Implied (risk neutral) probabilities, betting odds and prediction markets

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1 Impled (rsk neutral) probabltes, bettng odds and predcton markets Fabrzo Caccafesta (Unversty of Rome "Tor Vergata") ABSTRACT - We show that the well known euvalence between the "fundamental theorem of asset prcng" and the "dutch book argument" conssts, n realty, of two parts: the frst concernng the exstence of a rsk free nvestment, and the second the possblty of arbtrages. We argument that the "mpled probabltes" of an odds system may be used to rate the bookmaker's farness, and brefly dscuss the sgnfcance of the same probabltes as defned by the prces n the predcton markets. KEYWORDS - Dutch books; arbtrage; rsk neutral probabltes; mpled probabltes; bettng odds; predcton markets. Frst draft: May 20, 2006 Ths draft: September, Reference s to the well known analogy between the "dutch book" argument, accordng to whch a system of bettng odds allows no sure wns f and only f the "mpled probabltes" are formally coherent, and the fundamental theorem of asset prcng (FTAP). Ths last states that n a market where one euro may be alternatvely nvested to gve the random amount S or the certan amount r, no arbtrage s possble f and only f a (rsk neutral: "r.n.") probablty dstrbuton exsts for whch () E[S] = r and for whch the non zero probablty events are the ones consdered as really possble. Moreover, there s only one such dstrbuton f and only f the market s "complete". 2. We consder the typcal (although not the most general) case, n whch an odds system = ( ) correspondng to some "partton of realty" {E } s gven: a famly of n mutually exclusve events s defned, one of whch must necessarly occur; and someone offers to pay (>) euro f event E occurs to anyone that has bet one euro on ths fact. Our man concern wll be the (eventual) relaton between and Prob(E ), but let us start wth askng ourselves whch logc (f any does) underles the choce of the odds. Now, suppose that (for =,..., n) the bettors have altogether placed c euro on the event E. If the bookmaker wants to keep for hmself the fracton k of the total ncome, hs eulbrum wll be safe f the odds satsfy the eualtes: (2) k c = c =... = n c n. So, f the bookmaker has an opnon about what the ratos c /c wll be, he can use (2) to determne the odds he can offer. He wll namely put:

2 (3) h = k ch Note, that the odds may be progressvely adusted, accordng the way the bettors actually share ther bets among the events. From (3) (or (2)), t obvously follows that, for the eulbrum to hold, the rato between the odds must be eual to the nverse of that between the bets: c. (4) c = c (for every, ). 3. It s worthwhle notng that the parameter k has a value every one can calculate (as the nverse of the sum of the nverse of the odds), and may be used to rate the bookmaker's farness. Here are ust two examples. A. The odds the offcal talan bettng agency ("Sna") offered (on may 25, 2006) for the wnners of the three of the prelmnary round robns of the football world champonshp were: r.r. A r.r. B r.r. E Germany,25 England,50 Italy,80 Costarca 25 Paraguay 8,50 Ghana Poland 4,50 Trn. & Tobago 50 Usa 7,50 Ecuador 2 Sweden 3 Chech rep. 2,80 0,8729 0,8790 0,8796 In the last row, the correspondng value of k appears. B. For the sake of a comparson, here are the odds some australan bookmakers offered (on the same may 25) about the wnner of next year's poltcal electons: Coalton Labor k Centrebet,60 2,20 0,9263 IASbet,75 2,05 0,944 Sportngbet,70 2,05 0,9248 Sportsbet,67 2,0 0, We leave now the ueston of the bookmakerss' farness asde, and note that the bets portfolo made up by bettng, for every, (5) a = euro on the event E, has cost, and the wnnng that t yelds s n any case h h 2

3 (6) h h = ( ) = k. So, the same uantty k we saw before may also be seen as a rsk free accumulaton factor (albet, usually smaller than ). 5. We fnally approach to our man goal: nvestgatng the relaton (f any exsts) between odds and probabltes. The dea of nterpretng bet's odds n terms of subectve probabltes goes back to Ramsey and de Fnett: one should be wllng to sell or buy at the prce p the rght to receve one euro f an event occurs f p (or: because p) s, n hs opnon, that event's probablty. The dffcultes arsng from the fact that ths obvously mples the operators to be rsk neutral are usually gnored, or overcome by assumng the stakes small enough for ths to be approxmately true. In our framework, t s therefore natural to call the uanttes (7) p = / (=, 2,, n) mpled probabltes (we mean, of course: mpled by the odds system ). As >, every p s n (0, ), but nothng nsures that P = p s 2. Now, t s straghtforward to see that P = p = k; so we can use our k to normalze the mpled probabltes. The products (8) p * = kp sum up to, and are therefore, formally, probabltes. Puttng (7) and (8) together one gets: (9) p * = k ( =, 2,, n) whch expresses the result that k s the *-expected value of the wnnngs for a bettor that has placed euro on E (for every ). The classcal dutch book argument may now be stated by wrtng: (0) k = p * = p P =. Some comments are necessary. If /P = k > (the odds are "hgh") we are n presence of a "real" dutch book: whoever bets accordng the proportons (5) wll receve more than the amount pad. If, conversely, /P = k < (the odds are "low") thngs are much more uestonable, and one can thnk about a dutch book "su geners". By sellng the portfolo (a ), an operator receves more than what he wll have to pay. The problem s, that to adopt ths strategy, he must be allowed to accept bets n the exact uanttes he wshes: and ths s not possble for a Statng, as we have done, that the probabltes mpled n an odds system are ust the "rsk neutral" ones the FTAP deals about, s maybe an up-to-date way to express the same concept. 2 As we are consderng bets on mutually exclusve events, ths s the only condton of coherence we must care about. 3

4 professonal bookmaker (note that, when P s dfferent from, acceptng a bet on some event cannot be replcated by bettng on all the others). Ths possblty must therefore be specfcally postulated, ust as the one of sellng short the rsk free asset n a fnancal market Let us go back to the (9) and note that, as k may be seen as an accumulaton factor (the one assocated wth the odds system ) each of those n eualtes s perfectly analogous to the eualty (), whch expresses the FTAP. It looks therefore possble to generalze the "dutch book argument" n the followng terms. Just as n the framework of the FTAP, suppose there are at hand two possbltes to nvest money: bettng accordng the odds, or nvestng at the rskless rate ρ (ths could be, eventually assocated wth a dfferent system of bet odds; f ρ =, we get agan the classcal dutch book argument). Fndng the "rsk neutral" probabltes, that s some probabltes accordng to whch the two alternatves are - n expected value - ndfferent, means lookng for a soluton of the system of n+ euatons: () n p = p * = * = ρ ( =, 2,..., n) where ( ) and ρ are gven and (p *) are the unknown. A necessary and suffcent condton for t to admt a sngle soluton s gven by: (2) ρ = h h. (2) s also a condton of "no arbtrage": ths s obvous, snce t expresses the eualty between two rskless factors (n the rght, the one assocated wth appears). The analoges wth the FTAP may be completed as follows. If nstead of () we have to deal wth a system as: (3) n p = p * * = = ρ ( =, 2,..., m < n) then there s - generally - more than one soluton (t s the case of a non complete fnancal market): n m among the events E are consdered as actually possble, but no bet s accepted on them. No odd s gven for them, and the correspondng mpled probabltes cannot be defned. If, on the other hand, the system looks lke: 3 Note also, that n order to be able to speak of proper "arbtrages", one must (n both stuatons) add the hypothess that the tme doesn't flow: or, euvalently, that there s a rsk free rate of 0. Then, f k > one borrows the money to bet; f k <, one sells the bets, and ust keeps the money. 4

5 (4) m p = p * * = = ρ ( =, 2,..., m n) then the resultng mpled probabltes are not euvalent to the real ones. Even f (2) hold, arbtrages are possble, as the followng examples show. - Suppose m = 2, n = 3, = 2 = 2, ρ =, so that p = p 2 = /2. The bets portfolo ( on E, on E 2 ) costs 2 and yelds 2 f E or E 2 occurs, but nothng f E 3 does. By sellng t and holdng the money one realzes an arbtrage. - Suppose m = 3, n = 2, = 2 = 3 = 3, ρ =, so that p = p 2 = p 3 = /3. The bets portfolo ( on E, on E 2 ) costs 2 and yelds 3 f E or E 2 occurs: that s, n any really possble case. but nothng f E 3 does. By borrowng 2 and buyng that portfolo, one realzes an arbtrage. A last observaton concernng system (): f we don't consder the possblty of the alternatve nvestment, and wrte t ust as n p * = = p * = k ( =, 2,..., n) wth k gven by (6), then - trvally - the (unue) soluton always exsts, and s gven by the normalzed mpled probabltes. Our concluson s that the classcal formulaton of the "dutch book argument" may be slghtly mproved, by statng that the sad normalzed mpled probabltes always exst, and have exactly the meanng of the "rsk neutral" probabltes the FTAP deals wth: they are the probabltes, wth respect to whch bettng on any of the E 's gves the same mean result as nvestng at the rate k. 7. Intepretng, as we have done, the mpled probabltes as "rsk neutral" ones s probably the most modern way to express the hypothess of rsk ndfference we are forced to formulate, f we want to read odds as probabltes. The ueston remans open, of whch the nformatve, "real" content of such probabltes s; and t s an up-to-date ueston, as the recent brth of the "predcton markets" shows. We refer to the most common type of predcton markets: that n whch a contract payng f a gven event E occurs may be bought or sold at prce p. The stuaton s the same of that of a bets market, n whch a bookmaker operates free of charge (k = ) and sells bets on E at the odd /p. p s therefore the "mpled probablty" of E, and s - as one sees - gven by the market (predcton markets have been realzed to make the best of the dea that "odds are probabltes": after all, observng the real behavour of a bettor s much more drect and less based than ust askng hm at what prce he would be ndfferent between bettng on an event or on ts complementary). The consensus looks to be that the prce on such a market "represents the market's expectaton of the probablty that an event wll occur" (Wolfers-Ztzewtz, 2004). These authors are aware of the fact that rsk neutralty must be assumed, but argue that "the sums wagered n predcton markets are typcally small enough that assumng that nvestors are not averse to the dosyncratc rsk nvolved seems reasonable". Two obectons present themselves. 5

6 The frst. To make well ther duty, the predcton markets should attract people endowed wth better than average knowledge, and "force" them to reveal what they know. But ths only works f bettng s consdered as an nterestng nvestment opportunty. It s ndeed dffcult to accept the dea of an expert usng hs sklls for ust a symbolc remuneraton. The second. Ths s n our opnon more mportant, and concerns the very logc of the bettors' behavor. We llustrate t wth a smple example. Let 0,667 be the prce of a contract that pays f E occurs: to buy t means bettng 2/3 on E; sellng t means bettng /3 on none at the odd. Thngs go exactly as n a bets market where the bookes act free of charge, and apply the odds E =,5, none = 3. Can we really thnk that ths means that, n the market's opnon, that Prob(E) = 0,667 and Prob(nonE) = 0,333? Suppose that all the operators share the opnon that Prob(E) = 2Prob(nonE); on the other hand, none = 2 E. In such a stuaton, every operator who s wllng to bet on E s rsk neutral, and s therefore ndfferent between rskng hs money on ether of the events. It look necessary to deduce that as many euro wll be betted on E as on none. Ths contradcts the fact that, for the market's eulbrum, for every euro betted on none one must be on E. We are left wth the uncomfortable concluson that the (so called) market opnon cannot be the opnon of all the partcpants n the market. Bblography N.H. BINGHAM - R. KIESEL, Rsk Neutral Valuaton, Sprnger (998) M.L. EATON - D.A. FREEDMAN, Dutch Book aganst some 'Obectve' Pror, Bernoull (0) 2004 D.P. ELLERMAN, Arbtrage Theory: a Mathematcal Introducton; SIAM Revew (26) J. WOLFERS - E. ZITZEWITZ, Predcton Markets, J. of Economc Perspectves (8)

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