american options and replicating portfolios

Question 1

american option:

Answer 1

an option that can be exercised at any time before the expiry date

Question 2

value of american put option:

Answer 2

P(S,t)>=E-S(t)
essentially the value cannot be worth less than the payoff if it was exercised immediately, otherwise there’s potential arbitrage

Question 3

how to replicate a european call option:

Answer 3

only using stocks and bonds so Π=∆S-NB, and ΠT=CT=max(S-E,0)
by the no arbitrage principle the values at 0<=t<T would also be equal

Question 4

self financing:

Answer 4

a portfolio is self financing if any change in ∆ is offset by a change in N (numbers of shares changing means number of bonds changes to offset the portfolio)

Question 5

trading strategy:

Answer 5

the ∆ and N in Π=∆S-NB