Hi, I read that the delta of a call option is between 0 and 1.

But in situations, where the price of a call option decreases even as the price of the underlying stock increases it tends to imply that the delta of such an option is negative since delta is the sensitivity of the option price to the stock price.

Where am I going wrong in my reasoning? OR do we calculate delta assuming all other variables that can affect the option price to be constant? OR is it that the delta is possitive but other variables causing the stock price to decrease are stronger than delta?

UPDATE:

I was reading that the time value of an option cannot be negative. But I also found that it can be negative in the case of a deep ITM European Put option on a non-dividend paying stock? So, is the time value of such an option negative? Also, if it is negative can it be a cause for theta of such an option to be positive?

Thanks.

The other greeks have a larger impact.

Think of the price of a call as a function of many variables, which are typically stock price, vol, rate, etc so C = f(S,v,r). The greeks come from the Taylor expansion of the call price and you get something like C ~= C_0 + dSdel(f)/del(S) + dS^2/2del^2(f)/del(S) + ...

or more simply C ~= C_0 + dSdelta + dS^2/2gamma + dv*vega + ...

As the other post mentioned, you're not considering enough variables in the description you set up. You are attributing all of the change in call price to the delta component, ignoring the other terms, when in fact these can be primary drivers under certain conditions.

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It's just vol moving on you, most likely. It happens...

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Okay. Thanks for clearing it out. Also, I read that the delta of a call can is always between 0 and 1. Can someone explain why?

Yes, mathematically it can be shown that the first derivative is between 1 and 0. But I am searching for a general explanation to understand the why behind it.

"The markets are always changing , and they are always the same."

Delta can be actually greater than 1, under certain specific conditions, but that's a little esoteric and probably not particularly interesting.

%age delta lives between 0 and 1 by construction. Broadly speaking, in most cases, the value of an ITM option cannot move more than the underlying and the value of an OTM option cannot move less than 0. That's just a consequence of how the payoff is determined. If you think about the delta as an instantaneous correlation between the change in the value of the underlying and the change in the value of the option, you should have a sense of how it intuitively belongs between 0 and 1.

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Best Response

delta is the rate of change of the price of the option, as a % of the change in the price of the underlying (all else, such as implied vol and time decay, being equal). Since the option can be thought of as a proxy for the underlying, but with an expiration date and a price barrier where the option will expire worthless, it makes sense that delta should never exceed 1 (assumes the option will stay in the money), and could be as low as 0 (if the market believes the option will expire worthless).

However, as time marches on, theta (time decay) pulls the price of the option towards zero (theta bleed). Theta (time decay) increases as the option approaches expiration.

The other big driver of the option price is implied vol (vega). As the underlying's expected vol increases, vega increases (and vice versa). So depending on what is happening with the underlying, different factors will impact the price of the option...and these things change over time.

For example, when there is a surprise expected in the market (earnings for a company, or an important economic release that affects treasuries or FX rates) this will often make options dealers increase vega (implied volatility) and this raises option prices temporarily....that vega increase will then dissipate after the surprise period is over.

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Okay, so if the delta of an option is greater than 1 then it's possible at some point of time the price of the option will end up being greater than the underlying. But since that will violate the upper bound of the price of a call option so delta can never be > 1.

On the other hand if an options starts going OTM the delta keep keep decreasing until it reaches 0 but does not decrease after that since the option is already worthless.

Is that correct?

"The markets are always changing , and they are always the same."

JoyfulMonkey:

Okay, so if the delta of an option is greater than 1 then it's possible at some point of time the price of the option will end up being greater than the underlying. But since that will violate the upper bound of the price of a call option so delta can never be > 1.

On the other hand if an options starts going OTM the delta keep keep decreasing until it reaches 0 but does not decrease after that since the option is already worthless.

Is that correct?

This isn't entirely correct...

For call delta to be greater than 1, for a 1 unit increase in the value of the underlying, the value of the option would have to change by more than 1 unit, all else being equal. Apart from a few interesting, but somewhat esoteric, cases (there's a common interview question about this), this would contradict the definition of the call option contract.

BTW and in relation to an earlier post, another common interview question is "why is the delta of an ATM option not 0.5?"

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