Conclusions

This chapter is about the numerical valuation of both European and American options

based on Monte Carlo simulation. The chapter introduces a generic valuation class, called

valuation_class. This class provides methods, for example, to estimate the most

important Greeks (Delta, Vega) for both types of options, independent of the simulation

object (risk factor/stochastic process) used for the valuation.

Based on the generic valuation class, the chapter presents two specialized classes,

valuation_mcs_european and valuation_mcs_american. The class for the valuation of

European options is mainly a straightforward implementation of the risk-neutral valuation

approach presented in Chapter 15 in combination with the numerical estimation of an

expectation term (i.e., an integral by Monte Carlo simulation, as discussed in Chapter 9).

The class for the valuation of American options needs a certain kind of regression-based

valuation algorithm. This is due to the fact that for American options an optimal exercise

policy has to be derived for a valuation. This is theoretically and numerically a bit more

involved. However, the respective present_value method of the class is still concise.

The approach taken with the DX derivatives analytics library proves to be beneficial.

Without too much effort we are able to value a pretty large class of options with the

following features:

Single risk factor options

European or American exercise

Arbitrary payoff

In addition, we can estimate the most important Greeks for this class of options. To

simplify future imports, we will again use a wrapper module, this time called

dx_valuation.py, as presented in Example 17-5.

Example 17-5. Wrapper module for all components of the library including valuation

classes

DX Library Valuation

dx_valuation.py

import numpy as np
import pandas as pd

from dx_simulation import *
from valuation_class import valuation_class
from valuation_mcs_european import valuation_mcs_european
from valuation_mcs_american import valuation_mcs_american

Again, let us enhance the init file in the dx directory (see Example 17-6) to stay

consistent here.

Example 17-6. Enhanced Python packaging file

DX Library

packaging file

init.py

import numpy as np