(to account for different   conventions)



            Results

            =======

            delta_list  :   array

                            year    fractions

            ”’

start = date_list[ 0 ]
delta_list = [(date - start).days / day_count
for date in date_list]
return np.array(delta_list)

This function can then be applied as follows:

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 ]:  get_year_deltas(dates)

Out[ 4 ]:   array([ 0.                              ,       0.49589041,     1.                              ])

When modeling the short rate, it becomes clear what the benefit of this is.

Constant Short Rate

We focus on the simplest case for discounting by the short rate; namely, the case where the

short rate is constant through time. Many option pricing models, like the ones of Black-

Scholes-Merton (1973), Merton (1976), and Cox-Ross-Rubinstein (1979), make this

assumption.

[ 66 ]

We assume continuous discounting, as is usual for option pricing

applications. In such a case, the general discount factor as of today, given a future date t

and a constant short rate of r, is then given by D 0 (t) = e

–rt

. Of course, for the end of the

economy we have the special case D 0 (T) = e

–rT

. Note that here both t and T are in year

fractions.

The discount factors can also be interpreted as the value of a unit zero-coupon bond (ZCB)

as of today, maturing at t and T, respectively.

[ 67 ]

Given two dates t ≥ s ≥ 0, the discount

factor relevant for discounting from t to s is then given by the equation Ds(t) = D 0 (t) /

D 0 (s) = e

–rt

/ e

–rs

= e

–rt

· e

rs

= e

–r(t–s)

.

Example 15-2 presents a Python class that translates all these considerations into Python

code.

[ 68 ]

Example 15-2. Class for risk-neutral discounting with constant short rate

DX Library Frame

constant_short_rate.py

from get_year_deltas import *

class constant_short_rate(object):
”’ Class for constant short rate discounting.

            Attributes

            ==========

            name    :   string

                            name    of  the object

            short_rate  :   float   (positive)

                            constant    rate    for discounting



            Methods

            =======