[R-SIG-Finance] R-SIG-Finance Digest, Vol 41, Issue 8: American Basket Options

[Moshe Olshansky]

Hi Matt,

Thank you for your note.

In R things are much easier - one can use mvrnorm
function from MASS package to generate a sample from a
multivariate normal distribution with a given
covariance matrix. I did this to price European basket
options.
The main problem in using Monte Carlo for American
options is not knowing when to exercise prior to
expiration date.

Regards,

Moshe.

P.S. as to covariance matrix, does it make sense to
use a longer history to estimate the correlation
matrix and a shorter one to estimate individual
(marginal) variances?

--- Matt Slezak <nocman43202 at yahoo.com> wrote:

> American Basket Options
> 
> Here is an efficient method I used to value European
> options on a basket of securities.  Maybe you can
> piece together R code for the American basket
> option. 
> I have done something similar in Excel using
> QuantLibXL.  First calculate the correlation matrix
> for the assets in the portfolio.  Next, do a
> Cholesky
> decomposition on this matrix.  Generate 1000 random
> numbers for each asset in the portfolio. Transform
> these into correlated random numbers by multiplying
> each by its factors from the Cholesky matrix.  Next
> generate quasi-random sequences (Halton or other,
> range 0 to 1) and pull these numbers from the
> cumalative normal distribution.  Next you create
> Geometric Brownian Motions for each asset using the
> respective correlated random number, the asset's
> volatility, and the asset's drift.  Run the 1000
> simulations for each asset over the time horizon,
> then
> calculate the net present value (NPV) of the payoffs
> for the whole portfolio (which is the maximum of the
> postive NPV or 0, since the long option cannot have
> a
> negative return).  The average payoff of the
> portfolio
> is the value of the basket option.  
> 
> If someone can add how one determines whether the
> option is exercised in each stage of the Monte Carlo
> simulation for American exercise it would be
> appreciated - I used this method for European
> options.
> 
> Hope this is helpful -Matt Slezak
> --- r-sig-finance-request at stat.math.ethz.ch wrote:
> 
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> > When replying, please edit your Subject line so it
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> > than "Re: Contents of R-SIG-Finance digest..."
> > > Today's Topics:
> > 
> >    1. Re: American basket options (Dale Smith)
> >    2. Re: Burns on Cramer (Patrick Burns)
> >    3. Re: returns convention
> (david.jessop at ubs.com)
> >    4. Re: Burns on Cramer (BBands)
> >    5. Re: Burns on Cramer (Patrick Burns)
> >    6. Black -Litterman Model
> > (ngottlieb at marinercapital.com)
> >    7. Re: Black -Litterman Model (Brian G.
> Peterson)
> >    8. Re: Black -Litterman Model
> > (ngottlieb at marinercapital.com)
> >    9. Re: American basket options (Moshe
> Olshansky)
> >   10. Re: American basket options (Wojciech
> > Slusarski)
> > > From: "Dale Smith" <dsmith at viciscapital.com>
> > To: "Wojciech Slusarski"
> > <wojciech.slusarski at gmail.com>,
> > 	<r-sig-finance at stat.math.ethz.ch>
> > Date: Tue, 9 Oct 2007 07:05:07 -0400
> > Subject: Re: [R-SIG-Finance] American basket
> options
> > 
> > For baskets of stocks larger than three or so,
> Monte
> > Carlo methods outperform the best finite
> difference
> > code. As mentioned below, there are Monte Carlo
> > algorithms for the American case.
> > 
> > Dale Smith, Ph.D.
> > Vicis Capital, LLC
> > 
> > -----Original Message-----
> > From: r-sig-finance-bounces at stat.math.ethz.ch
> > [mailto:r-sig-finance-bounces at stat.math.ethz.ch]
> On
> > Behalf Of Wojciech Slusarski
> > Sent: Tuesday, October 09, 2007 6:00 AM
> > To: r-sig-finance at stat.math.ethz.ch
> > Subject: Re: [R-SIG-Finance] American basket
> options
> > 
> > There is an algorithm called OLS Monte Carlo, or
> > Longstaff-Schwarz
> > algorithm for valuation of american/bermudan
> options
> > using MC method,
> > though it can be a bit tricky to programm that for
> a
> > portfolio of 10
> > securities and be a bit unstable, though worth of
> > trying. If the
> > dividends are not high, it should not differ much
> > from a european
> > option priced using Monte Carlo. If dividends are
> > high, then the price
> > should be slightly higher.
> > 
> > Regards,
> > Wojciech ¦lusarski
> > 
> > 
> > 2007/10/9, Moshe Olshansky
> <m_olshansky at yahoo.com>:
> > > This is an OTC traded option.
> > >
> > > For a European option one can estimate the
> > covariance
> > > matrix and then use Monte Carlo (taking into
> > account
> > > the dividends for each stock). This is pretty
> > > straightforward (well, there may be many ways to
> > > estimate the covariance matrix but let's use the
> > > simplest one).
> > >
> > > Regards,
> > >
> > > Moshe.
> > >
> > > --- Krishna Kumar <kriskumar at earthlink.net>
> wrote:
> > >
> > > > I am just curious as to if this is being
> traded
> > in
> > > > some market ?.
> > > >
> > > > This is probably not very helpful but I don't
> > think
> > > > a European style
> > > > basket is there in the existing packages.
> > European
> > > > style baskets are
> > > > themselves tricky if you want to get the
> basket
> > > > smile right etc.
> > > > American style baskets will be messy.
> > > >
> > > > Cheers
> > > > Krishna
> > > >
> > > >
> > > >
> > > >
> > > > Moshe Olshansky wrote:
> > > > > Hello,
> > > > >
> > > > > Is there any R code which allows to
> calculate
> > the
> > > > > price of an American basket option (option
> on
> > a
> > > > price
> > > > > of a portfolio)?
> > > > > If yes, are there any references to how
> > accurate
> > > > these
> > > > > calculations are?
> > > > > If no, can anybody recommend a relatively
> easy
> > to
> > > > use
> > > > > software doing this?
> > > > >
> > > > > Are there any non Monte Carlo methods to
> > compute
> > > > (even
> > > > > roughly) the price on an American basket put
> > > > option on
> > > > > a portfolio of 10 dividend paying stocks
> with
> > 6
> > > > months
> > > > > maturity?
> > > > >
> > > > > Thank you in advance,
> > > > >
> > > > > Moshe.
> > > > >
> 
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