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Suppose that a certain stock is currently worth $S_0=\$61$. Consider an investor that buys one call with a strike price equal to $K_1=\$55$, that costs $c_1=\$10$, buys another call with strike price equal to $K_3=\$65$, paying for such a call $c_3=\$5$ and sells two calls with a $K_2=\$60$ strike price, receiving $c_2=\$7$ for each of such call (assume that all the options have the same underlying asset and same maturity).

a) Present a combination of puts, instead of calls, such that you have exactly the same payoff as you have with this combination of calls.

b) Derive the price of such spread, using only puts.

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  • $\begingroup$ In a) what I thought was to buy two puts $p_2=\$7$ and sell $p_1=\$10$ and $p_3=\$5$. But the problem with this is that your initial investment is different... $\endgroup$ – Papagaio_da_Fauna May 19 at 19:48
  • $\begingroup$ What I was thinking was to do the inverse strategy, but in that case the initial investment would be +1 (instead of -1) thus leading to different payoffs... $\endgroup$ – Papagaio_da_Fauna May 19 at 20:23
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For a given maturity, given three equally spaced option strikes $K_1,K_2,K_3$ a "butterfly" combination consists of shorting 2 of the middle strike calls and buying one each of the "wing" or lateral calls. This position has a positive cost i.e. $c_1+c_3-2 c_2 >=0$ (why? because it has a positive payoff for $S_T\approx K_2$ and zero payoff elsewhere).

In the example given we have $10+5-2*7=1>=0$

It can be shown (Gordon has already shown it above) that the same payoff can be obtained with puts: You short two of the middle strike puts and buy one each of the wing puts. By no arbitrage the cost $p_1+p_3-2 p_2$ will be the same as the cost with calls we found above.

However in general $p_1\ne c_1,p_2\ne c_2,p_3\ne c_3$. If we assume zero interest rates (as Zumba showed above) we will have $p_i=c_i-S_0+K_i$ instead (by Put Call Parity).

If the example given we have $p_1=10-61+55=4$, $p_2=7-61+60=6$, $p_3=5-61+65=9$. Notice that the call prices $10,7,5$ are decreasing with strike while the put prices $4,6,9$ are increasing in strike. Nevertheless $p_1+p_3-2 p_2=4+9-2*6=1$ is the same as the cost we found with calls. All as expected.

Hope this clarifies a few things. (Note that there is no need to reverse the sign of the positions, if we buy $c_i$ we also buy (not sell) $p_i$).


What if interest rates are non-zero? Then we have $p_i=c_i-S_0+PV(K_i)$. So $$p_1+p_3-2 p_2=c_3+c_1-2c_2-S_0-S_0+2S_0+PV(K_1)+PV(K_3)-2PV(K_2)$$ Because $K_2=(K_1+K_3)/2$ we have $2 PV(K_2)= PV(K_1)+PV(K_3)$. Simplifying the above we have that$$p_1+p_3-2 p_2=c_1+c_3-2 c_2$$ So the equal cost of the put butterfly and the call butterfly is true in general, for any level of interest rate.

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  • $\begingroup$ Thank you very much $\endgroup$ – Papagaio_da_Fauna May 23 at 11:37
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Note that \begin{align*} \max(S-K, 0) = S-K + \max(K-S, 0). \end{align*} Then, \begin{align*} &\ \max(S-K_1, 0)+ \max(S-K_3, 0) - 2 \max(S-K_2, 0)\\ =&\ S-K_1 + \max(K_1-S, 0) + S-K_3 + \max(K_3-S, 0)\\ &\qquad -2(S-K_2) - 2\max(K_2-S, 0)\\ =&\ 2K_2 - (K_1+K_3) + \max(K_1-S, 0)+\max(K_3-S, 0)- 2\max(K_2-S, 0)\\ =&\ \max(K_1-S, 0)+\max(K_3-S, 0)- 2\max(K_2-S, 0). \end{align*}

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  • $\begingroup$ I understand that. So the the price of the options doesn't matter? In the initial scenario, at $t=0$ you would have $2c_2-c_1-c_3=-1$ With the strategy you described, at $t=0$, we would have $p_1+p_3-2p_2=+1$ Therefore wouldn't the payoff be different? $\endgroup$ – Papagaio_da_Fauna May 19 at 20:42
  • $\begingroup$ That is correct. $\endgroup$ – Gordon May 19 at 23:18
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I'll assume rates to be 0, so any $\text{Call}=\text{Put}+S-K$ so in the end you need to have $p_1+p_3-2p_2$, theoretically

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