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There's a question in an interview book saying "when can hedging an options position make you take on more risk?"

The answer provided is that "Hedging can increase your risk if you are forced to both buy short-dated options and hedge them." And it gives an example that you short the stock to hedge, and the stock price rises up to strike so the option expires worthless, then you lose on both the options and the short stock position. Therefore you are worse off than if you had not hedged.

What I don't quite understand is that, in that example, if the stock price goes down, I would gain on my short stock position, why didn't it being taken into account? Also, I hope to have an analytical formula to see the "risk" more clearly.

Could anyone help me with this one?

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In this example, what the interviewer mean is that in this case, you increased the range of possible PnL values and you added a few more extreme values on the distribution of PnL. It is important, as a marketmaker, that you have an idea about what is your potential maximum loss and in this case, your hedge increase the poetntial loss in the tail event. – BlueTrin Nov 23 '12 at 14:03

E.g. on Monday you get forced to buy some Friday expiry OTM puts, say 95% strike S&P weeklies. Of course, you go and buy some delta against them to "hedge" yourself. Next thing you know, the the market tanks. Unfortunately, by Friday it's only down 3.5%, so it's does not fall far enough to reach the strike. So, on Friday expiration, you are out your premium and down money on your delta.

Overall, it's a pretty typical "painful moment" for a market maker. Usually, in this case you are better off selling some ATMish gamma and leaving the tinys to decay in peace. Should the market really take a dive, you got some lottery tickets.

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Could you kindly explain how to sell ATMish gamma? Thanks~ – Will Best Nov 23 '12 at 3:59
Sell a short dated option, in this case more or less matching the amount of theta and delta hedge it. – Strange Nov 23 '12 at 6:51

You are absolutely right, I would say that how the interview question was posed and the example given is very misleading, if not outright incorrect. Here is why:

Hedging does not increase your risk in this particular example: You take on delta exposure by buying the short dated option outright. Thus buying/selling underlying (put/call) in any case will reduce your delta exposure, hence risk of changes in the underlying, given you hedge the right amount and at the right timing (this i venture is impossible to generalize as it applies differently to each case). Now, you are long gamma but being long gamma does not guarantee at all that you end up better off not hedging initially. If your boss instructs you to be at all times almost perfectly delta hedged (most French bosses are anal about this, probably because they are horrible delta traders) then you hedge, period. It reduces your delta exposure, hence risk in moves in the underlying. It is complete nonsense to start arguing in retrospect that no hedge may have resulted in a better payoff because the underlying followed a price path not anticipated earlier.

I concur with Strange that there are often better ways to hedge than always going through the underlying but I disagree with him that it poses a "painful moment" to market markers. Market makers who are dependent on the market moving in specific ways are probably very bad volatility traders. Your job as market maker is to earn money from the bid/offer spread and to reduce your risk exposure to lower moment greeks, given it is feasible and cost-efficient. The other times you, as market maker, attempt to benefit from what you perceive as mispricings in the option valuation. Thus, hedging the long options position with the underlying reduces your risk, period. There are obviously exceptions to this, for example, when the underlying is so extremely illiquid that it would be prohibitive to hedge/re-hedge frequently. But it has to really be analyzed in context. But if the interview book looked for a straight forward answer which applies to most cases then hedging reduces your risk, simple as that.

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In theory, of course, a market maker is supposed to run a nice, balanced book and "earn money from the bid/offer spread" (good luck there, most trades print at mid). In real life you often get stuck in a lopsided position against your will. E.g. a large insurance company comes to roll a hedge or you "provided liquidity to an important client". "Stripping" decaying wing options is a pretty standard thing to do. As an alternative, you can manage it at a different implied volatility and many other tricks. No offence, but it does not sound like you have ever managed a large sell-side options book. – Strange Nov 24 '12 at 6:56
@Strange, you are correct, I have not managed a large sell-side options book, I only managed large prop options books within banks and hedge funds and that as desk head. You missed my point entirely, whatever a MM ends up with in the book, delta exposure can be in almost all cases cut to close to zero instantaneously (aside the fact that many options in the OTC already trade with the delta hedge attached). It reduces your risk (delta here) and makes my point that hedging delta in this instance always reduces risk...and – Matt Wolf Nov 24 '12 at 12:07
...if you disagree with this specific point then pls make your case but address the point rather than question others' pedigree – Matt Wolf Nov 24 '12 at 12:07
I'm with @Strange on this one. In the above example of the 95 put expiring friday, if you hedge with underlying to be delta flat, you are implicitly long delta, no doubt about it. The option will bleed delta rapidly and you'll be left holding hard stock. Moreover, the premium of wings is very sticky - if the market ticks up or down, the premium ain't gonna move with the delta, meaning your PL is solely being driven by your cash position. – user3316 Nov 26 '12 at 2:20
Market makers who are dependent on the market moving in specific ways are probably very bad volatility traders. Exactly - by hedging your delta with the cash, you're basically taking on a naked cash position, and thus are more dependent on the market moving in a specific fashion. IMO, short dated wings are best left unhedged, or leaning your deltas strongly towards them. – user3316 Nov 26 '12 at 2:22

My back testing has shown that in the case of really big market moves, dynamic delta hedging of short positions can increase risk in comparison to no hedging, and in fact cause large losses.

For confirmation, see http://sisla06.samsi.info/fmse/lp/Presentations/JumpHedging.pdf which concludes just that.

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