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**Extra info for A Problem in Card Shuffling**

**Example text**

Contingent claim 10 THE BINOMIAL MODEL then sell the stock on the market for su, thus making a net proﬁt of su − K. If S1 < K then the option is obviously worthless. In this example we thus have X= su − K, 0, if Z = u, if Z = d, and the contract function is given by Φ(u) = su − K, Φ(d) = 0. Our main problem is now to determine the “fair” price, if such an object exists at all, for a given contingent claim X. If we denote the price of X at time t by Π(t; X), then it can be seen that at time t = 1 the problem is easy to solve.

The proof is left to the reader. 21 Suppose that X is reachable using the portfolio h. Suppose furthermore that, at some time t, it is possible to buy X at a price cheaper than (or to sell it at a price higher than) Vth . Then it is possible to make an arbitrage proﬁt. We now turn to the completeness of the model. e. every claim can be replicated by a self-ﬁnancing portfolio. It is possible, and not very hard, to give a formal proof of the proposition, using mathematical induction. The formal proof will, however, look rather messy with lots of indices, so instead we prove the proposition for a concrete example, THE MULTIPERIOD MODEL 19 using a binomial tree.

M } and that the probabilities pj = P (ωj ), j = 1, . . , N are all strictly positive. The price vector S0 is assumed to be deterministic and known to us, but the price vector at time t = 1 depends upon the outcome ω ∈ Ω, and S1i (ωj ) denotes the price per unit of asset No. i at time t = 1 if ωj has occured. We may therefore deﬁne the matrix D by ⎤ ⎡ 1 S1 (ω1 ) S11 (ω2 ) · · · S11 (ωM ) ⎥ ⎢ ⎥ ⎢ 2 ⎢ S1 (ω1 ) S12 (ω2 ) · · · S12 (ωM ) ⎥ ⎥ ⎢ ⎥ ⎢ D=⎢ ⎥ ⎥ ⎢ .. .. ⎥ ⎢ . . ⎥ ⎢ ⎦ ⎣ N N N S1 (ω1 ) S1 (ω2 ) · · · S1 (ωM ) ABSENCE OF ARBITRAGE We can also write D as 27 ⎤ | | D = St = ⎣ d 1 · · · d M ⎦ | | ⎡ where d1 , .