A function f : R R is called convex if for every [0,1] and every x,y R, f(x + (1 )y) f(x) + (1 )f(y) holds.

(a) (6 pts) For a fixed constant s R, prove that the function f(x) = (s x)+ = max{s x,0} is convex.

(b) (10 pts) Denote by CE(K) = CE(K,T) the premium (price at time t = 0) of a European call option with strike price K and exercise date T. Show that the function K 7 CE(K) is convex. That is, show that for every [0,1] and every K1,K2 > 0, we have CE(K1 + (1 )K2) CE(K1) + (1 )CE(K2). (1) (Hint: try to find an arbitrage opportunity using 3 European call options)

Problem 5. A function f:RR is called convex if for every [0,1] and every x,yR, f(x+(1)y)f(x)+(1)f(y) holds. (a) (6 pts) For a fixed constant sR, prove that the function f(x)=(sx)+=max{sx,0} is convex. (b) (10 pts) Denote by CE(K)=CE(K,T) the premium (price at time t=0 ) of a European call option with strike price K and exercise date T. Show that the function KCE(K) is convex. That is, show that for every [0,1] and every K1,K2>0, we have CE(K1+(1)K2)CE(K1)+(1)CE(K2) Problem 5. A function f:RR is called convex if for every [0,1] and every x,yR, f(x+(1)y)f(x)+(1)f(y) holds. (a) (6 pts) For a fixed constant sR, prove that the function f(x)=(sx)+=max{sx,0} is convex. (b) (10 pts) Denote by CE(K)=CE(K,T) the premium (price at time t=0 ) of a European call option with strike price K and exercise date T. Show that the function KCE(K) is convex. That is, show that for every [0,1] and every K1,K2>0, we have CE(K1+(1)K2)CE(K1)+(1)CE(K2)