DEJDCharacteristicFunction

class quantmetrics.price_calculators.dejd_pricing.dejd_characteristic_function.DEJDCharacteristicFunction(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 DEJD model.

Parameters

unp.ndarray

Input array for the characteristic function.

Returns

np.ndarray

The characteristic function values.

Notes

The characteristic function of the DEJD under the risk-neutral measure is defined as follows:

  • If emm = "mean-correcting":

\[\Phi^{\mathbb{Q}}(u) = \exp\left\{T \left[i u b^\mathbb{Q} -\frac{u^2}{2}c + \lambda \left( \frac{p\eta_1}{\eta_1 - iu} + \frac{q \eta_2}{\eta_2 + iu} - 1 \right) \right]\right\},\]

where

\[b^\mathbb{Q}= r - \frac{\sigma^2}{2} -\lambda \kappa \quad c = \sigma^2\]
\[\kappa = \frac{p\eta_1}{\eta_1 -1} + \frac{q\eta_2}{\eta_2+1} - 1, \quad q = 1-p\]

  • If emm = "Esscher" and psi = 0:

\[\Phi^{\mathbb{Q}}(u) = \exp\left\{T \left[i u b^\mathbb{Q} -\frac{u^2}{2}c + \lambda^\mathbb{Q} \left( \frac{\frac{p\eta_1}{\eta_1 - (\theta +iu)} + \frac{q\eta_2}{\eta_2 + (\theta + iu)}}{\frac{p\eta_1}{\eta_1 - \theta} + \frac{q\eta_2}{\eta_2 + \theta}} - 1 \right) \right]\right\},\]

where

\[b^\mathbb{Q}= r - \frac{\sigma^2}{2} -\lambda \kappa + \theta \sigma^2 \quad c = \sigma^2\]
\[\lambda^\mathbb{Q} = \lambda \left( \frac{p\eta_1}{\eta_1 - \theta} + \frac{q\eta_2}{\eta_2 + \theta} \right)\]

  • If emm = "Esscher" and psi != 0:

\[\Phi^{\mathbb{Q}}(u) = \exp\left\{T \left[i u b^\mathbb{Q} -\frac{u^2}{2}c + \lambda^\mathbb{Q} \left(\frac{p\eta_1 I(\theta+iu, \eta_1, \psi) + q\eta_2 I(\theta +iu, -\eta_2, \psi)}{p\eta_1 I(\theta, \eta_1, \psi) + q\eta_2 I(\theta, -\eta_2, \psi)} - 1 \right)\right] \right\},\]

where

\[b^\mathbb{Q}= r - \frac{\sigma^2}{2} -\lambda \kappa + \theta(\psi) \sigma^2 \quad c = \sigma^2\]
\[\lambda^\mathbb{Q} = \lambda \frac{1}{2}\sqrt{\frac{\pi}{|\psi|}} \left[p\eta_1 I(\theta, \eta_1, \psi) + q\eta_2 I(\theta, -\eta_2, \psi) \right],\]
\[I(a,b,\psi):= \exp \left[-\psi \left(\frac{a-b}{2\psi} \right)^2 \right] \left\{1 - erf\left[\sqrt{|\psi|} \left(\frac{a-b}{2\psi} \right) \right] \right\}\]

with

\[\psi < 0.\]

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.

  • \(p,q\geq 0, p+q =1\) are the probabilities of upward and downward jumps.

  • The upward jump sizes are exponentially distributed with mean \(1/\eta_1\) with \(\eta_1>1\).

  • The downward jump sizes are exponentially distributed with mean \(1/\eta_2\) with \(\eta_2 >0\).

  • \(\lambda\) is the jump intensity rate.

References