Question

Suppose that the continuously compounded expected return on the stock is a and that the stock does not pay dividends. We denote by the appropriate discount rate for the option during the period h. The option price is then given by: Show (in a thorough manner) that this price is algebraically equivalent to the risk-neutral calculation.

Answer 1

rate = risk Free rate. s let the risk free rate be r. (=r=&] since unele Szüst neutral valucetion Cu s option payoff if stock goes up expected return = discoun ed - 1 i s all down. Son y current stock price u as up step factor dy down step factor Alco let co be the value of the option i 3 Sou- cu Co Soda Cd Y cast Cet w build a risk free portfolio with a shares and such that this portfolio replicates the option pay off. Therefore, at time to Portfolio Value (vol ² Aso

for the portfolio to be replicating, we have Un = Asou + Yoon - lu - Tasod & werh - ca - Alo (u-d) = Co-ed =) D = (Cu-Cd) Putting this in , A sou + ve = Cu yh Sou + yeth = cu I Solu-d) J =) 4 = thru Cd - deu ud Now, Vo: Aso + 4 h uld-dla = (cu - cd ) 96 + 8th Juld-dcu | Solu-d) and und rh V. = rh - e u-d Also, sinu vo is the replicating portfolio for the option, By the principle of no arbitrage their prie at time o must be the same . Vo = Co = erh ud