ARG model

The code is an implementation of ARG model given in [R1]. Its major features include:

• simulation of stochastic volatility and returns
• estimation using both MLE and GMM
• option pricing

References

[R1]	Stanislav Khrapov and Eric Renault (2014) “Affine Option Pricing Model in Discrete Time”, working paper, New Economic School. <http://goo.gl/yRVsZp>

[R2]	Christian Gourieroux and Joann Jasiak (2006) “Autoregressive Gamma Processes”, 2006, Journal of Forecasting, 25(2), 129–152. doi:10.1002/for.978

[R3]	Serge Darolles, Christian Gourieroux, and Joann Jasiak (2006) “Structural Laplace Transform and Compound Autoregressive Models” Journal of Time Series Analysis, 27(4), 477–503. doi:10.1111/j.1467-9892.2006.00479.x

Class documentation

class argamma.arg.ARG(param=None)

Class for ARG model.

\[E\left[\left.\exp\left\{ -uY_{t}\right\} \right|Y_{t-1}\right] =\exp\left\{ -a(u)Y_{t-1}-b(u)\right\}\]

Attributes

vol	Volatility series
ret	Asset return series
param	Parameters of the model
maturity	Maturity of the option or simply time horizon. Fraction of a year, i.e. 30/365
riskfree	Risk-free rate of return per day

Methods

afun(uarg, param)	Function a().
bfun(uarg, param)	Function b().
cfun(uarg, param)	Function c().
plot_abc(uarg, param)	Plot a() and b() functions on the same plot.
vsim([nsim, nobs, param])	Simulate ARG(1) process for volatility.
vsim2([nsim, nobs, param])	Simulate ARG(1) process for volatility.
rsim([param])	Simulate returns given ARG(1) process for volatility.
load_data([vol, ret])	Load data into the model object.
estimate_mle([param_start, model, bounds])	Estimate model parameters via Maximum Likelihood.
estimate_gmm([param_start, model])	Estimate model parameters using GMM.
cos_restriction()	Restrictions used in COS method of option pricing.
charfun(varg)	Risk-neutral conditional characteristic function.
option_premium([vol, moneyness, maturity, ...])	Model implied option premium via COS method.

afun(uarg, param)

Function a().

\[a\left(u\right)=\frac{\rho u}{1+cu}\]

Parameters:

uarg : array

Grid

param : ARGparams instance

Model parameters

Returns:

array

Same dimension as uarg

bfun(uarg, param)

Function b().

\[b\left(u\right)=\delta\log\left(1+cu\right)\]

Parameters:

uarg : array

Grid

param : ARGparams instance

Model parameters

Returns:

array

Same dimension as uarg

cfun(uarg, param)

Function c().

\[c\left(u\right)=\delta\log\left\{1+\frac{cu}{1-\rho}\right\}\]

Parameters:

uarg : array

Grid

param : ARGparams instance

Model parameters

Returns:

array

Same dimension as uarg

charfun(varg)

Risk-neutral conditional characteristic function.

Parameters:

varg : array

Grid for evaluation of CF. Real values only.

Returns:

array

Same dimension as varg

Notes

This method is used by COS method of option pricing

estimate_gmm(param_start=None, model='vol', **kwargs)

Estimate model parameters using GMM.

Parameters:

param_start : ARGparams instance

Starting value for optimization

model : str

Type of the model to estimate. Must be in:

• ‘vol’
• ‘ret’
• ‘joint’

uarg : array

Grid to evaluate a and b functions

zlag : int, optional

Number of lags in the instrument. Default is 1

Returns:

param_final : ARGparams instance

Estimated model parameters

mygmm.Results instance

GMM estimation results

estimate_mle(param_start=None, model=None, bounds=None)

Estimate model parameters via Maximum Likelihood.

Parameters:

param_start : ARGparams instance, optional

Starting value for optimization

model : str

Type of model to estimate. Must be in:

• ‘vol’
• ‘ret’
• ‘joint’

bounds : list of tuples

Bounds on parameters, i.e. [(min, max)]

Returns:

param_final : ARGparams instance

Estimated parameters

results : OptimizeResult instance

Optimization output

load_data(vol=None, ret=None)

Load data into the model object.

Parameters:

vol : (nobs, ) array

Volatility time series

ret : (nobs, ) array

Return time series

option_premium(vol=None, moneyness=None, maturity=None, riskfree=None, call=None, data=None, npoints=1024)

Model implied option premium via COS method.

Parameters:

vol : array_like

Current variance per day

moneyness : array_like

Log-forward moneyness, np.log(strike/price) - riskfree * maturity

maturity : float, optional

Maturity of the option or simply time horizon. Fraction of a year, i.e. 30/365

riskfree : float, optional

Risk-free rate of return per day

call : bool array_like

Call/Put flag

data : pandas DataFrame, record array, or dictionary of arrays

Structured data. Mandatory labels: vol, moneyness, maturity, riskfree, call

npoints : int

Number of points on the grid. The more the better, but slower.

Returns:

array_like

Model implied option premium via COS method

rsim(param=None)

Simulate returns given ARG(1) process for volatility.

Parameters:

param : ARGparams instance

Model parameters

Returns:

(nobs, nsim) array

Simulated data

vsim(nsim=1, nobs=100, param=None)

Simulate ARG(1) process for volatility.

\[\begin{split}Z_{t}|Y_{t-1}&\sim\mathcal{P}\left(\beta Y_{t-1}\right)\\ Y_{t}|Z_{t}&\sim\gamma\left(\delta+Z_{t},c\right)\end{split}\]

Parameters:

nsim : int

Number of series to simulate

nobs : int

Number of observations to simulate

param : ARGparams instance

Model parameters

Returns:

(nobs, nsim) array

Simulated data

vsim2(nsim=1, nobs=100, param=None)

Simulate ARG(1) process for volatility.

Uses non-central Chi-square distribution to simulate in one step.

Parameters:

nsim : int

Number of series to simulate

nobs : int

Number of observations to simulate

param : ARGparams instance

Model parameters

Returns:

(nobs, nsim) array

Simulated data