Why So Negative to Negative Probabilities?

Espen Gaarder Haug THE COLLECTOR: Why so Negative to Negative Probabilities?

What is the probability of the expected being neither expected nor unexpected?

1The History of Negative paper: “The Physical Interpretation of Quantum Feynman discusses mainly the Bayes formu- Probability Mechanics’’ where he introduced the concept of la for conditional probabilities negative energies and negative probabilities: In finance negative probabilities are considered P(i) P(i α)P(α), nonsense, or at best an indication of model- “Negative energies and probabilities should = | α breakdown. Doing some searches in the finance not be considered as nonsense. They are well- ! where P(α) 1. The idea is that as long as P(i) literature the comments I found on negative defined concepts mathematically, like a sum of α = is positive then it is not a problem if some of the probabilities were all negative,1 see for example negative money...”, Paul Dirac " probabilities P(i α) or P(α) are negative or larger Brennan and Schwartz (1978), Hull and White | The idea of negative probabilities has later got than unity. This approach works well when one (1990), Derman, Kani, and Chriss (1996), Chriss increased attention in physics and particular in cannot measure all of the conditional probabili- (1997), Rubinstein (1998), Jorgenson and Tarabay quantum mechanics. Another famous physicist, ties P(i α) or the unconditional probabilities P(α) (2002), Hull (2002). Why is the finance society so | Richard Feynman (1987) (also with a Noble prize in an experiment. That is, the variables α can negative to negative probabilities? The most like- in Physics), argued that no one objects to using relate to hidden states. Such an approach has ly answer is simply that we “all’’ were taught that negative numbers in calculations, although therefore been used in quantum physic to solve probabilities by definition must be between 0 “minus three apples” is no valid concept in real problems involving hidden variables. and 1, as assumed in the Kolmogorov measure life. Similarly he argued how negative probabili- There has since been a multitude of papers in theoretical axioms. Our negativity to negative ties as well as probabilities above unity can be theoretical physics that focuse on the use of neg- probabilities might be as short sighted as if we very useful in probability calculations: ative probabilities. I have included a few exam- decided to limit ourselves to consider only posi- “Trying to think of negative probabilities ples in the reference list, that can also be down- tive money. I am not the first one to think that gave me a cultural shock at first, but when I loaded from the web; Castro (2000), Cereceda negative probabilities can be useful. Negative finally got easy with the concept I wrote myself (2000), Curtright and Zachosy (2001) and probabilities where to my knowledge first intro- a note so I wouldn’t forget my thoughts...It is Peacock (2002). There seems to be a continuation duced by Paul Dirac. Dirac is probably best usual to suppose that, since probabilities of of interest in negative probabilities in physics: known for his mathematical prediction of anti- events must be positive, a theory which gives matter, a feat for which he was awarded the negative numbers for such quantities must be “I have done some work recently, on mak- Noble prize in 1933. In 1942 Paul Dirac wrote a absurd.’’, Richard Feyman, 1987 ing supergravity renormalizable, by adding

*Thanks to JØrgen Haug and Sebastian Gutierrez for useful comments. Any errors or views in this article are naturally my own responsibility. The ideas from this article was pre- sented at the ICBI Global Derivatives Conference, May 26-2004, Madrid, Spain.

34 Wilmott magazine ^ 35 unacceptable, and any model yielding negative probabilities as having broken down. The model should or be alternative- trashed, ly we should only use the model for parameters input that do not results in negative probabilities. replacing any transition probabilities that are negative or above unity with probabilities consistent with the standard axiomatic For framework. example Derman, Kani, and Chriss (1996) suggest this as a possible solution in their implied trinomial tree when probabilities. negative into running add more flexibility to ical tool the model. to Consider now case 3. The problems with the From From these choices I have only seen choice 1 Back to the CRR binomial tree example. In In case 2 we could override the few negative 1. We can consider negative probabilities as 3. probabilities as a mathemat- Look at negative 2.Basically probabilities. negative the Override general make the model loose way) some informationin market the to calibrated if particular (in and is not desirable, as also by indicated Derman, tri- implied their of use in (1996) Chriss and Kani, nomial tree. CRR tree is in my view actually not the negative probabilities, but the choice of sample space. The probabilities indicate negative thatsimply the forward price is outside the sample space. The forward price is in a “hidden state’’, not The covered in the tree. of nodes by location the sub-optimal certainly could space sample of choice sub-optimal be a problem when trying to value some options. will be outside zero and one. When we observe negative probabilities in a tree model we have at threeleast choices (most common) and 2 being discussed in the finance literature. I am not saying that choice 1 and 2 are wrong, but this should not automati- us reject choice 3. make cally this case what we do proba- the about negative bilities should depend on the use of the model. Let’s consider case 1. In this case we would simply say that for this numerical case the model is useless. In the CRR model case we can actually getaround the whole problem of negative probabilities, but let us ignore this the for moment. probabilities in some “smart’’ way. This will in i , p 1] , 02 . and − [0 0 1 . = 833 . σ 4 0 3689 , . , , 3689 = 1 . 0 12 6 . = / 006 9942 0 . . 5 . . . 1 0 | d =− = 0 t d 1 = = − # = 9942 b . t − t √ 0 , # 9942 i i u . 3689 # b 0833 0833 b 5 . . . | . 0 e − 0 0 1 0 √ √ − = − we get we = 02 02 i . . 0833 σ< i . 1 0 0 p T 0 p e − 006 × e . = , = 1 12 1 . More precisely we will have nega- = 0 p u e 3 100 d − = = 1 1 S p . From this we have have we this From . 6 = n tive probabilities in the when CRR tree tive equals equals unity. As mentioned by Chriss (1997) a low volatility and high relatively cost of carry in the CRR risk-neutral tree can lead to negative probabilities. In this context it is worth mentioning that even if the CRR binomial tree can give negative and higher than unity probabilities the sum of the one. sum to and up probability will still down 2.1.1 do with negative What to probabilities? Assume we are using a CRR binomial tree to value a derivative instrument, and consider the following numerical example. The asset price is 100, the time instrument is 6 to months, the volatility maturity of the underlying asset is of2%, the cost-of-carry theof the derivative underlying asset is 12% for the first month, and are increasing by 0.5% for every month there- after. For simplicity we use only six time steps. That is up probability of a risk-neutral Further get we probability of a down get and we probabilities. In this negative get we As expected, generalized version of the CRR tree we can have different probabilities for every time so steps, we interval the outside probabilities have could for some time steps and inside for othersteps, in numericaltime example all the probabilities subscript subscript indicates that we can have a different probability and also cost-of-carry at each time 1 for Solving step. The probability of going down must be since the probability of going either up or down i ij is is S (1) i S b The 2 is the set the is P , , t j , where # √ is the volatility σ apart is given by given is apart SdZ , − σ t ,..., e σ 1 dS # ) , = + i 0 p d = Sdt − i µ 1 ( , = t + # , dS √ i uS σ i − is the drift and j e p is the size of each time step, and is the size of each time step, d i µ n = magazine = / Su t u T # i b = is risk-neutral probability of the asset Se t i p # accuracy. If one can accept that, one can live accuracy. If one can accept that, Hawking quite happily with ghosts.’’ Stephen higher derivative terms to the action. This higher derivative with nega- apparently introduces ghosts, states found this is an tive probability. However, I have system in a illusion. One can never prepare a the presence state of negative probability. But with arbitrary of ghosts that one can not predict is the number of time steps. The layout of the Wilmott of the asset. In the CRR tree the asset price in any by node of the tree is given asset asset price (the geometry of the will call tree) we (the space the set sample of all price asset where price increasing at the next time step, and the asset price, price, asset the the cost-of-carry of asset. the the cost-of-carry underlying of the probabilities at the various nodes. The tree CRR the in node each to related probabilities follows from the arbitrage principle price can take at each time step step time each at take can price where n measure probability The values). node in Quantitative Finance In this section we look at a few examples where of negative probabilities in can show revolutionary up not are in My examples finance. solved never problems that solve they the sense before. I hope, that however, they will illustrate why negative probabilities are not necessary “bad’’, and that be useful. they might even 2.1 Probabilities Negative in the CRR binomial tree The well-known Cox, Ross, and Rubinstein (1979) binomial tree (CRR tree) is often used to price a variety of derivatives instruments, European and American options. The CRR binomi- including al tree can be seen as a discretization of geometric Brownian motion where the up and down jump size that the asset 2Negative Probabilities 2Negative ESPEN GAARDER HAUG

2 Actual realizations of the asset price of relevance This will lead to negative up probability pu if b σ Integer 3bT 1 1. to the option price could easily fall outside the − + σ 2 + 4b + % ' (& sample-space of the model. The strike price X can 2 2√1 6b#t σ> 2b + For very low volatility and high cost-of-carry this also be outside the sample space of the tree. This $ + 3#t + 3#t seems to be the main problem with the CRR tree, will typically require a lot of time steps, and can and not the negative probabilities. We could easily for this reason be very computer intensive. A bet- and negative down probability pd if think of an example where the user of the CRR ter way to avoid negative probabilities is to chose model simply used it to compute forward prices, a more optimal sample space, for example in a 2 2√1 6b#t taking into account a deterministic term structure σ< 2b + binomial tree this can be done using a geometry $ + 3#t − 3#t of the tree as suggested by Jarrow and Rudd of interest rates and dividends (bi). Computing the forward price can naturally be achived in a sim- (1983), setting For example with cost-of-cary 20% and 20 time (b σ 2 /2)#t σ √#t pler and more efficient way, but there is nothing u e − + , wrong with using a complex model to value a sim- steps and one year to maturity, then we will get a = ple product, except possibly wasting computer negative down probability if the volatility is and below 7.63%, this is far from a totally unrealistic (b σ 2 /2)#t σ √#t speed. Even with negative probabilities (and proba- d e − − , bilities larger than unity) the CRR tree will still case. Similarly we can also get probabilities high- = 7 give the correct forward price. Allowing negative er than unity . In the same way a CRR equivalent which gives equal up and down probability of trinomial tree can also give negative probabili- probabilities still leaves the model partly intact. p 0.5, 1 p 0.5. The negative probabilities actually seems to add ties, see appendix. = − = flexibility to the model. If you take a close look at the Collector car- Even if the CRR tree and the Jarrow-Rudd tree use It is also worth refreshing our memories that toon story accompanying this article you will see different sample space and probability measure most probabilities used in quantitative finance are it is surprisingly similar to the binomial and tri- they are both equivalent in the limit, for many so called risk-neutral probabilities, including the nomial tree just described. By believing he is time steps. For a binomial tree there is a almost binomial probabilities we just have looked at. As talking to James, the professor has no idea that an unlimited amount of sample spaces to choose every reader probably already know, risk-neutral the expected death of James is far outside his from, each with their own probability measure, probabilities5 should not be confused with real prob- sample space. The professor is selecting a sub- but all leading to the same result in the limit. abilities. Risk neutral probabilities sometimes also optimal sample space, only the future, as basis For more complex models and stochastic called pseudo-probabilities are simply computation- for his model. The Collector, admittedly having processes we will not necessary be able to avoid al devices constructed for a “fantasy world’’ (in this more information is recognizing that the profes- negative probabilities. Does this mean that the case a risk-neutral world) to simplify our calcula- sor uses a sub-optimal sample space, but is still model has to be trashed? In the introduction to tions. For this reason I can see nothing wrong with able to answer the question with remarkable this article I basically told you that all papers I at least starting to thinking about negative probabil- accuracy, by allowing negative probabilities have seen in the finance literature are negative to ities as a mathematical tool of convenience. This (pseudo probabilities). This is very much a paral- negative probabilities. This is not entirely true; in expand to outside the world of binomial trees, a lel to considering a node of a tree model outside a recent and very interesting paper by Forsyth, binomial model can simply be seen as a special case the geometry of the tree. Vetzal, and Zvan (2001): “Negative Coefficients in of a explicit finite difference model, see for example Two Factor Option Models’’ the authors illustrate Heston and Zhou (2000) or James (2003). In the same 3Getting the Negative how some finite difference/finite element models way a trinomial tree can also be shown to be equiva- Probabilities to Really Work with negative probabilities are still stable and lent to the explicit finite difference method (see for consistent. They moreover show how the require- example James (2003)) when the probability of the in Your Favor ment of positive pseudo-probabilities not only are unnecessary but can even be detrimental. asset price going up pu and down pd and stay at the In the case of the discretization of a geometric same are set to6 Brownian motion through a binomial or trinomi- Forsyth, Vetzal and Zvan seem to try to avoid the al model we can admittedly avoided negative term “negative probability’’, and are consistently 1 #t p (b σ 2/2) probabilities all together. In the CRR or trinomial using the term “negative coefficients’’. What they u = 6 + − 12σ 2 # tree just described this can be done simply by call negative coefficients are closely related to the binomial and trinomial probabilities we just dis- 1 2 #t respectively setting the number of time steps n pd (b σ /2) cussed (as also indicated by the authors). That the = 6 − − 12σ 2 equal to or higher than # authors possibly try to avoid the term negative 2 T probability seems to be somewhat a parallel to p Integer , m = 3 (σ/b)2 % & how many physicist treat the Wigner distribution. 36 Wilmott magazine Wigner and Szilard developed a distribution func- 4.1 Negative Probabilities such a scenario. Finance Professors’ assumptions of tion which for the first time was applied by Hidden in Waste from Model “no free lunch’’ is probably only a reality at the uni- Wigner to calculate quantum corrections to a gas Break Downs? versity campuses. pressure formula. Wigner original called this a My main point is that there are almost unlim- probability function. The Wigner distribution Personally I don’t know of any financial model ited examples of cases where the real uncertainty gives apparently negative probabilities for some that at some point in time has not had a break- is outside the sample space of our quantitative quantum states. For this reason many physicists down—-there’s a reason we call them “models’’. models. A whole new field of finance has grown prefer to call it the Wigner function, this even if it Just as an example, any derivatives valuation out from model errors, going under the name has no other physical interpretation than a proba- model I am aware of assume that the current “model risk’’. Can there be that negative probabili- bility distribution. Khrennikov (1999) is giving a asset price is known. When valuing for example ties are hidden in “data waste’’ from such model detailed mathematical description of why it a stock option we take the latest traded stock breakdowns? There is obvious many ways we can should be considered a probability distribution, price from our computer screen as input to the improve our models and still stay inside the stan- and also explain why the Wigner distribution model. In reality this price is not known. Every dard assumption of probabilities between zero results in negative probabilities. year there are multiple incorrectly reported and one, but could it be that building our models Even if Forsyth, Vetzal and Zvan seems to prices, due to a human error, or a hardware or from the ground up to allow negative probabilities avoid the term negative probabilities, their paper software problem. One can naturally reduce the in the first place could help us make the models are in a excellent way illustrating just what I have uncertainty surrounding the current price by more robust, closer to reality? been looking for, a demonstration that allowing double-checking several independent providers negative probabilities really can make a differ- etc. Still there are uncertainty concerning any ence in quantitative finance models. 5The Future of Negative price, even the current one. In some sense almost The binomial, trinomial and finite difference Probabilities in Quantitative any asset in financial markets can be affected by models we have touched upon so far were origi- “hidden variables’’. For example in the interbank Finance nally developed by assuming probabilities in the Foreign Exchange markets, if you physically The main reason that negative probabilities have interval [0, 1]. So even if negative probabilities trade an option at a price over the phone you can not found much use in quantitative finance yet is seems to add flexibility to some of these models not with 100% certainty know that you actually probably that the researchers that have developed we must be careful with the interpretation of traded at that price. It is not uncommon that the these models have been doing this under the any such negative pseudo-probabilities. counterpart calls you up hours later to cancel or belief that any probability must lie between zero adjust the price of a deal, because they claim and one, limiting their view to Kolmogorov prob- 4Hidden Variables in Finance they priced it using wrong input parameters, abilities. So far, I have just done a feeble attempt Feynman’s idea of applying negative probabilities human error or whatever. Few or no models in to trace the footsteps of Dirac and Feynman. That to hidden variables can possibly have some paral- quantitative finance take such situations into is we have considered negative probabilities as lels in finance. Expected return, and expected risk account. just formal quantities that can be useful in cer- (expected volatility, correlation etc.) are hidden Others examples of model break downs we have tain calculations. Most people in quantitative variables that can not be observed directly. Such when asset prices takes values completely outside finance are interested in finance and not the hidden variables in finance still play an important our expected sample space. I remember some years foundation of probability theory and have proba- role and can affect other observable variables. back in time, when the electricity price in a local bly for this reason ignored that the Komologrov Derman (1996) describe these as hidden variables area of Norway actually went negative for a few model is not necessary the complete picture of in the context to the somewhat related topic of hours during the night. The electricity was pro- stochastic reality. To take the next step we need to model risk, where he indicate that such variables duced by hydropower using large turbines. When use a rigid mathematical foundation for a proba- simply may be individuals’ opinion. Much of mod- there is an over supply of electricity on the grid one bility theory that also allows for negative proba- ern financial modeling deals with uncertainty for can naturally shut down some of the turbines, bilities. Andrei Khrennikov has developed such a such hidden variables, still it looks like nobody however this can be a costly process. In this case, theory, he has found the root of negative proba- have considered including negative probabilities instead of shutting down the turbine the power bilities in the very foundation of probability theo- in such calculations. Inspired by Dirac, Feynman, producer was willing to pay someone to use the ry: ensemble and frequency. With background in and Hawking could there be that negative proba- electricity. The Scandinavian electricity markets his theory described in mathematical detail in his bilities also could be of use to model hidden finan- are probably the most well functioning electricity brilliant book: “Interpretations of Probability” cial ghosts? Ultimately one would naturally like to markets in the world, including futures, forwards Khrennikov (1999), see also Khrennikov (1997), it ^ see research that proves that using negative proba- and options. We can easily think about a derivative is reason to believe that negative probabilities bilities gives a clear advantage. model assuming positive prices breaking down in applied to finance could take it’s next step? The

Wilmott magazine 37 ESPEN GAARDER HAUG

Kolmogorov probability theory that is the basis of FOOTNOTES & REFERENCES http://arxiv.org/PS_cache/quant-ph/pdf/0010/ all mathematical finance today can be seen as a 0010091.pdf. limited case of a more general probability theory 1. At least in the sense that they urge one to avoid nega- I CHRISS, N. A. (1997): Black-Scholes and Beyond. Irwin developed by a handful of scientists standing on tive probabilities. Professional Publishing. the shoulder of giants like Dirac and Feynman. 2. For stocks (b r), stocks and stock indexes paying a I COX, J. C., S. A. ROSS, and M. RUBINSTEIN (1979): “Option = The probability that the future of quantitative continuous dividend yield q (b r q), futures (b 0), Pricing: A Simplified Approach,’’ Journal of Financial = − = Economics, 7, 229–263. finance will contain negative probabilities is pos- and currency options with foreign interest rate rf I CURTRIGHT, T., AND C. ZACHOSY (2001): “Negative Probability sibly negative, but that does not necessary exclude (b r rf ). = − negative probabilities. 3. Negative risk-neutral probabilities will also lead to nega- and Uncertainty Relations,’’ Modern Physics Letters A, tive local variance in the tree, the variance at any time step 16(37), 2381–2385, http://arxiv.org/PS_cache/ If you eliminate the impossible, whatever 1 hep–th/pdf/0105/0105226.pdf. in the CRR tree is given by σ 2 p(1 p)(ln(u2 ))2 . remains, however improbable, must be the i = #t − I DERMAN, E. (1996): “Model Risk,’’ Working Paper, Goldman truth. Sherlock Holmes Negative variance seems absurd, there is several papers Sachs. discussing negative volatility, see Peskir and Shiryaev IDERMAN, E., I. KANI, AND N. CHRISS (1996): “Implied Trinomial 6Appendix: Negative probabilities (2001), Haug (2002) and Aase (2004). Trees of the Volatility Smile,’’ Journal of Derivatives, 3(4), 7–22. 4. This is also described by Hull (2002) page 407, as well as I DIRAC, P. (1942): “The Physical Interpretation of Quantum in CRR equivalent trinomial tree in the accompanying solution manual page 118. Mechanics,’’ Proc. Roy. Soc. London, (A 180), 1–39. 5. Risk-neutral probabilities are simply real world probabili- Trinomial trees introduced by Boyle (1986), are IFEYNMAN, R. P. (1987): “Negative Probability,’’ First published ties that have been adjusted for risk. It is therefore not similar to binomial trees and are also very popu- in the book Quantum Implications : Essays in Honour of David necessary to adjust for risk also in the discount factor for lar in valuation of derivative securities. One possi- Bohm, by F. David Peat (Editor), Basil Hiley (Editor) Routledge cash-flows. This makes it valid to compute market prices as & Kegan Paul Ltd, London & New York, pp. 235–248, bility is to build a trinomial tree with CRR equiva- simple expectations of cash flows, with the risk adjusted http://kh.bu.edu/qcl/pdf/feynmanr19850a6e6862.pdf. lent parameters, where the asset price at each probabilities, discounted at the riskless interest rate— I FORSYTH, P. A., K. R. VETZAL, and R. ZVAN (2001): node can go up, stay at the same level, or go down. hence the common name “risk-neutral’’ probabilities, “Negative Coefficients in Two Factor Option Pricing Models,’’ In that case, the up-and-down jump sizes are: which is somewhat of a misnomer. Working Paper, http://citeseer.ist.psu.edu/435337.html. σ √2#t σ √2#t 6. Hull (2002) also shows the relationship between trino- I HAUG, E. G. (2002): “A Look in the Antimatter Mirror,’’ u e , d e− , = = mial trees and the explicit finite difference method, but Wilmott Magazine, www.wilmott.com, December. and the probability of going up and down using another set of probability measure, however as he I HESTON, S., AND G. ZHOU (2000): “On the Rate of respectively are: points out also with his choice of probability measure one Convergence of Discrete-Time Contigent Claims,’’ can get negative probabilities for certain input parameters. 2 Mathematical Finance, 10, 53-75. b#t/2 σ √#t/2 e e− 7. We will get a up probability pu higher than unity if I HULL, J. (2002): Option, Futures, and Other Derivatives, pu − = σ √#t/2 σ √#t/2 5th Edition. Prentice Hall. ) e e− * 50 10√25 6b#t − σ< 2b + , I HULL, J., AND A. WHITE (1990): “Valuing Derivative Securities 2 $ + 3#t − 3#t eσ √#t/2 eb#t/2 Using the Explicit Finite Difference Method,’’ Journal of pd − . = σ √#t/2 σ √#t/2 and down probability of higher than unity if Financial and Quantitative Analysis, 25(1), 87–100. ) e e− * − I JAMES, P. (2003): Option Theory. John Wiley & Sons. 50 10√25 6b#t I The probabilities must sum to unity. Thus the prob- σ> 2b + . JARROW, R., AND A. RUDD (1983): Option Pricing. Irwin. ability of staying at the same asset price level is $ + 3#t + 3#t I JORGENSON, J., AND N. TARABAY (2002): “Discrete Time Tree Models Associated to General Probability I AASE, K. (2004): “Negative Volatility and the Survival of pm 1 pu pd. Distributions,’’ Working Paper, City College of New York. = − − the Western Financial Markets,’’ Wilmott Magazine. I KHRENNIKOV, A. (1997): Non-Archimedean Analysis: When the volatility σ is very low and the cost-of- I BOYLE, P. P. (1986): “Option Valuation Using a Three Quantum Paradoxes, Dynamical Systems and Biological carry is very high p and p can become larger Jump Process,’’ International Options Journal, 3, 7–12. u d Models. Kluwer Academic Publishers. p I BRENNAN, M. J., AND E. S. SCHWARTZ (1978): “Finite than one and then naturally m will become neg- I –––––– (1999): Interpretations of Probability. Coronet Difference Methods and Jump Processes Arising in the ative. More precisely we will get a negative proba- Books. Pricing of Contigent Claims: A Synthesis,’’ Journal of bility pm < 0 when I PEACOCK, K. A. (2002): “A Possible Explanation of the Financial and Quantitative Analysis. b2#t Superposition Principle,’’ Working paper, Department of I CASTRO, C. (2000): “Noncommutative Geometry, σ< . Philosophy University of Lethbridge, http://arxiv.org/ 2 Negative Probabilities and Cantorian-Fractal Spacetime,’’ # PS_cache/quant-ph/pdf/0209/0209082.pdf. Center for Theoretical Studies of Physical Systems Clark From this we can also find that we need to set the I PESKIR, G., AND A. N. SHIRYAEV (2001): “A note on the Put- b2 T Atlanta University”,, http://arxiv.org/PS_cache/ number of time steps to n Integer 2 1 to Call Parity and a Put-Call Duality,’’ Theory of Probability ≥ 2σ + hep-th/pdf/0007/0007224.pdf. avoid negative probabilities. Alternatively+ , we and its Applications, 46, 181–183. I CERECEDA, J. L. (2000): “Local Hidden-Variable Models W could avoid negative probabilities by choosing a I RUBINSTEIN, M. (1998): “Edgeworth Binomial Trees,’’ and Negative-Probability Measures,’’ Working paper, more optimal sample space. Journal of Derivatives, XIX, 20–27.

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