Investment Portfolio

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3.

a.

The two possible values of the index in the first period are:

uS0 = 1.06 × 50 = 53

dS0 = 0.95 × 50 = 47,5

The possible values of the index in the second period are:

uuS0 = (1.06)2 × 50 = 56,18

udS0 = 1.06 × 0.95 × 50 = 50,35

duS0 = 0.95 × 1.06 × 50 = 50,35

ddS0 = (0.95)2 × 50 = 45,125

b.

The call values in the second period are:

Cuu = 56,18 − 53 = 3,18

Cud = Cdu = Cdd = 0

Since Cud = Cdu = 0, then Cd = 0.

To compute Cu, first compute the hedge ratio:

[pic]

Form a riskless portfolio by buying five shares of stock and writing nine calls.

The cost of the portfolio is: 5S – 9Cu = 5 x $53 – 9Cu

The payoff for the riskless portfolio equals $252:

|Riskless Portfolio |S = 50,35 |S = 56,18 |

|Buy 5 shares |215,75 |280,9 |

|Write 9 calls |0 |-28,62 |

|Total |215,75 |252,28 |

Therefore, find the value of the call by solving:

5 x $53 – 9Cu = $252/1.0125 ( Cu = $ 1,79

To compute C, compute the hedge ratio:

[pic]

Form a riskless portfolio by buying 0.3255 of a share and writing one call.

The cost of the portfolio is: 0.3255S – C = $16.275 – C

The payoff for the riskless portfolio equals $15.4615:

|Riskless Portfolio |S = 47,5 |S = 53 |

|Buy 0.3255 share |15,4615 | 17.2515 |

|Write 1 call | 0.000 | −1,79 |

|Total |15.4615 |15.4615 |

Therefore, find the value of the call by solving:

$16.275 – C = $15.4615/1.0125 ( C = $1.00438

d.

The put values in the second period are:

Puu = 0

Pud = Pdu = 53 – 50,35 = 2,65

Pdd = 53 – 45,125 = 7,875

To compute Pu, first compute the hedge ratio:

[pic]

Form a riskless portfolio by buying four share of stock and...