Least/Most Favorable Paths when dynamically hedging a short straddle

[tooriginal]

Assuming Realized vol = Implied vol, what are the best and worst paths?

Best:
Stock moves the same direction every day at less than IV until gamma is negligible at which point it moves in the same direction but by a very large amount in order to make up the realized vol. It's important that the stock moves enough in the beginning such that the back loaded realized vol happens far from the strike.
Ex. IV = 16%, 1 year to expiry, stock returns +.1%,+.1%, ...., +.1% (251 times) and then +1267% (approximately exp(sqrt(252)*0.16)) on the last day. In this case you keep 43% of the straddle premium as profits:

Worst:
Same IV, DTE. Stock goes no where for the first 251 days and then moves 1267% up. Then your losses are 89 times the straddle premium you sold. It's neat to see that you lose much more even though the final value of the straddle is less than in the first case.

Doubtless there are even more contrived paths with more extreme pnl. Can you name any?

 

[BlueWaterSailor]

Doubtless there are even more contrived paths with more extreme pnl. Can you name any?

Best: stock pins at the strike and stays there; 100% of premium collected.
Worst: stock goes to infinity, and your short call with it.

Unless you're postulating a scenario where the trader is watching it every moment with his finger on the "get out" button, I don't understand why it would need to be anything as complex as you're describing.

 

[tooriginal]

Best: stock pins at the strike and stays there; 100% of premium collected.
Worst: stock goes to infinity, and your short call with it.

Unless you're postulating a scenario where the trader is watching it every moment with his finger on the "get out" button, I don't understand why it would need to be anything as complex as you're describing.

You are not dynamically hedging. With a static position comes a static terminal PnL diagram but when hedging the final PnL is path dependent.

 

[BlueWaterSailor]

You are not dynamically hedging. With a static position comes a static terminal PnL diagram but when hedging the final PnL is path dependent.

Ah - you didn't mention that. Gotta establish context and parameters (theoretical exercise, dynamic hedging/costs ~= 0, etc.) if you want meaningful answers.

 

[taowave]

I like your example,and hope I never live thru it...So much for continuously hedging


I have a more fun question.....

You sell a 30% IV straddle,believe HV will be 20%,do you make more money hedging at the implied or forcast vol???

Assuming Realized vol = Implied vol, what are the best and worst paths?

Best:
Stock moves the same direction every day at less than IV until gamma is negligible at which point it moves in the same direction but by a very large amount in order to make up the realized vol. It's important that the stock moves enough in the beginning such that the back loaded realized vol happens far from the strike.
Ex. IV = 16%, 1 year to expiry, stock returns +.1%,+.1%, ...., +.1% (251 times) and then +1267% (approximately exp(sqrt(252)*0.16)) on the last day. In this case you keep 43% of the straddle premium as profits:
View attachment 296868

Worst:
Same IV, DTE. Stock goes no where for the first 251 days and then moves 1267% up. Then your losses are 89 times the straddle premium you sold. It's neat to see that you lose much more even though the final value of the straddle is less than in the first case.
View attachment 296869


Doubtless there are even more contrived paths with more extreme pnl. Can you name any?

 

[tooriginal]

I'm working on the mathematical proof. Had to review ito's lemma and such. But for now, the expected value is independent of the hedging vol. Hedging at 0 vol (step function delta) is the same as at infinite vol (no hedging at all as gamma is zero everywhere). However standard deviation of the pnl in one dt time step is proportional to the difference in the actual vs implied deltas so in the interest of maximizing your sharpe ratio you need to hedge at the actual vol. For martingales, E[PnL_0] = E[PnL_t] for all t so this argument should hold when integrated in time.

pnl after dt time (dX = Weiner):

https://www.cqfinstitute.org/sites/...ned-About-Volatility-Over-The-Years-Notes.pdf

Simulation:

 

[Kevin Schmit]

...standard deviation of the pnl in one dt time step is proportional to the difference in the actual vs implied deltas so in the interest of maximizing your sharpe ratio you need to hedge at the actual vol.

You have it exactly backwards, hedge at IV for minimal daily marked-to-market pnl sd. As explained in the slide-show you linked to. Did you actually read it?

Full paper version of the slideshow attached.

Also, there is no guarantee that your forecast vol is going to match the actual (realized) vol.

 

[taowave]

Hi Kevin,going to go thru the papers,but am curious to your opinion.Assuming your forecast vol does match actual, would you make more heding with forcast than hedging using implied vol?? Are you " paid more" for accurately predicting realised vol and hedging at forecast vol as opposed to implied.

I'm assuming continuous/discrete hedging and do not care about the discrepancy in deltas and minimising daily P and L fluctuations..

As an example,if I sell a 30 day 90 percent spot put at 40 vol,but predict 25 percent realized, do I theoretically make the most money continuously hedging at 25 percent assuming the underlying realized vol is 25 percent??

Not sure if you were trading way back when,but guys used to get paid selling 40 vol,hedging and MARKING it at 25...
We know how that turned out

You have it exactly backwards, hedge at IV for minimal daily marked-to-market pnl sd. As explained in the slide-show you linked to. Did you actually read it?

Full paper version of the slideshow attached.

Also, there is no guarantee that your forecast vol is going to match the actual (realized) vol.

 

[tooriginal]

You have it exactly backwards, hedge at IV for minimal daily marked-to-market pnl sd. As explained in the slide-show you linked to. Did you actually read it?

Full paper version of the slideshow attached.

Also, there is no guarantee that your forecast vol is going to match the actual (realized) vol.

I think there must be a miscommunication. My figure exactly matches his figure 14 on the article you gave and both conclude that hedging at the actual vol minimizes the the standard deviation of the final pnl. However to minimize pnl fluctuations after dt one step, you are correct the implied will accomplish that.

 

[Kevin Schmit]

Assuming your forecast vol does match actual, would you make more hedging with forecast vol... Are you " paid more" for accurately predicting realized

In a risk adjusted sense, yes, your expectated terminal profit over the stdev of that profit will be higher. Your expected profit, before transaction costs of the hedging, is price of the option (or the spread in this case) minus the price of the option (spread) priced at your forecast vol.

The thing is, that is the same expectation as not hedging at all. If you sell an option at price a, and it is fairly priced at b, then your expectation is a minus b. So unless you can delta hedge at mid and pay no comms/fees, your expectation is actually higher with no hedge (but your sd is worse, much worse). Also, note the expected profit line in the plots in the posts above is essentially flat, your exact hedging scheme affects sd, but not terminal profit much (ignoring hedge transaction costs)

As an example,if I sell a 30 day 90 percent spot put at 40 vol,but predict 25 percent realized, do I theoretically make the most money continuously hedging at 25 percent assuming the underlying realized vol is 25 percent??

No, but you do have the lowest terminal standard deviation (highest sharpe). You'll also change the shape of the terminal distribution -- the short strangle no-hedge distro is negative skewed and fat tailed negative also. The hedge will make the terminal distro much closer to normal. Sharpe doesn't account for skew and kurtosis, but Kelly does. A four moment Kelly Fraction function is below, if anyone is interested:

kFf4M <- function(mu,sigma,skew,kurt)
{ mu <- max(mu,0)
  rawSigma <- sqrt(sigma^2 + mu^2)
  rawSkew <- skew * sigma^3
  rawSkew <- rawSkew + 3 * mu * rawSigma^2 - 2 * mu^3
  rawKurt <- kurt * sigma^4
  rawKurt <- rawKurt + 4 * mu * rawSkew - 6 * rawSigma^2 * mu^2 + 3 * mu^4
  f = mu * (rawSigma^4 - mu * rawSkew) /
  (rawSigma^6 + rawKurt * mu^2 - 2 * mu * rawSkew * rawSigma^2)
  return(f)
}