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Question: Given two call options $c_1$ and $c_2$ with strike price $30$ and $40$ respectively. If $c_1$ costs \$3 and $c_2$ costs \$4, is there an arbitrage opportunity?

My attempt:

Short $c_2$ and long $c_1.$ Then we make a profit of $\$4 -\$3 = \$1.$ At expiration, we have $$(S(T) - 30)^+ - (S(T) - 40)^+ = \begin{cases} 0 & \text{ if } S(T)\leq 30, \\ S(T) - 30 & \text{ if } 30\leq S(T)\leq 40, \\ 10 & \text{ if } S(T)\geq 40. \end{cases}$$ Since there is a positive probability that the payoff is nonnegative, so we have an arbitrage opportunity.

Is my attempt above correct?

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No you need to subtract the cost of entering your position as well as the financing costs thereof. In this case you actually receive net \$1 option premiums which yields additional interest at maturity: $$+1\cdot e^{rT}$$ Hence the net value is always positive and represents an arbitrage.

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  • $\begingroup$ I do not understand your first sentence. What do you mean by 'subtract the cost of entering'? $\endgroup$ – Idonknow Dec 8 '19 at 4:22
  • $\begingroup$ @Idonknow I think he's just saying that if you want to be more accurate you should grow your $1$ earned at the beginning at the risk free rate. Now your payoff at maturity will include the future value of $1$. In other cases it could be that you need to pay money to enter the position and therefore at maturity you would need to include the future value of the debt when calculating whether or not there's an arbitrage. $\endgroup$ – Slade Dec 8 '19 at 6:34
  • $\begingroup$ @Slade Why are there debt? By selling and buying option at the beginning, we have \$1 surplus, thus no debt? $\endgroup$ – Idonknow Dec 8 '19 at 6:36
  • $\begingroup$ @Idonknow Yeah in this particular case there's a surplus. But in another example, you may have debt. The point is to include that debt or surplus in the final accounting of the payoff. It may affect whether or not there is an arbitrage. $\endgroup$ – Slade Dec 8 '19 at 6:38
  • $\begingroup$ @Slade Under what circumstances will we get a debt straight after we long and short options, if we start with the aim of obtaining arbitrage? $\endgroup$ – Idonknow Dec 8 '19 at 7:05

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