Risk-Neutral Discounting

Obviously, risk-neutral discounting is central to the risk-neutral valuation approach. We

therefore start by developing a Python class for risk-neutral discounting. However, it pays

to first have a closer look at the modeling and handling of relevant dates for a valuation.

Modeling and Handling Dates

A necessary prerequisite for discounting is the modeling of dates (see also Appendix C).

For valuation purposes, one typically divides the time interval between today and the final

date of the general market model T into discrete time intervals. These time intervals can be

homogenous (i.e., of equal length), or they can be heterogenous (i.e., of varying length). A

valuation library should be able to handle the more general case of heterogeneous time

intervals, since the simpler case is then automatically included. Therefore, we work with

lists of dates, assuming that the smallest relevant time interval is one day. This implies that

we do not care about intraday events, for which we would have to model time (in addition

to dates).

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To compile a list of relevant dates, one can basically take one of two approaches:

constructing a list of concrete dates (e.g., as datetime.datetime objects in Python) or of

year fractions (as decimal numbers, as is often done in theoretical works).

For example, the following two definitions of dates and fractions are (roughly)

equivalent:

In  [ 1 ]:  import datetime as dt

In  [ 2 ]:  dates   =   [dt.datetime( 2015 ,     1 ,     1 ),   dt.datetime( 2015 ,  7 ,     1 ),

dt.datetime( 2016 ,  1 ,     1 )]

In  [ 3 ]:  (dates[ 1 ] -   dates[ 0 ]).days    /   365.

Out[ 3 ]:   0.4958904109589041

In  [ 4 ]:  (dates[ 2 ] -   dates[ 1 ]).days    /   365.

Out[ 4 ]:   0.5041095890410959

In  [ 5 ]:  fractions   =   [0.0,   0.5,    1.0]

They are only roughly equivalent since year fractions seldom lie on the beginning (0 a.m.)

of a certain day. Just consider the result of dividing a year by 50.

Sometimes it is necessary to get year fractions out of a list of dates. The function

get_year_deltas presented in Example 15-1 does the job.

Example 15-1. Function to get year fractions from a list or array of datetime objects

DX Library Frame

get_year_deltas.py

import numpy as np

def get_year_deltas(date_list, day_count=365.):
”’ Return vector of floats with day deltas in years.
Initial value normalized to zero.

            Parameters

            ==========

            date_list   :   list    or  array

                            collection  of  datetime    objects

            day_count   :   float

                            number  of  days    for a   year