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In chapter 4 of Natenberg's "Option and Volatility and pricing", he discusses how to draw parity graphs for option positions. These are defined as a plot of the intrinsic value of the position against the price of the underlying at expiration.

One of the examples given involves a position of long 2 calls and short one underlying in the image below. Parity Graph Natenberg

However, why isn't the parity graph be shifted down? If $x$ is the underlying price at expiry and $K$ is the strike price of the call, wouldn't the intrinsic value at maturity be following? The graph given looks like the graph of a regular straddle of long one call and short one put rather than long 2 calls and short one underlying. $$f(x) =2(x-K)^{+}-x= \begin{cases}x-2K &\quad x\geq K\\ -x & \quad x < K\end{cases}$$

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  • $\begingroup$ I think you are missing the initial value of the short position. If you set $K=S_t$ (buy at-the-money options) and short the stock at price $S_t=K$, then your payoff will be $2(x-K)^++(K-x)$, so $x-K$ if above $K$ and $K-x$ if below $K$. $\endgroup$
    – mmencke
    Jan 7, 2022 at 14:04
  • $\begingroup$ I don't think parity graphs include the premiums paid or received for buying/selling options. I think that would be a P & L graph. Also, there is no indication in the book that the stock is shorted at $K$. $\endgroup$
    – BaroqueFreak
    Jan 10, 2022 at 12:06
  • $\begingroup$ That is a good point. Maybe the underlying is assumed to be a futures contract? $\endgroup$
    – mmencke
    Jan 10, 2022 at 17:47

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