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Why is the delta of a short call option negative? In Black-Scholes-Merton equation the delta of a call option is always a probability function therefore it does not imply such a consequence. How do I interpret this fact from a mathematical/quantitative point of view?

Edit: My bad. I thought a long/short call refers to a call with long/short maturity time. Please disregard this question.

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closed as off-topic by Quantuple, Richard, LocalVolatility, Luigi Ballabio, vanguard2k Feb 7 '17 at 13:01

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  • $\begingroup$ For a given S,T,E,r and $\sigma^2$ the Delta is a real number, not a "probability function" AFAIK. $\endgroup$ – Alex C Jan 20 '17 at 2:09
  • $\begingroup$ @AlexC What I meant is that it is always positive and always between 0 and 1. $\endgroup$ – Ethan Jan 20 '17 at 2:14
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    $\begingroup$ Well, when you are long the call it is between 0 and 1, when you are short the call it takes the opposite sign, and therefore is between -1 and 0. That is the convention regarding short positions. $\endgroup$ – Alex C Jan 20 '17 at 2:29
  • $\begingroup$ Is it clear why the delta of a long option is positive and (!) what this means? If yes, then it should be clear why the delta of the short position is negative ... $\endgroup$ – Richard Jan 20 '17 at 9:12
  • $\begingroup$ I'm voting to close this question because the OP stated it comes from a misunderstanding. $\endgroup$ – Luigi Ballabio Jan 25 '17 at 9:15
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The delta of any position, $\Delta_P$, is the number of units you hold, $N$, multiplied by the delta of each unit, $\Delta$

$$ \Delta_P = N\times \Delta $$

You are correct that for a call option you have $0\leq \Delta \leq 1$. If you are short a call option, then you have a negative position (that's what being short means) so $N<0$ and therefore $\Delta_P < 0$.

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    $\begingroup$ This is spot on. Just to be very pedantic though, under BS assumptions, we should have $0 \leq \Delta \leq e^{-q\tau}$ with $q$ the continuously compounded div yield and $\tau$ the time to maturity (because $\Delta = e^{-q\tau}\Phi(d_1)$) $\endgroup$ – Quantuple Jan 20 '17 at 13:07
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    $\begingroup$ Yes, thanks for the refinement! We might also note that this applies to non-equity instruments too, assuming you interpret $q$ appropriately (the foreign interest rate for a currency option, storage rate for a commodity option or yield for a treasury option). $\endgroup$ – Chris Taylor Jan 20 '17 at 14:16
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A couple of different ways to look at it

  • Delta is the change in price/value of the option per unit change in price of the underlying
    • For a long call option - any +ve change in the price of underlying, regardless of ATM/OTM/ITM, can only increase the value of the option. Hence, the delta of a long call option is always positive
    • A short call position is the mirror/counterparty of the long call position. So if the long increases in value - the short can only decrease in value. Hence the short call position always has a negative delta
  • Another way to look at this would be in terms of replicating a stock with options
    • Long a Put, and Short a call replicates a Short position in the Stock. Now a short position in the stock has a delta of -1. Long Put has a delta between -1 and 0. Hence the short call needs to have a negative delta
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