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I have rigorously stated my problem here.

The task at hand is to express a butterfly spread [no transaction fees] as a sum of long and short call options.

I have found the solution on Wikipedia: $$\big(K-|V-a|\big)^+ = \big(V-(a+K)\big)^+ + \big(V-(a-K)\big)^+ - 2\big(V-a\big)^+,$$ for the underlying $V$, $K>0$ and $a\in \mathbb{R}$. However, in the book [Föllmer, Schied] it's an exercise to come up with the long and short call options [i.e. the righthand side] without any hint. For me, it turned out to be insolvable in a constructive way.

Would anyone even be able to come up this by themselves? What am I missing?

Crossposting in the hopes of finding people who have thought about this before.

Thanks for reading.

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If you're reconstructing a payoff as a linear sum of call options, then the procedure is quite simple -> since the payoff of a call is zero up to the strike, and then linear, you start on the left (i.e. most negative value) and move to the right.

example 1: long butterfly spread, strikes of 80, 100, 120.

  1. start at zero. gradient of payoff is zero: hold no zero strike calls.
  2. Move to next change of gradient: 80. Gradient is now 1, so we must be long 1 call at strike 80.
  3. Move to next change of gradient: 100. Gradient is now -1, but the gradient of our current portfolio of calls is 1 since we are already holding a call at a lower strike. Solution: sell 2 calls, portfolio gradient is now -1.
  4. Move to next change of gradient: 120. Gradient is now zero, but the gradient of our current portfolio of calls is -1 since we are already holding (net) negative one call at a lower strikes. Solution: buy 1 call, portfolio gradient is now 0.

result: long one 80 strike call, short two 100 strike calls, long one 120 strike call.

You can apply the same logic to any (continuous) european payoff.

silly example: long put, strike 100.

  1. start at zero. gradient of payoff is -1. short 1 zero strike call.
  2. move to next change of gradient: 100. payoff is now flat, so we must buy a 100 strike call to flatten out the gradient of our call option portfolio.

result: short 1 zero strike call, long a 100 strike call. note that a zero strike call is just the same as holding the stock - we're saying that a put is equivalent to a call plus a short position in the stock -> i.e. put call parity.

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