LJDCharacteristicFunction
- class quantmetrics.price_calculators.ljd_pricing.ljd_characteristic_function.LJDCharacteristicFunction(model: LevyModel, option: Option)[source]
Bases:
object
Implements the characteristic function for a lognormal jump-diffusion (LJD) model.
Parameters
- modelLevyModel
A LevyModel object specifying the underlying asset’s model and its parameters.
- optionOption
An Option object specifying the option parameters: interest rate, strike price, time to maturity, dividend yield and the equivalent martingale measure.
- calculate(u: ndarray) ndarray [source]
Calculate the characteristic function for the LJD model.
Parameters
- unp.ndarray
Input array for the characteristic function.
Returns
- np.ndarray
The characteristic function values.
Notes
The characteristic function of the LJD under the risk-neutral measure is defined as follows:
If
emm = "mean-correcting"
:
\[\Phi^{\mathbb{Q}}(u) = \exp\left\{T \left[i u b^\mathbb{Q} -\frac{u^2}{2}c + \lambda \left( e^{iu \mu_j - \frac{u^2}{2}\sigma_J^2} - 1 \right) \right]\right\},\]where
\[b^\mathbb{Q}= r - \frac{\sigma^2}{2} -\lambda \kappa \quad c = \sigma^2\]\[\kappa = \exp \left(\mu_J + \frac{\sigma_J^2}{2} \right) - 1\]If
emm = "Esscher"
:
\[\Phi^{\mathbb{Q}}(u) = \exp\left\{T \left[i u b^\mathbb{Q} -\frac{u^2}{2}c + \lambda^\mathbb{Q} \left( \left[e^{iu \mu_j - \frac{u^2}{2}\sigma_J^2} e^{iu \theta \sigma_J^2}\right]^{\frac{1}{g(\psi)}} - 1 \right) \right]\right\},\]where
\[b^\mathbb{Q}= r - \frac{\sigma^2}{2} -\lambda \kappa + \theta(\psi) \sigma^2 \quad c = \sigma^2\]\[\lambda^\mathbb{Q} = \lambda f(\theta)\]\[f(y) = \frac{1}{\sqrt{g(\psi)}} \exp \left[\frac{1}{g(\psi)} \left(\mu_J y + \frac{\sigma_J^2}{2} y^2 + \psi \mu_J^2 \right) \right] \quad \text{and} \quad g(\psi) = 1 - 2\psi \sigma_J^2\]with
\[\psi < \frac{1}{2\sigma_J^2} \]The first-order Esscher parameter \(\theta\) is the risk premium (market price of risk) and which is the unique solution to the martingale equation for each \(\psi\) which is the second-order Esscher parameter. See the documentation of the
RiskPremium
class for the martingale equation and refer to [1] for more details.\(\mathbb{Q}\) is the risk-neutral measure.
\(T\) is the time to maturity.
\(i\) is the imaginary unit.
\(u\) is the input variable.
\(r\) is the risk-free interest rate.
\(\sigma\) is the volatility of the underlying asset.
\(\mu_J\) is the mean of the jump sizes.
\(\sigma_J\) is the standard deviation of the jump sizes.
\(\lambda\) is the jump intensity rate.
References