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I'm having a hard time understanding the solution to the following problem from Dan Stefanica's book "A Primer for the Mathematics of Financial Engineering":

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?

To model the situation, we assume $x_1, x_2, x_3$ units of the three options are purchased (where the $x_i$ may be negative as well). Then, the cost of the portfolio is $8x_1 + 5x_2 + 3x_3$, which we want to be negative for an arbitrage opportunity to exist. So we get our first equation

$$8x_1 + 5x_2 + 3x_3 < 0$$

Then we model the payoff of the portfolio at maturity as $$C(T) = x_1 max(S(T) - K_1, 0) + x_2 max(S(T) - K_2, 0) + x_3 max(S(T) - K_3, 0)$$. This gives us a piecewise description of the payoff -

\begin{equation} C(T) = \left\{\begin{array}{lr} 0, & S(T) \le K_1 \\ x_1(S(T) - K_1), & K_1 \le S(T) \le K_2\\ (x_1 + x_2)S(T) - x_1K_1 - x_2K_2, & K_2 \le S(T) \le K_3\\ (x_1 + x_2 + x_3)S(T) - x_1K_1 - x_2K_2 - x_3K_3, & K_3 \le S(T) \end{array}\right. \end{equation}

Each of these should be non-negative if an arbitrage opportunity is to exist. If we set $S(T) = K_3$, then we get $(x_1 + x_2)K_3 - x_1K_1 - x_2K_2 \geq 0$. Substituting the actual numbers in, we get $30x_1 + 10x_2 \geq 0$.

Now, here comes my confusion - the solution states that from the payoff description, we can also come up with a third condition $x_1 + x_2 + x_3 \geq 0$. I'm not sure how to arrive at this condition based on the current information we have. Any thoughts?

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  • $\begingroup$ Why don't you try with brute force and ask the Excel solver to find the values $x_1, x_2, x_3$ to create an arbitrage? Add a few constraints to ensure the payoff is positive for values like 90, 91, 92, .. 138, 139, 140. $\endgroup$ Commented Apr 25 at 6:48
  • $\begingroup$ You need each payoff to be positive in each area of the terminal payoff function. So the first condition implies that $x_1>0$, the next condition implies that $x_1+x_2>0$ etc $\endgroup$ Commented Apr 25 at 8:05
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    $\begingroup$ Look at your last equation: if $x_1+x_2+x_3<0$, then for sufficiently high $S(T)$ the payoff will be negative. $\endgroup$ Commented Apr 26 at 12:52
  • $\begingroup$ Hi @dm63, Thank you! I do see that for sufficiently high S(T) the third expression will become negative. This helps a lot. $\endgroup$ Commented Apr 29 at 5:34

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