August 2009
Intermediate to advanced
244 pages
9h 5m
English
Content preview from Financial Modelling in Python
Become an O’Reilly member and get unlimited access to this title plus top books and audiobooks from O’Reilly and nearly 200 top publishers, thousands of courses curated by job role, 150+ live events each month,
O’Reilly covers everything we've got, with content to help us build a world-class technology community, upgrade the capabilities and competencies of our teams, and improve overall team performance as well as their engagement.I wanted to learn C and C++, but it didn't click for me until I picked up an O'Reilly book. When I went on the O’Reilly platform, I was astonished to find all the books there, plus live events and sandboxes so you could play around with the technology.I’ve been on the O’Reilly platform for more than eight years. I use a couple of learning platforms, but I'm on O'Reilly more than anybody else. When you're there, you start learning. I'm never disappointed.I'm always learning. So when I got on to O'Reilly, I was like a kid in a candy store. There are playlists. There are answers. There's on-demand training. It's worth its weight in gold, in terms of what it allows me to do.
P1: JYS
app04 JWBK378-Fletcher April 24, 2009 8:20 Printer: Yet to come
Appendix D
Pickup Value Regression
The seminal paper by Longstaff and Schwartz [15], on estimating the early exercise premium,
uses regressions of the holding value, i.e. the value of holding on to the option. In this short
appendix we develop a simple alternative regression scheme for determining the early exercise
premium of a callable structure when pricing using Monte-Carlo.
Consider times T
1
< T
2
<...<T
N
and denote B
t
as the numeraire at time t. The holding
value H
n−1
(T
n−1
) at time T
n−1
is given by the relation below
HV
n−1
(T
n−1
) = B
T
n−1
E
B
−1
T
n
max(IEV
n
(T
n
), HV
n
(T
n
))|F
T
n−1
= B
T
n−1
E
B
−1
T
n
(IEV
n
(T
n
) −HV
n
(T
n
))
+
|F
T
n−1
+ B
T
n−1
E
B
−1
T
n
HV
n
(T
n
)|F
T
n−1
(D.1)
where IEV
n
(T
n
) denotes the immediate exercise values at time T
n
. Setting HV
N
(T
N
) = 0 and
using the above recursion relation, we obtain
HV
n−1
(T
n−1
) = B
T
n−1
N
m=n
E
B
−1
T
m
(IEV
m
(T
m
) −HV
m
(T
m
))
+
|F
T
n−1
(D.2)
Let’s denote the exercise region at time T
n
by R
n
,
R
n
=
{
ω ∈ : H
n
(T
n
,ω) ≤ IEV
n
(T
n
,ω)
}
(D.3)
=
{
ω ∈ : IEV
n
(T
n
,ω) − H
n
(T
n
,ω) ≥ 0
}
(D.4)
The stopping time is then (T
N+1
denoting no exercise)
τ (ω) = min
{
T
n
, n ≥ 1|ω ∈ R
n
}
∧ N +1(D.5)
The pickup value PKV
n
(T
n
) at time T
n
is defined by
PKV
n
(T
n
):= IEV
n
(T
n
) −HV
n
(T
n
)(D.6)
At each time T
n
we approximate the the pickup value by the following sum:
G
n
(T
n
,ω) =
M
k=1
α
k
(T
n
)X
k
x
1
(T
n
,ω), x
2
(T
n
,ω),...,x
p
(T
n
,ω)
(D.7)
where {X
1
(...), X
2
(...), ..., X
m
(...)} are the basis ...
Become an O’Reilly member and get unlimited access to this title plus top books and audiobooks from O’Reilly and nearly 200 top publishers, thousands of courses curated by job role, 150+ live events each month,
and much more.
Read now
Unlock full access
More than 5,000 organizations count on O’Reilly
O’Reilly covers everything we've got, with content to help us build a world-class technology community, upgrade the capabilities and competencies of our teams, and improve overall team performance as well as their engagement.Julian F.
I wanted to learn C and C++, but it didn't click for me until I picked up an O'Reilly book. When I went on the O’Reilly platform, I was astonished to find all the books there, plus live events and sandboxes so you could play around with the technology.Addison B.
I’ve been on the O’Reilly platform for more than eight years. I use a couple of learning platforms, but I'm on O'Reilly more than anybody else. When you're there, you start learning. I'm never disappointed.Amir M.
I'm always learning. So when I got on to O'Reilly, I was like a kid in a candy store. There are playlists. There are answers. There's on-demand training. It's worth its weight in gold, in terms of what it allows me to do.