A.4 The Black-Scholes model Problem A.4.1 Consider Black-Scholes models with interest rates 2% and 4%, and volatility 20%. Assuming that initial stock and bank account values are 100 and 1 , respectively, determine fair price of call/put options with strike price 110 and maturities T=108/365,215/365. Comparing these prices, is it possible to say something about their behavior as functions of interest rate? Give similar comparison of call prices for a dividend paying stock with dividend rates 1% and 2%. Problem A.4.2 Consider the Black and Scholes model: dSt(dt+dWt) with interest rate r. Derive the following heuristic formula for a call option with strike price K : CTheauristice()=e(r)TS0(d+(,T))KerT(d(,T)) where d(,T)=Tln(S0/K)+(2/2)T () is a standard normal distribution function and S0 is the initial price of stock. Problem A.4.3 Consider a Black-Scholes market with parameters r=0.02, S0=100,=r,=0.2. Find prices of call and pul options wilh K=110, T=365108, and T=365215. Determine also the prices of call and put options for other values of volatility: =0.1,=0.4, and give an explanation of existing differences in prices. Problem A.4.4 Consider a Black-Scholes market with parameters == 2%,S0=100,=0.2. 282 Risk Analysis in Finance and Insurance (a) Find prices of call and put options with strike price $110 and maturities T1=365108 and T2=365215. (b) Compare these prices with the corresponding prices (only for call option with maturity T2=215/365 ) in the model with a proportional dividend 10%. (c) How these results will change for the market volatility 80% (only for call option with maturity T2=215/365) ? (d) How these results will change for the strike $80