UH Master program Mathematics and Statistics MAST31801 Mathematical Finance I, Fall Solutions to problems 1 ( ) Notation: R + = [0, ).

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1 UH Master program Mathematics and Statistics MAST31801 Mathematical Finance I, Fall Solutions to problems 1 ( ) Notation: R + = [0, ). 1. A gambling website offers these odds for next week F.C Barcelona- Juventus football game: for F.C. Barcelona s win, 3.65 for a tie, and 4.30 for Juventus win. It means that for example by placing 1eon Barcelona win, the gambler will get ein case Barcelona will win, with a profit of 0.80e, otherwise he will lose the money he gambled. Is this price system free of arbitrage? Does a gambler, who is constrained to place only non-negative bets ( only long positions are allowed, short-selling at the same rates is not possible), have arbitrage possibilities? Use the separating hyperplane theorem to give a proof. 2. Table (1) shows the odds for next week F.C. Barcelona-Juventus UEFA champions league game picked from different gambling websites. Check whether by picking the best odds offered by the bookmakers for each outcome, it is possible to create an arbitrage opportunity, without short positions (Hint: pick the best offer for each result, and consider gambling portfolio with positive coefficients). This example is based on real data. Sometimes an arbitrage opportunity between different markets may appear, but it should vanish rapidly as soon as arbitrageurs take advantage of it forcing the prices in different markets back to joint equilibrium. Trading spreads spreads information across the markets. 1

2 Website 10Bet 18bet Asianodds bet-at-home bet365 Bethard bwin Jetbull Marathonbet Pinnacle Tempobet TonyBet Unibet Barcelona win Draw Juventus win Table 1: odds offered for the UEFA Champions League football game F.C. Barcelona vs Juventus on Solutions Let s denote W (ω) {1, x, 2} the outcome of the football game. the events {W = 1},{W = x},{w = 2} are disjoint and their union is a certain event. The prices of the indicators 1(W = 1) 1(W = x) 1(W = 2) are respectively 5/9, 20/73, 10/43, but their sum is , so there is an arbitrage, by placing the bets at the given prices the betting website gets eand pays out 1ewith a profit of e. Note that any portfolio with coefficients (c, c, c) with c < 0 is an arbitrage portfolio which correspond to a profit for betting company is selling the bets. Note that the customers are allowed only to buy the bets at the given prices (the portfolio coefficients are constrained to be positive, under this positivity constaint there is no arbitrage. Let 1 c(1) c(1) c(1) A = c(x) 1 c(x) c(x) c(2) c(2) 1 c(2) 2

3 be the 3 3 profit matrix for the gambler, where (c(1), c(x), c(2)) = (5/9, 20/73, 10/43) are the prices of the bets paying 1efor each of the three result. By using the separating hyperplane theorem we show that one and only one of these two alternatives is always true (a) Either there exists an arbitrage strategy with non-negative coefficients corresponds to a vector ξ R 3 + \ { 0} such that ξa R 3 + \ { 0}. (b) Or there is an equivalent supermartingale measure, which a vector π (0, 1) 3 such that 3 π i = 1 and (Aπ) R 3 +. (By normalizing this is equivalent to finding a vector y (0, ) 3 such that (Ay) R 3 +. Consider the convex cone C = { Ay : y (0, ) 3 }. If (b) does not hold, for every R 3 + C =. By the separating hyperplane theorem, there exists a separating hyperplane { v R 3 : v ξ = 0 } with ξ R 3 separating the two convex sets C and R 3 + such that ξay 0 y (0, ) 3, ξay > 0 for some y (0, ) 3 and ξ x 0 x R 3 +. This means that ξa R 3 + and ξ R + 3, which is an arbitrage strategy with positive coefficients. Here we used another version of the separating hyperplane theorem, which tells if C and C are two convex sets C C =, there is a separating hyperplane between them. We show that Then by Gordan lemma either there is a strategy ξ with ξa R 3 \ {0} and ξ R 3 \ 0, or there is y (0, ) 3 with Ay = 0. But the latter cannot be. In practice the gambling websites in order to implement their arbitrage strategy need to find customers placing their bets are competing for customers and have to keep their profit margin low. We also observe empirically that the prices fluctuate (according to the flow of customers placing their bets on different event) the gambling prices offered by the different websites offer occasionally an arbitrage opportunity for the customer who picks the best price on the market for each event. The best odds for the results 1, x, 2 in Table 1 are respectively 1.93, 3.94, 4.61 with = , meaning that the strategy (1, 1, 1) is an arbitrage portfolio for the customer, paying with certainity 1efor the initial price of ewith a profit of e. 3

4 3. Consider a vector space V, for example V = R d. We say that a subset C V is convex if and only if x, y C, 0 α 1 = αx + (1 α)y C (every convex combination of two points from the subset is in the subset). (a) Show that n N, x i C, α i 0, i = 1,..., n and = α i x i C α i = 1, (every convex combination of n-points from the subset is in the subset). (b) For a subset A V, its Convex hull is defined as { C(A) := x = α i y i : n N, y i A, α i > 0, } α i = 1 Show that C(A) is convex. Solution if for β [0, 1] x and x are convex combinations of A-elements, then x = α iy i, x = α i y i, with α i = α i = 1, α i, α j > 0, x = βx + (1 β)x = βα iy i + (1 β)α i y i is a convex combination convex combinations of A-elements, with and positive weights. β α i + (1 β) α i = 1 4

5 (c) Show that C(A) is closed when A is a finite set. For A = {y 1,..., y n } V, { C(A) := x = α i y i : n N, y i A, α i 0, } α i = 1 Solution If x (k) = n α(k) i y i C(A) is convex combination of A elements and x (k) x V, then the vector α (k) = (α (k) 1,..., α n (k) ) is contained in the compact simplex S = { α = (α 1,..., α n ) : α i 0, α i = 1 ) and by the Heine Borel theorem there is an α S such that α (k) α. Necessarily x = αi y i C(A) Show also a counterexample of an infinite set A such that C(A) is not closed. Solution For A = {1/n : N N \ {0}} C(A) = (0, 1] which is not closed lim n 1/n = 0 C(A). (d) Show that if { C i : i I} is an arbitrary collection of convex sets in V, then the interesection C = i I C i is also convex. Solution If x, y C i i and α [0, 1], then αx + (1 α)y C i i. If C and C are convex in V, is always C C necessarily convex? Solution No, just take C = { x} and C = { y} with x y, then {x, y} { αx + (1 α)y; α [0, 1] } (e) Show that the convex hull of A can be also defined as C(A) = C C convex : A C V 5

6 Note that C(A) is convex since is the (arbitrary) intersection of convex set, and since it contains A it must contain all the convex combinations of A elements. On the other hand the set of convex combintations of A elements is a convex set containing A, therefore it contains this arbitrary intersection. 4. Farkas lemma. For a d n-matrix A, and a vector b = (b 1,..., b d ) R d, one and only one of these two alternatives is always true: (1) There exists x = (x 1,..., x n ) (0, ) n such that Ax = b (2) There exists y = (y 1,..., y d ) R d such that ya R d \ {0} ja b y < 0. Prove Farkas lemma by using the separating hyperplane theorem. Hint Look at the following geometric interpretation: denote by a 1,..., a n R d the column vectors of the matrix A. Show that { } C = α i a i : α i > 0 R d which is the cone closed in R d. generated by the vectors a 1,..., a n, is convex and (1) and (2), correspond to the alternatives b C and b / C. Solution: the proof is in the lecture note, Lemma Gordan lemma. Prove that, for a matrix A R d n, (1) either Ax > 0 for some x (0, ) n, ( r = (r 1,..., r d ) > 0 meaning r i > 0 i). (2) or ya = 0 for some y R d + \ {0}. Solution The set C = { ya : y R d + \ {0} } is a convex cone in V = R n. By the separating hyperplane theorem if 0 / C, there exists ξ R n such that yaξ 0 y R d + and y R d +\{0} with y Aξ > 0. But this means that Aξ (0, ) n. If 0 C, then C contains a linear subspace of R n. In such case there cannot be an hyperplane {v R n : (v, x) = 0} with x R n separating strictly C and {0}, with (Ax) i > 0 i = 1,..., n. 6