When are you suggesting he buys the call spread?? At inception? Thats entirely different position When the stock moves x percent??? With your hedging,he will be in a loose 1x2,i.e + 1x-1x-1 If he is delta hedging,hes definetly over hedging at short strike.... And why should one magically "overhedge", or delta hedge at short strike as opposed to taking a loss at strike or having a stop loss in place? The strategy stinks,and the risk management is worse
I have many questions but mostly I'd like to confirm my understanding of the article [p.19 (labeled 17) of the PDF here]. So, in order to have you help me, I should probably first give my understanding of the article. Lol... @anon9812 writes an OTM call and when the underlying moves against him (and goes to the strike) he's going to cover by buying the underlying. This sounds silly and premature since he could just buy the calls back. Otherwise, he'd really want to Δ hedge and simply buy 1/2 of the shares. Right? In fact, it may have been wiser to Δ hedge from the start but, for whatever reason... He didn't so let's assume he wanted the risk at the time and now he wants "out". Again, I'm not sure why he wouldn't just buy the calls back. So maybe the first minor question is exactly what's the difference between Δ hedging at this point and simply closing the short call position? Ok, I'll try to answer myself... if you buy the calls back all the risks go to 0, you take a loss and move on, duh! However, after Δ hedging, other risks still remain and so you cling to some fantasy of changing winds and reaping a reward. I'd certainly like to hear more discussion on this though. Anyway, back to the article. It suggests that Δ-Γ hedging would be a better choice than Δ hedging alone. I guess this is because, ideally, we'd like to eliminate as many risks as possible and simply identify options that are badly priced in order to get a free lunch by pulling off a great arb. Personally I'm happy with an inexpensive lunch, but whatever. Thus, Δ-Γ hedging gets us closer and can be done by buying some shares, as well as buying some OTM calls further away. Something like a bear call vertical but with more math and shares thrown in for good measure. How many though? Well... we calculate our Δ and Γ values for each call (the one we sold and the one we're going to use as a hedge) and plug those values, along with the number of calls we sold, into the equations and then solve. Bada-bing bada-boom, some Gaussian elimination or LU-decomposition or maybe just plug in random guesses until it works, and we get the number of shares and calls we need to buy. An example is probably in order at this time. To keep it simple let's assume @anon9812 decided to Δ-Γ hedge from the start (although it could be done at any time by using the Δ & Γ values at that moment). Here are some real #'s I just picked off nasdaq.com for Cigna @ $254.88: Call #1 (CI 230428C00252500) Delta 0.68556 Gamma 0.06223 Call #2 (CI 230428C00262500) Delta 0.13746 Gamma 0.03369 @anon9812 decides to sell 5 contracts of #1 while simultaneously buying some CI stock and some #2s according to the following (refer to equations #8 & #9 from the article): Code: # note: this is pseudo-code, not real code η(s) == number of CI shares needed = unknown η(1) == number of type #1 calls sold = 500 [note, multiply contracts by 100] η(2) == number of type #2 calls needed = unknown Δ(s) == delta of shares = 1 [price of share moves in unison with itself] Δ(1) == delta of type #1 calls sold = 0.68556 Δ(2) == delta of type #2 calls needed = 0.13746 Γ(s) == gamma of the shares = 0 [rate of change in share price remains 1 at all prices] Γ(1) == gamma of type #1 calls sold = 0.06223 Γ(2) == gamma of type #2 calls needed = 0.03369 0 = η(s)*Δ(s) + η(1)*Δ(1) + η(2)*Δ(2) # equation 8 0 = η(s)*Γ(s) + η(1)*Γ(1) + η(2)*Γ(2) # equation 9 # after a few substitutions [note, we're *short* call contracts so use -1*η(1)] 0 = η(s) + -500*0.68556 + η(2)*0.13746 # equation 8 0 = -500*0.06223 + η(2)*0.03369 # equation 9 # which, if my calculator knows what it's doing means: η(s) = 216 η(2) = 924 In this case, as well as selling the first 5 calls, you'd buy 216 shares of CI and 9.24 contracts of the second call (4/28 exp w/ strike 262.5) thereby profiting more handsomely. Where the profit comes from I'm not exactly sure so maybe someone would like to elaborate on that too, theta decay or vol crush maybe? Potentially many places I guess. All told, is that fair? Am I way off base? Did I make any arithmetic mistakes? Given the type of hedging, wouldn't this scale rather well? It does seem to be more expensive up front but presumably the extra cost is going towards the extra protection, right? Finally, please feel free to nitpick this completely apart. I think the specific language matters a lot when it comes to getting a full understanding. Thanks for playing options with me.
I can't speak to your math but the idea of gamma hedging is to reduce your risk further. But since gamma is tied to theta and theta/gamma are tied to vega, you will have some basis risk even if these numbers are flat at current spot. As a retail trader this is impractical to profit from. Bank desks want to do this because they are receiving bid/offer and want to keep as much of it. A proper bank trader will spend the majority of his time figuring out how to keep his delta, gamma and vega (skew, term, etc) as flat as possible. He'll trade a billion dollars of notional to make $300,000. Delta hedging is obviously reducing the delta risk. But in theory an option is a basket of stock and bonds that is dynamically traded. So the point of delta hedging is to create a replicating option to trade against your current option. The expected cost of delta hedging = the premium you receive. At any given time, the cost of delta hedging is the theta of the option. The gamma refers to how much you have trade to maintain replicating option. The gamma and theta are linked. the OP is on a completely different level than you. If you are thinking as a graduate level options trader, he's thinking like a pre-schooler. He doesn't want to buy the call back because then he's admitting he was wrong. I wouldn't try to figure out what he's doing.
Lol, yeah.... I'm just trying to figure out what I'm doing; @anon9812 is on his own. Thanks for the reply
honestly because I didn’t want to go through it. I’ll take your word it works. It doesn’t really matter for a retail size trader as it’s not practical to trade third order options risks and when it might make sense it’s easier just to brute force the calculation using an option pricer and a solver.
Wait, an option "pricer" I'm familiar with... but an option "solver"? What exactly does that thing do... take the greeks for different options and tell you how to minimize them with different baskets of options? That sounds useful, and in a sense, what the article is describing. But I haven't seen a tool specifically for that... I've just been using sympy/numpy/emacs-calc/perl-pdl/etc. It would be nice, however, if I didn't need the domain expertise to translate the [symbolic] algebra myself though. Can you suggest an existing tool (either proprietary or open-source)? Just to get a sense of the functionality these things provide? Or maybe these things are very strictly proprietary and I should consider building something on my own. Ah... weapons of math destruction; aren't they great!?!?