Consider a portfolio II of European at-the-money options that holds two long call options, 3 short put options, and is long a units of the stock S. Mathematically we define this as follows:

II = 2CE(S,t) 3PE(S,t) + Sa

Suppose we would like to choose a such that the value of the portfolio II will be unaffected by small changes in the underlying stock price S.

Assuming the annual risk-free r = 2%, the stock volatility o = 0.2, and there are 4 years to expiration, find the optimal numerical value for a.

Please round your numerical answer to 2 decimal places.

Q51;

In this problem we assume the stock price S(t) follows Geometric Brownian Motior described by the following stochastic differential equation:

dS = ySdt +oSdw,

where dw is the standard Wiener process and u = 0.13 and o = 0.20 are constants. The current stock price is $100 and the stock pays no dividends. Consider an at-the-money European call option on this stock with 1 year to expiration.

What is the most likely value of the option at expiration?

Please round your numerical answer to 2 decimal places.

Consider a portfolio II of European at-the-money options that holds two long call options, 3 short put options, and is long a units of the stock S. Mathematically we define this as follows:

II = 2C#(S.) - 3P (S. t) + Sa

Suppose we would like to choose a such that the value of the portfolio II will be unaffected by small changes in the underlying stock price S.

Assuming the annual risk-free r = 2%, the stock volatility a = 0.2, and there are 4 years to expiration, find the optimal mumerical value for a. Please round your numerical answer to 2 decimal places.