Economic Theory (2007) 30: 191–193 DOI 10.1007/s00199-005-0050-x
E X P O S I TA N OT E
Anna Battauz · Fulvio Ortu
Dynamic versus one-period completeness in event-tree security markets
Received: 21 May 2005 / Accepted: 4 October 2005 / Published online: 8 November 2005 © Springer-Verlag 2005
Abstract We show that in event-tree security markets dynamic completeness does not coincide with one-period completeness unless the law of one price is explicitely assumed. We do so by means of a simple example of a dynamically complete market with an incomplete one-period sub-market. Keywords Event-tree security markets · Dynamic completeness · One-period completeness · Law of one price JEL Classification Numbers G10 · G12 1 Introduction A result commonly found in the financial economics literature 1 states that multiperiod event-tree security markets are dynamically complete if and only if they are one-period complete at every node of the event-tree. While completeness of all one-period sub-markets implies anyway dynamic completeness, we show here by means of a simple example that the converse implications fails unless the law of one price is explicitly imposed in all one-period sub-markets following the first one. Our example displays a two-period dynamically complete event-tree market whose first-period sub-market is incomplete. To obtain dynamic completeness we construct the event-tree so that in a time 1 node the securities violate the law of A. Battauz Istituto di Metodi Quantitativi, Universit`a Bocconi, Milano, Italy F. Ortu (B) Istituto di Metodi Quantitativi and IGIER, Universit`a Bocconi, V. le Isonzo 25, 20135, Milano, Italy, E-mail:
[email protected] 1 See in particular Dothan (1990), Theorem 3.3 and LeRoy and Werner (2001), Theorem 23.2.1. See also (Pliska 1997, Proposition 4.17) and (Duffie 1988, Proposition II.12.G).
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one price, generating free money that allows to replicate any time 1 pay-off. The usual equivalence between dynamic and one-period completeness holds however in the vastly employed special class of multinomial models, since in these models dynamic completeness inhibits violations of the law of one price. 2 Dynamic versus one-period completeness An event-tree security market is characterized by a time set T = {0, 1, ..., T }, a finite set = {ω1 , ..., ωK } of possible terminal states and an information structure described by a family P = {Pt }Tt=0 of partitions of with P0 = {}, Pt+1 finer than Pt for all t < T and PT = {{ω1 } , {ω2 } , ..., {ωK }} . We denote with fht the generic element of Pt . There are J long-lived securities traded, whose prices are collected in the J -valued, P-adapted vector process S = {S(t)}Tt=0 . Intermediate dividends are disregarded without loss of generality. Dynamic trading is formalized −1 . The j th by means of J -valued, P-adapted stochastic processes ϑ = {ϑ (t)}Tt=0 component ϑj (t) of ϑ (t) constitutes the position in security j established at time t and liquidated at time t + 1 . To each dynamic investment strategy ϑ we associate a cash-flow process Cϑ = {Cϑ (t)}Tt=0 where Cϑ (0) = −ϑ (0) · S (0) , Cϑ (t) = ϑ (t − 1) · S (t) − ϑ (t) · S (t) for t = 1, ..., T − 1 and Cϑ (T ) = ϑ (T − 1) · S (T ). An event-tree security market is dynamically complete if for every adapted future cash-flow X = {X(t)}Tt=1 there is an investment strategy ϑ that replicates X, that is Cϑ (t) = X(t),
t = 1, ..., T .
(1)
To any node fht ∈ Pt , t = 0, ..., T − 1 we associate the one-period sub-market with terminal states the immediate successors of fht , i.e. flt+1 ⊂ fht and with J securities that are traded at the initial prices Sj (t)(fht ) and liquidated at the prices Sj (t + 1)(flt+1 ). The generic one-period sub-market mht is complete if for any payoff on the nodes flt+1 ⊂ fht there exists a strategy established in the node fht whose liquidation value replicates the given payoff. Therefore the one-period sub-market mht is complete if and only if the number of immediate successors of fht equals the number of securities whose prices are linearly independent in the immediate successors of fht . Consider now a two-period market with J = 2 long-lived securities whose prices S1 , S2 evolve as follows: mht
t =0 1 1 , 2 2
t =1 t =2 1, 0 → 1, 0
→ 0, 1 → 0, 1 1, 1 → 21 , 1
The initial one-period sub-market m10 is incomplete since there are fewer securities than time 1 states, while all one-period final sub-markets m11 , m21 and m31 are complete. Nonetheless, the violation of the law of one price in m31 allows for
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dynamic completeness. In fact, given a generic cash-flow X = {X(t)}2t=1 , equation (1) is solved for t = 2 by ϑ X (1)(f11 ) = [X(2)(ω1 ), 0]T , ϑ X (1)(f21 ) = [0, X(2)(ω2 )]T and ϑ X (1)(f31 ) = [2X(2)(ω3 ) − 2α, α]T for any α ∈ . Choosing α = X(1)(f31 ) + 2X(2)(ω3 ) − X(1)(f11 ) − X(1)(f21 ) − X(2)(ω1 ) − X(2)(ω2 ) equation (1) for t = 1 is feasible as well, hence the market is dynamically complete. In our example dynamic trading in redundant securities that violate the law of one price permits to complete the market even if the initial one-period sub-market is incomplete. Therefore, the link between dynamic and one-period completeness commonly stated in the literature should be amended as follows: Proposition 1 An event-tree security market is dynamically complete if every oneperiod sub-market is complete. Conversely, if the market is dynamically complete and the law of one price is satisfied in every one-period sub-market mht for t = 1, ..., T − 1, then every one-period sub-market is complete. To conclude, we observe that the equivalence between dynamic and one-period completeness holds for the special class of multinomial event-tree markets. We recall that an event-tree market is called multinomial if the number of immediate successors is the same for every node, and coincides with the number of longlived securities. These models are vastly employed in the financial engineering literature and constitute also a common pedagogical device. In this case dynamic completeness forces all the terminal one-period sub-markets to be complete and the terminal securities’ prices to be linearly independent, so that they satisfy the law of one price. The equivalence between dynamic and one-period completeness in multinomial markets is then established by backward induction (see Battauz and Ortu 2005, for a formal proof). References Battauz, A., Ortu, F.: Dynamic versus one-period completeness in event-tree security markets. Working paper N. 34, Istituto di Metodi Quantitativi, Universit`a Bocconi (2005) Dothan, M.U.: Prices in financial markets. New York: Oxford University Press 1990 Duffie, D.: Security markets: stochastic models. New York: Academic Press 1988 LeRoy, S.F., Werner, J.: Principles of financial economics. London: Cambridge University Press 2001 Pliska, S.R.: Introduction to mathematical finance: discrete time models. Oxford: Blackwell 1997
