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Cuchulainn
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PDE Question: Black Scholes Basket with 100 underlyings

March 12th, 2005, 3:30 pm

I have a question on BS PDE with 100 (well, 15) S1, S2, S3, ....This is impossible to solve with FDM/FEM.Is there some change of variables that could reduce this to essentially a one factor PDE? albeit it with some lower-order coupling?Buzz words ...Dimemsion reductionReduction to heat equationUse n-dim Fourier transform to get a simple ODE and retransform back to get the original solution?We might need to use some inherent symmetry property.I have absolutely no idea. Some maths type might be able to answer this.thx for any help.
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Alan
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PDE Question: Black Scholes Basket with 100 underlyings

March 12th, 2005, 3:55 pm

The transition density is known -- just an N-dim Gaussian inx_i = log S_i. So the real problem is evaluating the integral ofthis times the payoff. I would try to develop some kind of 'large N' expansionif the problem has a closed-form solution, for a given payoff,at N = Infinity. In others words, a power series in (1/N). regards,
 
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Cuchulainn
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PDE Question: Black Scholes Basket with 100 underlyings

March 12th, 2005, 4:10 pm

AlanSo we could maybe do some kind of asymtotic (quasi exact) analysis?My somewhat pessimistic idea is that there is no exact solution. But in practice I am worrying about nothing?
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Alan
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PDE Question: Black Scholes Basket with 100 underlyings

March 12th, 2005, 4:27 pm

My guess is that it depends upon the payoff.For example, consider the S&P500 index. In principle,you could analyze the options on that with the BS PDEwith 500 securities. In practice, people just "assume" thatthe index value is lognormally distributed even thougha weighted sum of lognormals *cannot* have a lognormaldistribution. My guess is that the 'assumption' is actually the correctleading term of an expansion. For example, consideran index with N securities, each security with someweights w_i -> 0 in some appropriate way as N -> Infinity.Then, I am speculating that there is a formal N-dim PDE solutionfor the call option C = C(infinity) + (1/N) C(1) + (1/N)^2 C(2) + ...,where C(infinity) is the usual 'lognormal' index solution.If the correction terms are small, then this provides ajustification for the usual (technically wrong) lognormal assumption.Just guessing, though ...
 
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Cuchulainn
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PDE Question: Black Scholes Basket with 100 underlyings

March 12th, 2005, 6:26 pm

Alan,As you can see I do not know so much about this. Is there some way to average a basket into a single Canonical option and then 1-factor PDE? That kind of thing.I suppose indeed that the payoff function is the crux.A kind of inverse problem? Find payoff for 1-factor that gives the same price as a basket?Another problem still are cross derivatives.Cannot seem to find any literature on this, tried Zhang but that's 2 factor
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PDE Question: Black Scholes Basket with 100 underlyings

March 12th, 2005, 8:06 pm

I seem to remember that there is some literature claiming that the sum of lognormals can be reasonably approximated in many circumstances by a lognormal in which the drift and volatility are obtained by matching the first two moments. That might serve as a basis to then estimate the next order term.A variation on this might be to note that if you had a product instead of a sum, then you would get exactly a log normal. You could then look into the difference between arithmetic and geometric means as a basis to get correction terms. But I've no idea if either approach will work.I also have a vague recollection of someone using a hypergeometric function of some sort. Sorry to be so vague.Googling "sum of lognormal" or "basket of lognormal" will, I think, give an abundance of references.
 
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Chukchi
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PDE Question: Black Scholes Basket with 100 underlyings

March 12th, 2005, 8:30 pm

Cuchulainn, there is a chapter on this subject in the excellent book Pricing and Managing Exotic and Hybrid Options by Vineer BhansaliThe book is out of print but you should be able to find it in any good library.
 
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PDE Question: Black Scholes Basket with 100 underlyings

March 12th, 2005, 8:40 pm

QuoteOriginally posted by: CuchulainnAlan,As you can see I do not know so much about this. Is there some way to average a basket into a single Canonical option and then 1-factor PDE? Well, I don't know about it much either but am mostly guessing how it would go.If the expansion I am suggesting for index option payoffs works, itis probably equivalent to manipulations with the PDE. For example, suppose youhave the BS-type PDE with N spatial variables S_1, S_2, ..., S_N. You could makea transformation to a new set of variables X_n, where X_1 = w_1 S_1 + w_2 S_2 + ... + w_N S_N is the "index" variable and the X_i (i > 1) are some other convenient variables (don't ask me what, since I haven't done it).Then, because the expansion "works", the PDE spatial operator reducesto the BS operator in X_1 plus terms (including cross-terms) in the other X_i. The drift and variance terms associated with X_1 would be the ones that arecomputed from the individual S_i and their covariances (see Fermion's comments earlier)But all those other terms are a perturbation and generate the other terms of my series.Having said all that, I would suspect that while you 'could' manipulate thePDE this way, it's probably more convenient to work with the integral form ofthe solution that I talked about earlier.
 
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PDE Question: Black Scholes Basket with 100 underlyings

March 13th, 2005, 6:03 am

QuoteOriginally posted by: ChukchiCuchulainn, there is a chapter on this subject in the excellent book Pricing and Managing Exotic and Hybrid Options by Vineer BhansaliThe book is out of print but you should be able to find it in any good library.ChikchithxI have the book! Do you know which chapter?
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Chukchi
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PDE Question: Black Scholes Basket with 100 underlyings

March 13th, 2005, 6:18 am

page 108 - Baskets and the Edgeworth Expansionand References on Risk Magazine articles
 
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PDE Question: Black Scholes Basket with 100 underlyings

March 13th, 2005, 7:04 am

thxthis looks certainly like a good start, it contains some of the intuition that has been posted here.see ja
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exotica
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PDE Question: Black Scholes Basket with 100 underlyings

March 14th, 2005, 11:01 pm

What about Monte Carlo methods? They are very competitive in high dimension problems.
 
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mrblue1978
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PDE Question: Black Scholes Basket with 100 underlyings

March 15th, 2005, 8:35 am

Edit : deleted (cf latest post)
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Cuchulainn
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PDE Question: Black Scholes Basket with 100 underlyings

March 15th, 2005, 4:35 pm

QuoteOriginally posted by: exoticaWhat about Monte Carlo methods? They are very competitive in high dimension problems.And very slow!
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yomi
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PDE Question: Black Scholes Basket with 100 underlyings

March 15th, 2005, 5:44 pm

QuoteOriginally posted by: CuchulainnQuoteOriginally posted by: exoticaWhat about Monte Carlo methods? They are very competitive in high dimension problems.And very slow!Take a look at this paper.
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