Summary of last lecture Primary objectives: Arbitrage: Definition - PowerPoint PPT Presentation

Summary of last lecture Primary objectives: � Arbitrage: Definition and example ✔ � Duality and complementary slackness ✔ � Fundamental theorem of asset pricing ✔ � Arbitrage detection using linear programming 86

Scenario � Portfolio of derivate securities (European call options) S i , i = 1,..., n of one security S is determined by vector ( x 1 ,..., x n ) � Payoff of portfolio is Ψ x ( S 1 ) = � n i = 1 Ψ i ( S 1 ) x i , where Ψ i ( S 1 ) = max{( S 1 − K i ),0}, where K i is strike price K i (piecewise linear function with one breakpoint!) � Cost of performing portfolio at time 0: n S i � 0 x i . i = 1 Determine arbitrage possibility � Negative cost of portfolio with nonnegative payoff (type A) � Cost zero and positive payoff (type B) 87

Observation Nonnegative payoff Payoff is piecewise linear in S 1 with breakpoints K 1 ,..., K n . Payoff is nonnegative on [0, ∞ ), if nonnegative at 0 and at all breakpoints and right-derivative at K n is nonnegative (assume K 1 < K 2 < ··· < K n ). Formally: Ψ x (0) � 0 Ψ x ( K j ) � 0, j = 1,..., n Ψ x ( K n + 1) − Ψ x ( K n ) � 0. 88

Linear program min � n i = 1 S i 0 x i � n i = 1 Ψ i (0) x i 0 � � n i = 1 Ψ i ( K j ) x i 0, j = 1,..., n � � n i = 1 ( Ψ i ( K n + 1) − Ψ i ( K n )) x i 0. � 89

Proposition There is no type A arbitrage if and only if optimal objective value of LP is at least 0 Proposition Suppose that there is no type A arbitrage. Then, there is no type B arbitrage if and only if the dual of LP has strictly feasible solution. 90

Constraint matrix � Ψ i ( K j ) = max{ K j − K i ,0} � Constraint matrix A of LP has the form   K 2 − K 1 0 0 0 ··· K 3 − K 1 K 3 − K 2 0 0 ···     . . . . ...   . . . . A = . . . .       K n − K 1 K n − K 2 K n − K 3 0 ···   1 1 1 1 ··· Theorem Let K 1 < K 2 < ··· < K n denote strike prices of European call options on the same underlying security with same maturity. There are no arbitrage opportunities if and only if prices S i 0 satisfy 1. S i 0 > 0, i = 1,..., n 2. S i o > S i + 1 , i = 1,..., n − 1 0 3. C ( K i ) : = S i 0 defined on { K 1 ,..., K n } is strictly convex function 91

Summary of lectures on asset pricing Primary objectives: � Arbitrage: Definition and example ✔ � Duality and complementary slackness ✔ � Fundamental theorem of asset pricing ✔ � Arbitrage detection using linear programming ✔ 92