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Hi All,It seems that a definition of exact/closed form is not totally agreed upon. What does everyone think 'closed form', 'exact', and 'explicit' mean in the maths context? I ask the question in relation to the paper by Songping Zhu, who writes that his formula for the American option is called 'exact and closed form'. The fact that it is closed form seems to be laboured over in the paper, so much so that I begin to get suspicious...I guess this paper is relatively well known, and perhaps pretty important (thoughts?), even if it doesnt justify the press release from Wollongong University (Australia) - "A holy grail of mathematics appears solved!" (I dont know if its quite a Riemann hypothesis problem, but anyway...). So perhaps it is a good bench mark for deciding when a solution to a given problem is closed form. The solution is in terms of an infinite series expansion, with 3 single integrals and 2 double integrals. What about the European option solution written in terms of the normal CDF? Im interested in everyones thoughts on the topic, and maybe on the claim in this paper?!Exact and Explicit Solution for the Valuation of American Put Options, Quantitative Finance, Vol. 6, No. 3, June 2006, 229-242

QuoteOriginally posted by: outrunclosed form -> there is an equation for an exact solution with a finite number or terms, using well established sub-equations (like the cumulative Normal probability function)explicit -> Cuchulainn know's all about that... I don't know of a good definition outside FD for thatWe have regular discussions here at wilmott about whether infinite series are closed form solutions.I say "no", like outrun, but some take the opposite view.

The big discussion was on constructivism, that you not only prove that you have a solution but also that you can construct it. The most famous proponent was Brouwer's (zeta's great-uncle) fixed point theorem/Banach fixed point theorem Solve x = T(x) where T is a contractionby iterationx(n+1) = T(x(n))x(0) arbitraryThat's both a closed solution (but x on both sides of equation) and computable. In my view existence proofs are not enough, especially when algorithms are needed. BTW this scheme is so general that it contains linear and nonlinear algebra solvers as special cases, integral equations etc. Gauss knew FTP as well. Kind of amazing result that you iterate no matter what T is..Here is also a discussionQuoteWe have regular discussions here at wilmott about whether infinite series are closed form solutions.I say "no", like outrun, but some take the opposite view.I agree. An infinite series is the limit of a finite series so it is not closed; is closed always a non-discrete solution?BTW is the bespoke infnite series computable?

To be or not to be... exactThe answer to this question will probably elude us for a few more eons. Actually some knowledgeable people know that the answer is 42, althogh they don't know if it's the answer to this particular question. Otherwise, without tryint to shed any more light into this enigma, I'd say that the price of European options in Merton's jump diffusion model are exact enough for my taste. The series converges extremely fast, and you can take derivatives, so for all practical purposes you can use it as if it was a closed form solution. The series in the Zhu approximation does not seem to converge nearly as fast: in this presentation he gives an example of a (dimensionless) price after 24, 25 ... all the way to 29 iterations. The difference between consecutive iterations does not go down, so I wouldn't say that convergence was achieved. This being said, I think this is a remarkable result. Exact or not. Once you have a series that converges to the correct number, you can try to accelerate it, so it may become usable for pricing purposes (and by this I mean *calibration*, which is the first step in pricing). Anyway, Vandervolt, thanks for pointing out this article. I personally find it very interesting, I'll spend some time with it. Best,V.

QuoteOriginally posted by: outrun...but a FD grid solver converges as the grid-size increases, what's the difference between number of terms vs number of gridpoints?You mean 'decreases'?When you increase the number of gridpoints the discrete solution becomes continuous solution because the FD scheme is consistent. Then the discrete independent variable becomes a continuus independent variable.With a series, the indepdendent variable is always continuous. The issues later are 1) truncation of series 2) choosing discrete independent variable. I do not see these points addresed in bespoke paper.Does this make sense?Another answer is to confer with Costeanu's answer: 42 QuoteThe series in the Zhu approximation does not seem to converge nearly as fast: in this presentation he gives an example of a (dimensionless) price after 24, 25 ... all the way to 29 iterations. The difference between consecutive iterations does not go down, so I wouldn't say that convergence was achieved. Normally, one uses the root test or ratio test to prove convergence but this series on page 18 looks real hard to show this. edit: I see on page 28 the ratio test. BTW the ratio test as cited should beV(m+1) * p / (V(m) * m+1) < 1 imoAnd to compute looks awful hard???

QuoteOriginally posted by: outrunYes, indeed, oeps, 'decreases'You can write don *any* algorithm as a long single-lined formula. The fact that FD converges as the discretization step 'decreases' means that FD is part of the 'infinite series exact solutions'. People who say 'infinite series solution' is a 'closed form solutions' thus should accept FD as closed form! Tsk tsk tskWell, I have never really thought about like this until joining Wilmott.

QuoteOriginally posted by: outrunQuoteOriginally posted by: CuchulainnQuoteOriginally posted by: outrunYes, indeed, oeps, 'decreases'You can write don *any* algorithm as a long single-lined formula. The fact that FD converges as the discretization step 'decreases' means that FD is part of the 'infinite series exact solutions'. People who say 'infinite series solution' is a 'closed form solutions' thus should accept FD as closed form! Tsk tsk tskWell, I have never really thought about like this until joining Wilmott.you should write a book, "The first and complete almanac of closed form solution of exotic derivatives of infinite dimensional stochastic processes"Such a book would never converge

QuoteOriginally posted by: outrunQuoteOriginally posted by: CuchulainnQuoteOriginally posted by: outrunQuoteOriginally posted by: CuchulainnQuoteOriginally posted by: outrunYes, indeed, oeps, 'decreases'You can write don *any* algorithm as a long single-lined formula. The fact that FD converges as the discretization step 'decreases' means that FD is part of the 'infinite series exact solutions'. People who say 'infinite series solution' is a 'closed form solutions' thus should accept FD as closed form! Tsk tsk tskWell, I have never really thought about like this until joining Wilmott.you should write a book, "The first and complete almanac of closed form solution of exotic derivatives of infinite dimensional stochastic processes"Such a book would never converge hahaha,strange though... Who's the one that states 'infinite series solution' is a 'closed form solutions'? *Any* computer program can be written down as a long single-lined formula. A 10 day QED simulation on the upcoming NCCS exaflop machine can easily be written down as a 'closed form solution' with only 1.000.000.000.000.000.000.000.000 terms , ..far less that infinite!Well, me for one. Its impossible to define closed form vs non-closed form in any non-arbitrary way, so I call pretty much anything that isn't FD, FE, or MC as closed form. Unless I decide not to. A good question is, does the 'closed form' solution teach me anything that isn't obvious from the PDE equation. Can I differentiate it to get the greeks? Can I use it to find limits? Some infinite series are understandable, and meaningful - think of all the intuition about fourier series that has been built up by engineers working in frequency space. Others are not. This understanding is the main benefit of a closed form solution (although speed of execution is another distinguishing factor).In this particular case, I'd much rather use one of the (pretty accurate afaik) analytical approximations to the American put option, than some massive intractable series expansion.

"close" form doesn't always translate to faster computation, particularly an infinite series whose terms all contain double integrals. Think about binomial tree or FD solutions. Let a_n be the option price when maturity is partitioned into n equal sections. a_n, written explicitly, is just a series of matrix multiplication and max operations, which cannot be "closer". Then an infinite series can be constructed with terms b_n = a_n - a_{n-1}.

Hello,I'm currently working on my Bachelor Thesis in Mathematics (at the University ofCologne, Germany) on Zhu's paper (reference see below). I saw your entries andfound it interesting what you have been discussing in 2009 about this paper.Now I am wondering if anyone knows about the current state of this "explicit andexact solution" ? Any pointers will be appreciated.Songping Zhu, Exact and Explicit Solution for the Valuation of American PutOption, Quantitative Finance, Vol. 6, No. 3, June 2006, 229-242