Question

A bond and a stock have the following values and possible outcomes:

Bond values: A(0) = 100, A(1) = 120
Stock price: S(0) = 100, with two possible outcomes at S(1):
S(1) = 150 with probability p
S(1) = 90 with probability 1 - p
A call option has the following payoff at S(1):

C(1) = 50 if S(1) = 150
C(1) = 0 if S(1) = 90
Now, we want to create a portfolio with x shares of stock and y bonds:

(a) Find the values of x and y so that V(1) = C(1) with probability 1.

(b) Find the value of the portfolio V(0).

Answer 1

To achieve V(1) = C(1) with probability 1, set x = 0 and y = 1.

Step-by-step explanation:

In this scenario, a call option's payoff is contingent on the stock's performance at S(1). The call option has a value of 50 if S(1) = 150 and 0 if S(1) = 90. To ensure V(1) = C(1) with probability 1, we need to eliminate stock exposure as the bond value is fixed. This is achieved by setting x (number of shares of stock) to 0. With x = 0, the call option's payoff becomes irrelevant, and the portfolio value is solely determined by the bond.

The bond values are A(0) = 100 and A(1) = 120. Therefore, the value of the portfolio at time 0, V(0), is the sum of the bond and stock values:

`\[ V(0) = x * S(0) + y * A(0) \]`

Since x is set to 0:

`\[ V(0) = 0 * S(0) + y * A(0) = y * A(0) = 1 * 100 = 100 \]`

Thus, the value of the portfolio at time 0 is 100.