dstefan
03-22-2008, 01:19 PM
Chapter 1, Problem 14:
Call options with strikes 100, 120, and 130 on the same underlying asset and with the same maturity are trading for $8, $5, and $3, respectively (there is no bid--ask spread).
Is there an arbitrage opportunity present? If yes, how can you make a riskless profit?
Solution:
For an arbitrage opportunity to be present, there must be a portfolio made of the three options with nonnegative payoff at maturity and with a negative cost of setting up.
Denote by K_1 < K_2 < K_3 the strikes of the options. If x_1, x_2, x_3 are the positions taken in the options (which can be either negative or positive), then the values of the portfolio at time 0 and at maturity T are
V(0) ~=~ x_1 C_1(0) ~+~ x_2 C_2(0) ~+~ x_3 C_3(0)
V(T) ~=~ x_1 C_1(T) ~+~ x_2 C_2(T) ~+~ x_3 C_3(T) ~=~ x_1 \max(S(T)-K_1,0) ~+~ x_2 \max(S(T)-K_2,0) ~+~ x_3 \max(S(T)-K_3,0)
\begin{tabular}{|c|c|c|c|c|}\hline & S(T) < K_1 & K_1 < S(T) < K_2 & K_2 < S(T) < K_3 & K_3 < S(T) \\ \hline V(T) & 0 & x_1 S(T) - x_1 K_1 & (x_1 + x_2) S(T) - x_1 K_1 - x_2 K_2 & (x_1 + x_2 + x_3) S(T) - x_1 K_1 - x_2 K_2 - x_3 K_3 \\ \hline \end{tabular}
Note that V(T) \geq 0 for any value of S(T) if and only if x_1 \geq 0, the value of the portfolio when S(T) = K_3 is nonnegative, i.e., (x_1 + x_2) K_3 - x_1 K_1 - x_2 K_2 \geq 0, and x_1 + x_2 + x_3 \geq 0.
Thus, an arbitrage exists if and only if the values C_1(0), C_2(0), C_3(0) are such that we can find x_1, x_2, x_3 with the following properties:
x_1 C_1(0) + x_2 C_2(0) + x_3 C_3(0) ~<~ 0
x_1 \geq 0
(x_1 + x_2) K_3 - x_1 K_1 - x_2 K_2 \geq 0
x_1 + x_2 + x_3 \geq 0
For C_1(0)=8, C_2(0)=5, C_3(0)=3 and K_1=100, K_2=120, K_3=130, the problem becomes finding x_1 \geq 0 and x_2, x_3 such that
8 x_1 + 5 x_2 + 3 x_3 ~<~ 0
30 x_1 + 10 x_2 \geq 0
x_1 + x_2 + x_3 \geq 0
(For these option prices, arbitrage is possible since the middle option is overpriced relative to the other two options.)
The easiest way to find values of x_1, x_2, x_3 satisfying the constraints above is to note that arbitrage can occur for a portfolio with long positions in the options with lowest and highest strikes, and with a short position in the option with middle strike (note the similarity to butterfly spreads). Then, choosing x_3 = - x_1 - x_2 would be optimal, and the constraints become
5 x_1 + 2 x_2 ~<~ 0
3 x_1 + x_2 \geq 0
These are satisfied, e.g., for x_1 = 1 and x_2 = -3, which corresponds to x_3 = 2.
Buying one option with strike 100, selling three options with strike 120, and buying two options with strike 130 will generate a positive cash flow of $1, and will result in a portfolio that will not loose money regardless of the value of th eunderlying asset at the maturity of the options.
Call options with strikes 100, 120, and 130 on the same underlying asset and with the same maturity are trading for $8, $5, and $3, respectively (there is no bid--ask spread).
Is there an arbitrage opportunity present? If yes, how can you make a riskless profit?
Solution:
For an arbitrage opportunity to be present, there must be a portfolio made of the three options with nonnegative payoff at maturity and with a negative cost of setting up.
Denote by K_1 < K_2 < K_3 the strikes of the options. If x_1, x_2, x_3 are the positions taken in the options (which can be either negative or positive), then the values of the portfolio at time 0 and at maturity T are
V(0) ~=~ x_1 C_1(0) ~+~ x_2 C_2(0) ~+~ x_3 C_3(0)
V(T) ~=~ x_1 C_1(T) ~+~ x_2 C_2(T) ~+~ x_3 C_3(T) ~=~ x_1 \max(S(T)-K_1,0) ~+~ x_2 \max(S(T)-K_2,0) ~+~ x_3 \max(S(T)-K_3,0)
\begin{tabular}{|c|c|c|c|c|}\hline & S(T) < K_1 & K_1 < S(T) < K_2 & K_2 < S(T) < K_3 & K_3 < S(T) \\ \hline V(T) & 0 & x_1 S(T) - x_1 K_1 & (x_1 + x_2) S(T) - x_1 K_1 - x_2 K_2 & (x_1 + x_2 + x_3) S(T) - x_1 K_1 - x_2 K_2 - x_3 K_3 \\ \hline \end{tabular}
Note that V(T) \geq 0 for any value of S(T) if and only if x_1 \geq 0, the value of the portfolio when S(T) = K_3 is nonnegative, i.e., (x_1 + x_2) K_3 - x_1 K_1 - x_2 K_2 \geq 0, and x_1 + x_2 + x_3 \geq 0.
Thus, an arbitrage exists if and only if the values C_1(0), C_2(0), C_3(0) are such that we can find x_1, x_2, x_3 with the following properties:
x_1 C_1(0) + x_2 C_2(0) + x_3 C_3(0) ~<~ 0
x_1 \geq 0
(x_1 + x_2) K_3 - x_1 K_1 - x_2 K_2 \geq 0
x_1 + x_2 + x_3 \geq 0
For C_1(0)=8, C_2(0)=5, C_3(0)=3 and K_1=100, K_2=120, K_3=130, the problem becomes finding x_1 \geq 0 and x_2, x_3 such that
8 x_1 + 5 x_2 + 3 x_3 ~<~ 0
30 x_1 + 10 x_2 \geq 0
x_1 + x_2 + x_3 \geq 0
(For these option prices, arbitrage is possible since the middle option is overpriced relative to the other two options.)
The easiest way to find values of x_1, x_2, x_3 satisfying the constraints above is to note that arbitrage can occur for a portfolio with long positions in the options with lowest and highest strikes, and with a short position in the option with middle strike (note the similarity to butterfly spreads). Then, choosing x_3 = - x_1 - x_2 would be optimal, and the constraints become
5 x_1 + 2 x_2 ~<~ 0
3 x_1 + x_2 \geq 0
These are satisfied, e.g., for x_1 = 1 and x_2 = -3, which corresponds to x_3 = 2.
Buying one option with strike 100, selling three options with strike 120, and buying two options with strike 130 will generate a positive cash flow of $1, and will result in a portfolio that will not loose money regardless of the value of th eunderlying asset at the maturity of the options.