Problem A.4.8 Consider the Black-Scholes model of a (B,S)-market, and compare the optimal investment strategy with the minimal hedge of an European call option with fT=(STK)+. Problem A.4.9 In the framework of the Black-Scholes model of a (B,S) market, consider an investment portfolio with the initial capital x. Estimate the asymptotic profitability of : limsupTT1lnE(XT(x)),(0,1]. Problem A.4.10 Let C=C(S0,T,K,,r) be the fair price of a call option in the Black-Scholes model. It is a function of S0 (initial stock price), T (exercise time), K (strike price), (volatility), r (interest rate). Prove that (a) C(S0)0 as S00,C(S0) as S0; (b) C(K)S0 as K0,C(K)0 as K; (c) C(S0,T,K)S0 as T,C(S0,T,K)S0K as T0(S0>K); (d) C(S0,T,K,,r)S0KerT as 0(S0>KerT), C(S0,T,K,,r)S0 as (S0>KerT) (e) C(S0,T,K,,r)S0 as r. Problem A.4.11 Let P=P(S0,T,K,,r) be the fair price of a put option in the Black-Scholes model. Prove the following properties of P as a function of S0,T,K,,r : (a) P(S0,T)KerT as S00,P(S0,T)0 as S0; (b) P(K)0 as K0,P(K) as K; 284 Risk Analysis in Finance and Insurance (c) P(T)0 as T,P(T)0 as T0(S0>K); (d) P()0 as 0(S0>KerT),P()KerT as (S0> KerT) (e) P(r)0 as r