1. Black and Scholes pioneered the use of dynamic hedging in trading options. Suppose a nancial institution sells a European call option whose terminal payoff is maX(ST X , 0), where ST is the terminal stock price and X is the strike price. (a) (i) Explain why the number of units of stock held to hedge against the market risk of stock price uctuation is given by ans, t) BS ? where V(S, t) is the value of the European call option. (ii) Why A ) 1 when the stock price is high and A > 0+ when the stock price is low? (b) Suppose the delta is 0.5 at the current stock price level $101] and the delta falls to 0.45 when the stock price drops to $97. The nancial institution has sold 1, 000 units of European call option. What action should be taken by the hedger if the stock price does fall from $100 to $97? (c) Can dynamic hedging procedure using delta hedging eliminate all market risks? If not1 what other types of market risks that remain even delta hedging has been implemented? A: (d) We employ minimum variance hedge when the direct hedging instrument on the underlying asset is not available. Suppose we hedge the price uctuation of one unit of asset S using N units of another related asset F. The change in value in the hedged portfolio V over the hedging period is 6V 2 65' + NiF. (i) Show that the optimal number of units of hedged instrument is given by \"E 0%. 1 where 05;: = cov(|:'581 5F) and 0%. = va.r(F). (ii) At the optimal hedge N", show that the variance of 6V becomes N*= 2 2 _ 2 _ \"m JV, (IS 0% , where 0% = var(S). (iii) We dene the effectiveness of hedge to be R2 _ 0.29 (IE2 _ 02 S Show that R2 2 Pg}?- Explain using nancial intuition why R } 1 when .05.? } 1