𝐸1 (𝑢𝐵,2 )𝛽𝛬1 𝛽𝛬1 2 𝛽𝛬1 2 𝑢𝐵,2 𝑃0 = − 𝑞0 + 𝑞0 + 2 2 1 + 𝛽𝛬1 (1 + 𝛽𝛬1 ) (1 + 𝛽𝛬1 ) 1 + 𝛽𝛬1 Փ𝐵 𝑞0 Փ𝐵 𝑞0 − − 2 (1 + 𝛽𝛬1 ) (1 + 𝛽𝛬1 )2
(iv)
𝐸1 (𝑢𝐵,2 )𝛽𝛬1 2𝛽𝛬1 2 𝐸1 𝑢𝐵,2 2Փ𝐵 𝑃0 = − 𝑞0 + − 𝑞 2 1 + 𝛽𝛬1 (1 + 𝛽𝛬1 ) 1 + 𝛽𝛬1 (1 + 𝛽𝛬1 )2 0
(v)
1 2𝛽𝛬1 2 + 2Փ𝐵 𝑃0 = [𝐸 (𝑢 )𝛽𝛬1 + 𝑢𝐵,2 ] − 𝑞0 [ ] 1 + 𝛽𝛬1 1 𝐵,2 (1 + 𝛽𝛬1 )2
(vi)
Imposing 𝐸1 (𝑢𝐵,2 )𝛽𝛬1 = 𝑢𝐵,2 = ∑𝑛𝑖=1 𝑢̅𝑖 the expected value of underlying basket is equal to the sum every component of its portfolio. 2𝛽𝛬1 2 + 2Փ𝐵 𝑃0 = 𝑢̅𝐵 − 𝑞0 [ ] (1 + 𝛽𝛬1 )2
(vii)
Recalling that 𝛬1 = 2Փ𝐵 + 𝐴𝜎𝐵2
𝑃0 = 𝑢̅𝐵 − 𝑞0 [
2𝛽(2Փ𝐵 + 𝐴𝜎𝐵2 )2 + 2Փ𝐵 ] [1 + 𝛽 (2Փ𝐵 + 𝐴𝜎𝐵2 )]2
𝑛
𝑃0 = 𝑢̅𝐵 − 𝑞0 𝛬0 = ∑ 𝑢̅𝑖 − 𝑞0 [𝐴𝜎𝐵2 + 𝑖 =1
90
(1 + 4𝛽Փ𝐵 )(2Փ𝐵 + 𝐴𝜎𝐵2 ) ] [1 + 𝛽(2Փ𝐵 + 𝐴𝜎𝐵2 )]2
(vii)
(25)