GBMCharacteristicFunction

class quantmetrics.price_calculators.gbm_pricing.gbm_characteristic_function.GBMCharacteristicFunction(model: LevyModel, option: Option)

Bases: object

Implements the characteristic function for a Geometric Brownian Motion (GBM) 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

Calculate the characteristic function for the GBM model.

Parameters

unp.ndarray

Input array for the characteristic function.

Returns

np.ndarray

The characteristic function values.

Notes

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

\[\Phi^{\mathbb{Q}}(u) = \exp\left\{T \left[i u b^\mathbb{Q} -\frac{u^2}{2} c \right]\right\},\]

Where:

\[b^{\mathbb{Q}} = r - \frac{\sigma^2}{2}, \quad c = \sigma^2\]

• \(\mathbb{Q}\) 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.

References

[1]

Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of political economy, 81(3), 637-654.

[2]

Matsuda, K. (2004). Introduction to option pricing with Fourier transform: Option pricing with exponential Lévy models. Department of Economics The Graduate Center, The City University of New York, 1-241.