By Bouchaud J.-P.

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**Sample text**

This is a very natural step towards reality, where we face a large number of stocks traded on markets. Even when we restrict to a single stock, we may wish to add an international perspective, since if the stock is traded on some foreign market we should include in our model the home currency on this market. Finally, some derivative securities may involve many underlying assets. The initial prices of our two stocks will be denoted by S 1 (0) and S 2 (0). Their future prices are determined by random returns, denoted by K1 , K2 and assumed to be defined on the probability space Ω = {u, d} for simplicity, with obvious modifications if a general space is needed.

The subspace W contains the vector 1 = 1 then V(x,y) (1) = 1 for all scenarios. (1, . . 35 with the subspace W and the compact convex set A = {D}. 39 Suppose A ⊂ Rn is convex and compact, and W is a vector subspace of Rn disjoint from A. Then there exists z ∈ Rn such that z, a Q > 0 for all a ∈ A, and z, w Q = 0 for all w ∈ W, where x, y Q = i=1 xi yi qi , qi > 0, is an inner product in Rn . Proof See page 47. From the lemma we obtain an R M -vector Z = (z1 , . . , z M ) such that Z, D Q > 0 and Z, V Q = 0 for V ∈ W.

Proof If H(0) < H (0), buy H, sell H , the diﬀerence is the arbitrage profit (more precisely, take x = 0, y = H (0)−H(0) , zH = 1, zH = −1 and see A(0) that this is an arbitrage). 44 Law of one price If H(1) = H (1) then H(0) = H (0). Now the call-put parity relation immediately follows from the relation between the payoﬀs. Arbitrage bounds on option prices We would like to discuss the dependence of option price on today’s value of the underlying stock. This is quite a natural idea but it turns out to be a bit awkward.

### An introduction to statistical finance by Bouchaud J.-P.

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