Consider a risk free asset that can be bought (lent) or sold (borrowed) with an interest rate of 75% such that $1 today is worth$1.75 a period later |
Assemble a portfolio of X shares of stock and Y dollars of the risk free asset (quantities may be negative) to replicate the payoff from the option
WhenS1=$200: X*$200 + Y*$1.75 = $75 WhenS1=$50: X*$50 + Y*$1.75 = $0
Two equations in two unknowns
X * $50 + Y * $1.75 = $0
X = -1.75/50 Y = -0.035 Y
X*$200 + Y*$1.75 = $75 200 * -0.035 Y + Y * 1.75 = $75 -5.25 Y = 75
Y = 75/-5.25 = -14.29 (rounded )
1a. What is the number of shares (X) to be held in addition to the -$14.29 in the replicating portfolio?
1b. Demonstrate that this portfolio (X shares and Y cash) replicates the payouts of the option in both boom and bust.1c3. What is the price at time 0 of the replicating portfolio? By the Law of One Price this must be the premium of the option?