Theta is NOT the P/L of an option

[Amahrix]

You're correct, and your point?

What’s the source of the P/L then if it’s not theta? Another individual wrote that it simply is price -- in hindsight, my head must've been too deep in the sand to realize this simple fact.

But I do have a follow up concern if you can assist me..

Taleb continues to write "One way to look at the representation of theta is that it goes hand in hand with gamma. The alpha (gamma per theta ratio) will be the same regardless of the number of days to expiration, and so on. Selling very short-term options, a sport that is periodically practiced by newcomers, would be an attractive breadwinner except that the risks are exactly the same as selling longer options, unless the trader sells an expensive strike."

I'm having trouble understanding the hand-in-handness of theta and gamma. I see I can divide theta/gamma and get the ratio. What does this ratio signify?

Apparently the ratio is constant whether it is short term options, or long term options, unless short term option has an expensive strike.. but I am having trouble reconciling this because I always thought the P/L of shorter term options are higher than longer term options for the seller. Taleb is telling me the risks are the same, implying newcomers make the mistake of selling short term options and subsequently give up 'time diversification' while hunting for higher P/L.

And does capturing the alpha on longer termed options take longer than shorter term?

 

[Magic]

You’re selling vol regardless of expiration. Strip away vega changes and path dependency from imperfect hedging and you profit if you realize less vol than you sold short and vice versa. Theta is inverse of gamma; it’s like gamma in dollar terms. Profits will come faster via realizing less vol than you sold closer to expiration since the gamma exposure is higher. The losses will come faster as well if spot goes against your strike closer to expiration.

 

[drm7]

P/L is the difference between implied and realized vol at expiration (net of hedging costs).

 

[Amahrix]

You’re selling vol regardless of expiration. Strip away vega changes and path dependency from imperfect hedging and you profit if you realize less vol than you sold short and vice versa. Theta is inverse of gamma; it’s like gamma in dollar terms. Profits will come faster via realizing less vol than you sold closer to expiration since the gamma exposure is higher. The losses will come faster as well if spot goes against your strike closer to expiration.

Thank you for your reply.

Can you elaborate or say differently this part "Theta is inverse of gamma; it’s like gamma in dollar terms. Profits will come faster via realizing less vol than you sold closer to expiration since the gamma exposure is higher."

 

[Magic]

Thank you for your reply.

Can you elaborate or say differently this part "Theta is inverse of gamma; it’s like gamma in dollar terms. Profits will come faster via realizing less vol than you sold closer to expiration since the gamma exposure is higher."

Theta is inverse of gamma is pretty straight forward. If you’re positive theta that means you’re short gamma. If you’re positive more theta that means you’re short more gamma. That’s the whole point, probably why Taleb says options are the same regardless of expiration. Theta is the fair value compensation for the amount of gamma you’re on the hook for, or vice versa if you’re long options.

The edge comes in selling options that are priced at higher vol than what is realized, or buying lower implied than what is realized. In the former case you’ll collect more theta than you pay out in gamma, etc.

As you get closer to expiration, the vol you realize has a comparatively greater impact on the moneyness of the option, so gamma losses (gains) if you’re short (long) are more violent. You lose (gain) more for every consecutive dollar price goes against your position if you’re short (long) than you would with more time on the option since the terminal moneyness is affected to a greater degree. Hence compensation i.e theta for that reality rises (falls) as we approach expiration when short (long).

 

[newwurldmn]

Can you elaborate for me? And make me aware of theta and gamma relationship.

You want to be aware of theta and gamma relationship? Read a proper options book, not some pretentious tripe from Taleb.

 

[Wheezooo]

Thank you for your reply.

Can you elaborate or say differently this part "Theta is inverse of gamma; it’s like gamma in dollar terms. Profits will come faster via realizing less vol than you sold closer to expiration since the gamma exposure is higher."

It is really quite simple. Unlike most parameters that can go higher or lower, time only goes one way. All other parameters remaining the same. As a result of time, tomorrow an option will be worth less than today - The owner of the option loses over time.

In return. A long option position that is hedged benefits from any price movement that occurs where whether price goes up or down you make money (if this doesn’t make sense, then the problem is not you understanding the relationship between the two but rather you not understanding why gamma is beneficial to the options owner).

So basically, every day an options owner loses on decay, and in return will benefit from price movement and capturing the p+l created by gamma. Where that point of indifference is. Where the market maker sees no benefit to owning gamma or attempting to capture theta is approximately the vol where the option should be priced.

What Taleb is saying about the forward curve warrants about 30 caveats, but basically nowhere along the curve should there be a gamma/theta advantage if the product in question is perfectly identical regardless of time. So many other things wrong with it or limited in its scope that I wouldn't know where to begin typing.

 

[raf_bcn]

Sometimes is good to review what one thinks he knows. I am doing this in relation to option prices.
Will read Taleb.

Is the value of an option the cost of hedging that position ?

The probabilities that are embedded in the option prices are risk neutral probabilities.
The risk neutral valuation of an option presumes that the option seller will delta hedge his position.
So the risk neutral distribution is the one we get after we hedge away the directional risk.

Strip away vega changes and path dependency from imperfect hedging and you profit if you realize less vol than you sold short and vice versa.

How is the model expecting that we are going to hedge ?
What is imperfect hedging ?

Given a duration, iv, and moneyness , the model sets a value for that option. And also sets a delta for it, which determines how we are going to hedge with the underlying.
Does the model consider that the delta will change ? How to value those changes ? Is the model considering that the underlying will move by random walk.

E.g. XYZ spot at 300 , iv 17 % , i. rate 2.5 % , no dividends .
1 year 340 call value is 9.20 , delta 0.25

The iv is implying that the underlying will move +/- 40

If we sell the option and delta hedge buying 25 XYZ shares , and mantain that for one year, the cost of hedging will be 25 * 300 * 0.25 = 187.5
So there will be a profit of 920 - 187.5 . But that is considering that the underlying won't move.
If we consider , in the worst case, that we will need to hedge with delta 1, 100 shares per short option, the cost of hedging will be 100 * 300 * 0.25 = 750.
We also have profit in this case.

How is the model calculating the cost of hedging.

Thanks.

 

[Magic]

As Wheezoo said the time value of an option is the benefit the user will receive from realizing the amount of volatility that is implied. Volatility is profit for the gamma holder because of the options convexity.

I don't get too far into the weeds of theory and academics, but I believe the models work off the assumption of continuous hedging, which is impossible in IRL and even if implemented would have astronomical costs. Continuous hedging is the equivalent of perfectly replicating the option with shares.

Real hedging is a trade-off between transaction costs & tying up capital vs. accepting PnL variance. You could short 20% IV and the market could realize 12% but you could still have negative PnL if the market went perfectly in a single direction to realize that vol, if you were completely un-hedged.

Likewise the perma-premium sellers can get confused into thinking they have more expectancy than they actually do, with selling wide un-hedged strangles and stuff, when really they just caught a good clip of terminal distributions where vol was realized but they just didn't have to pay for it yet.

 

[Amahrix]

You want to be aware of theta and gamma relationship? Read a proper options book, not some pretentious tripe from Taleb.

Thanks for your absolutely useless comment.