# Why so Negative to Negative Probabilities?

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2 higher derivative terms to the action. This apparently introduces ghosts, states with negative probability. However, I have found this is an illusion. One can never prepare a system in a state of negative probability. But the presence of ghosts that one can not predict with arbitrary accuracy. If one can accept that, one can live quite happily with ghosts. Stephen Hawking 2 Negative Probabilities in Quantitative Finance In this section we look at a few examples of where negative probabilities can show up in finance. My examples are not revolutionary in the sense that they solve problems never solved before. I hope, however, that they will illustrate why negative probabilities are not necessary bad, and that they might even be useful. 2.1 Negative Probabilities 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, including European and American options. The CRR binomial tree can be seen as a discretization of geometric Brownian motion ds = µsdt + σ SdZ, where S is the asset price, µ is the drift and σ is the volatility of the asset. In the CRR tree the asset price in any node of the tree is given by Su i d j i, i = 0, 1,...,j, where the up and down jump size that the asset price can take at each time step t apart is given by u = e σ t, d = e σ t, where t = T/n is the size of each time step, and n is the number of time steps. The layout of the asset price (the geometry of the tree) we will call the sample space (the set of all asset price S ij node values). The probability measure P is the set of the probabilities at the various nodes. The probabilities related to each node in the CRR tree follows from the arbitrage principle Se bi t = p ius + (1 p i)ds, (1) where p i is risk-neutral probability of the asset price increasing at the next time step, and b i is the cost-of-carry of the underlying asset. 2 The i subscript indicates that we can have a different probability and also cost-of-carry at each time step. Solving 1 for p i we get p i = ebi t d u d. The probability of going down must be 1 p i since the probability of going either up or down equals unity. As mentioned by Chriss (1997) a low volatility and relatively high cost of carry in the CRR tree can lead to negative risk-neutral probabilities. 3 More precisely we will have negative probabilities in the CRR tree when 4 σ< b i t. 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 down and up probability will still sum to one What to do with negative 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 to maturity of the derivative instrument is 6 months, the volatility of the underlying asset is 2%, the cost-of-carry of the underlying asset is 12% for the first month, and are increasing by 0.5% for every month thereafter. For simplicity we use only six time steps. That is S = 100, T = 0.5, b 1 = 0.12, σ = 0.02, n = 6. From this we have t = 0.5/6 = and u = e = 1.006, d = e = , Further we get a risk-neutral up probability of p 1 = e and we get a down probability of = p 1 = = As expected, we get negative probabilities. In this generalized version of the CRR tree we can have different probabilities for every time steps, so we could have probabilities outside the interval [0, 1] for some time steps and inside for other time steps, in numerical example all the probabilities will be outside zero and one. When we observe negative probabilities in a tree model we have at least three choices 1. We can consider negative probabilities as unacceptable, and any model yielding negative probabilities as having broken down. The model should be trashed, or alternatively we should only use the model for input parameters that do not results in negative probabilities. 2. Override the negative probabilities. Basically replacing any transition probabilities that are negative or above unity with probabilities consistent with the standard axiomatic framework. For example Derman, Kani, and Chriss (1996) suggest this as a possible solution in their implied trinomial tree when running into negative probabilities. 3. Look at negative probabilities as a mathematical tool to add more flexibility to the model. From these choices I have only seen choice 1 (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 automatically make us reject choice 3. Back to the CRR binomial tree example. In this case what we do about the negative probabilities 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 get around the whole problem of negative probabilities, but let us ignore this for the moment. In case 2 we could override the few negative probabilities in some smart way. This will in general make the model loose information (in particular if calibrated to the market in some way) and is not desirable, as also indicated by Derman, Kani, and Chriss (1996) in use of their implied trinomial tree. Consider now case 3. The problems with the CRR tree is in my view actually not the negative probabilities, but the choice of sample space. The negative probabilities simply indicate that the forward price is outside the sample space. The forward price is in a hidden state, not covered by the sub-optimal location of nodes in the tree. The sub-optimal choice of sample space could certainly be a problem when trying to value some options. ^ Wilmott magazine 35

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

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