0.1)))[source]

Fit the variance Gamma model to the data using Maximum Likelihood Estimation (MLE).

Parameters

datanp.ndarray: The data points to fit the model.
methodstr: The minimization method, defualt is “Nelder-Mead”. Other options are the same as for the minimize function from scipy.optimize.
init_paramsnp.ndarray: A 5x1-dimensional numpy array containing the initial estimates for the drift (mu) and volatility (sigma).
brute_tupletuple: If initial parameters are not specified, the brute function is applied with a 5x3-dimensional tuple for each parameter as (start value, end value, step size).

Returns

minimize: The result of the minimization process containing the estimated parameters.

property kappa: float

property m: float

property model_params_conds_valid

property mu: float

pdf(data: ndarray, est_params: ndarray) → ndarray[source]

Probability density function for the Variance Gamma model.

Parameters

datanp.ndarray: The data points for which the PDF is calculated.
est_paramsnp.ndarray: Estimated parameters (mu, m, delta, kappa).

Returns

np.ndarray: The probability density values.