Serving the Quantitative Finance Community

 
User avatar
Alan
Posts: 2958
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

Re: American options with two free boundaries

February 2nd, 2021, 7:41 pm

So, for anybody: a plot of V(S) vs S with H=10 would be nice. Although it's built in to the solution, I'd like to visually see the two `smooth pastes'.
 
User avatar
Cuchulainn
Topic Author
Posts: 20252
Joined: July 16th, 2004, 7:38 am
Location: 20, 000

Re: American options with two free boundaries

February 2nd, 2021, 7:52 pm

So, for anybody: a plot of V(S) vs S with H=10 would be nice. Although it's built in to the solution, I'd like to visually see the two `smooth pastes'.
Yes. it should be a smooth paste (the solution was built on it as you say). I will get back.
BTW do you want delta at free boundaries like fig 9.13 of your book?

Of course, I am expecting nice graphs from bearish and Paul. Whenever you're ready. Just wondering of Jamshidian's approach is [$]C^1[$].
 
User avatar
Paul
Posts: 6604
Joined: July 20th, 2001, 3:28 pm

Re: American options with two free boundaries

February 2nd, 2021, 10:01 pm

I point out where your formulation is lacking, I then solve for a more general case, and still you aren't happy!!!
 
User avatar
Cuchulainn
Topic Author
Posts: 20252
Joined: July 16th, 2004, 7:38 am
Location: 20, 000

Re: American options with two free boundaries

February 3rd, 2021, 11:19 am

Using bearish's original data.
(Greeks by one-sided divided differences).
Attachments
3.jpg
2.jpg
1.jpg
 
User avatar
Alan
Posts: 2958
Joined: December 19th, 2001, 4:01 am
Location: California
Contact:

Re: American options with two free boundaries

February 3rd, 2021, 2:17 pm

Thanks  -- I guess the smooth paste is best seen by the continuity of delta.
 
User avatar
Cuchulainn
Topic Author
Posts: 20252
Joined: July 16th, 2004, 7:38 am
Location: 20, 000

Re: American options with two free boundaries

February 3rd, 2021, 3:30 pm

Thanks  -- I guess the smooth paste is best seen by the continuity of delta.
You're welcome. I took FDM to get delta which is OK since it is applied to an exact solution (it is _not_ a divided difference of a divided difference).
Another CSE approach (as in Matt Robinson's thesis 2019) is to differentiate the ODE wrt [$]S[$] to get an ODE for [$]\Delta[$] 

\[0 +  \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 \Delta}{\partial S^2} + (\sigma^2 + r )S \frac{\partial \Delta}{\partial S} =  0, \]

We can probably solve this analytically as before. Of course, we need 2 extra conditions at the free boundaries.

I suspect [$]\Delta[$] is not [$]C^1[$] continuous because we differentiate it at the boundaries wrt payoff (gamma, thus).
Or .. we can just differentiate [$]V[$] all over the place to see where it becomes discontinuous, if at all.
 
User avatar
bearish
Posts: 5186
Joined: February 3rd, 2011, 2:19 pm

Re: American options with two free boundaries

February 3rd, 2021, 6:50 pm

It’s trivial, if a tad tedious, to analytically differentiate the Jamshidian/Sidenius solution I posted. Some care is needed to interpret your graphs to the right of [$] S^* [$], since at that point the option has ceased to be. It’s an ex-option or, perhaps more accurately, an exercised option.
 
User avatar
Paul
Posts: 6604
Joined: July 20th, 2001, 3:28 pm

Re: American options with two free boundaries

February 3rd, 2021, 6:57 pm

This is depressing.
 
User avatar
DavidJN
Posts: 242
Joined: July 14th, 2002, 3:00 am

Re: American options with two free boundaries

February 3rd, 2021, 7:37 pm

"This is depressing".

Using a hammer to swat a fly that was killed long ago.
 
User avatar
Paul
Posts: 6604
Joined: July 20th, 2001, 3:28 pm

Re: American options with two free boundaries

February 3rd, 2021, 8:29 pm

Very nicely put!

And this particular fly was swatted in a classroom during remedial lessons for the weaker students.
 
User avatar
Cuchulainn
Topic Author
Posts: 20252
Joined: July 16th, 2004, 7:38 am
Location: 20, 000

Re: American options with two free boundaries

February 4th, 2021, 8:15 am

The current problem could be seen as a generalisation of PWQF vol 1. page 153. There the approach states it is a 'maximise' by differentiating to solve for [$]V' = 0[$] (under that assumption then you need to check [$]V'' < 0[$]? What's the motivation? How would this approach pan out for the 2-boundary case? That's not clear to me.

The smooth pasting condition gives the same results and is more in keeping with PDE methods.

All in all, it's about 4 lines of C++ code. The 'more general case' (div..) is easy as well.

Smooth pasting (also called high-contact condition) is a kind of boundary condition used to model the American option.[1] It tells that the American option value is maximized by an exercise strategy that makes the option value and option delta continuous. [2]
 
User avatar
Paul
Posts: 6604
Joined: July 20th, 2001, 3:28 pm

Re: American options with two free boundaries

February 6th, 2021, 12:58 am

 
User avatar
Cuchulainn
Topic Author
Posts: 20252
Joined: July 16th, 2004, 7:38 am
Location: 20, 000

Re: American options with two free boundaries

February 6th, 2021, 10:21 am

Thanks.
 
User avatar
Cuchulainn
Topic Author
Posts: 20252
Joined: July 16th, 2004, 7:38 am
Location: 20, 000

Re: American options with two free boundaries

February 8th, 2021, 8:50 pm

"This is depressing".

Using a hammer to swat a fly that was killed long ago.
You are probably right. But what is useful to know is the implicit assumptions inherent in these models and approximations. The original problem's solution exists, is regular and is unique (see Friedman's book on free boundaries). These approximation work in practice (mostly), do they work in theory, But there is no law of gravity that states that the approximation is [$]C^1[$]  regular. "Approximations of approximations" don't a;ways work. My solution is.
So, always flag assumptions.

// See the fly on my avatar.  A swat would ruin the banana.
 
User avatar
jherekhealy
Posts: 20
Joined: December 11th, 2017, 2:25 pm

Re: American options with two free boundaries

February 22nd, 2021, 1:56 pm

Just saw this thread late. It turns out the double exercise boundary happens naturally for American options under negative interest rates. And you may use the integral formulation to solve the exercise boundary, using similar techniques as in the single boundary case (see Andersen an Lake for example). A paper of mine on this has been accepted for publication in JCF (not sure when it will be actually published with the backlog).