Pricing American Put Options: Early Exercise Boundary (LSM etc)

[Hob]

Hi all,

I have a quick question about American put option pricing. For an option that pays no dividends, the CALL option price is the same for both American an European. Due to the early exercise, however, the American PUT option commands a higher price. My question is where the boundary exists (and why).

As I understand it, if exercising the option at some point in time, (t=t1) prior to expiry (t=T) means the time value of money (FV(T) = PV(t1) * (1+i)^n) exceeds or meets the strike price (plus costs etc) then the option should be exercised. This I am guessing sets up a boundary condition to exercise?

Now (on the assumption the above is correct...) does the problem with Monte Carlo simulations relate to how long to stay within the boundary? i.e. optimal exercise is unknown once you have passed the boundary?

Many thanks,

Hob

 

[Daniel Duffy]

Which specific method are you using?

If you really want a sound solution I would use PDE methods and not MC. It's slow and not accurate.

 

[Hob]

Which specific method are you using?

If you really want a sound solution I would use PDE methods and not MC. It's slow and not accurate.

Monte Carlo method, for a PDE method the optimal stopping point is still present? i.e. at what point do you exercise a put option?

 

[Daniel Duffy]

Monte Carlo method, for a PDE method the optimal stopping point is still present? i.e. at what point do you exercise a put option?

Sorry, could you explain? Which method do you use?

In PDE it is not a stopping point but a moving boundary B(t) which can be computed as part of the nonlinear PDE formulation.

http://www.emis.de/journals/AMUC/_vol-70/_no_1/_meyer/meyer.pdf


BTW Glasserman's book on MC has some methods for early exercise.

 

[aaronhotchner]

Why would you use Monte Carlo for a path-dependent valuation? You should be using a tree if you're simulating, unless you have all the time in the world for the Monte Carlo to run.

As far as what your question seems to be asking -- what is the optimal optimal exercise -- there's no model in the world that could tell you that. The best case scenario is S = 0, where it's obvious to everyone that exercising immediately could not get any better. Everything else is subject to risk tolerance, unless you truly believe short-term stock behavior behaves according to Brownian motion, in which case good luck to you.

 

[Hob]

Why would you use Monte Carlo for a path-dependent valuation? You should be using a tree if you're simulating, unless you have all the time in the world for the Monte Carlo to run.

It is for a research topic, for an asset with multiple variables on the underlying so simulation on an HPC isn't an issue, it is more the methodology than the result (for the moment) I am interested in.

As far as what your question seems to be asking -- what is the optimal optimal exercise -- there's no model in the world that could tell you that. The best case scenario is S = 0, where it's obvious to everyone that exercising immediately could not get any better. Everything else is subject to risk tolerance, unless you truly believe short-term stock behavior behaves according to Brownian motion, in which case good luck to you.

I see!

 

[C.Y.]

There are different types of Monte Carlo Simulations and Numerical methods for calculating optimal exercise, but a quick back of the mental method to know quickly:
Exercise Put if Interest Earned > Cost of call + Any future divs
The call premium has embedded the market implied vol