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PurpleRain
Topic Author
Posts: 2
Joined: January 22nd, 2022, 9:11 pm

### How does an Options Market Maker (OMM) deal with an asymmetric inventory?

Hello all, for simplicity let this question pertain only to vanilla Options on American equities.

Let us use an example of a market maker quoting the ATM straddle.

Under Black-Scholes:

S = 100
K = 100
DTE = 3
IV: 20
r = 0
q = 0

No rates or dividends for simplicity. This ATM straddle is worth $0.72 with a theo IV of 20%. Let's say the MM turns his quotes on now and has a crystal ball and is sure the stock will realize no more than the annualized 20% vol in the next 3 days. This implies the stock will move no less and more than a total realized volatility of 2.18%. This number was obtained by sqrt(3/252) * 0.20 * 100. The MM quotes a bid @$0.65 (18% vol) and offer @ $0.80 (22% vol) Whenever a customer purchases a straddle or sells a straddle to him, he delta-hedges respectively to pocket the spread as he continuously delta-hedges over the next couple of days. Now let's say there is 1 day left till expiration. The MM is currently long 100 straddles @ 0.65 and short 300 straddles @ 0.80. His Vega is -797.8 but the stock's realized volatility insofar has stayed in line with the MM's expectation of 2.18% vol over the next 3 days. With 1 day left, realized vol has been 1.78% and stock is back at$100.

The stock in a couple seconds suddenly shoots up 4% to $104 and stays like that, far out of line of the MM's 2.18% vol expectation. He is still only long 100 straddles at an 18% vol and short 300 straddles which he sold at a vol of 22%. With a day left the stock has so far realized 5.78% volatility (1.78% + 4%). This is in line with a 53% theo vol for 3 days instead of his old theo vol of 18%. sqrt(3/252) * 0.53 * 100 = 5.78 The MM re-hedges his delta. He is now down -$20,900. He has made +$12,700 from the straddles he bought below theo (and delta-hedging), but has lost -$33,600 from the straddles he sold far below actual theo vol which he got wrong.

Now the questions I have are:
• 1. With the market still open and more customers coming in, what does he do next?
• 2. Does he come up with an implied vol input of say 53% now and decide to quote the ATM straddle with a bid @ 40% vol and an offer @ 70% vol, thereby creating a wide bid/ask spread to cushion himself from further high amounts of realized volatility?
• 3. What if his inventory remains asymmetric and no buys from him or sells to him, and he eats the huge loss?
• 4. What if people do end up buying his straddle quoted @ 70% vol alleviating his asymmetric inventory and the market stays flat, he pockets some profit off of selling the straddle above theo vol since the market stays flat, but not enough contracts have been to minimize his -$20k loss. He would still be down in PnL. I know this is a crude example that avoids things like skew, higher order Greeks, and different real world heuristics, but this question is for the sake of simplicity and knowledge. If someone would like to provide another answer with real world examples involving skew, other higher order Greeks, heuristics or technical information etc that would be great too. Marsden Posts: 140 Joined: August 20th, 2001, 5:42 pm Location: Maryland ### Re: How does an Options Market Maker (OMM) deal with an asymmetric inventory? With the caveat that I'm not a professional trader and I know as little as I can get away with about market makers, some clarifications, please. I think given your assumptions the Black-Scholes price for each of a put and a call is the$0.72 you note, so the Black-Scholes price of a straddle would be $1.44. What do you mean, exactly? You imply that the market maker is strictly trading straddles, but you also write that he is delta hedging. The straddles are pretty nearly delta hedged already, but not quite. How is he supposed to be delta hedging? You write that with one day left, "The MM is currently long 100 straddles @ 0.65 and short 300 straddles @ 0.80." Do you mean that he bought 100 straddles at$0.65 and sold 300 at $0.80? The Black-Scholes value of a straddle with one day left, and assuming implied volatility is still 20%, is$0.84, or continuing your apparent half valuing, $0.42, so -- ignoring whatever he did to delta hedge -- he's got$350 or $175 given your half valuing in cash from transactions, and is net short 200 straddles worth$168 or $84 half valued. If the stock rises instantaneously to$104, a call with one day left and still assuming 20% volatility is worth $4.00 and a put only a fraction of a cent, so a straddle is worth$4.00 and the market maker is net short $800 (or$400) on his straddle position, with only $168 (or$84) in cash to show for it. How do you figure the massive losses that you note? What did he do to "re-hedge his delta?"

Basically, given the information you have given and ignoring whatever other trades were made to delta hedge, the market maker was ignoring gamma and lost money because of it. He ought to have considered this an acceptable cost of how he was trading, and just eaten his loss and carried on; if he can't do that, he should change how he trades and maybe reconsider his chosen profession.

PurpleRain
Topic Author
Posts: 2
Joined: January 22nd, 2022, 9:11 pm

### Re: How does an Options Market Maker (OMM) deal with an asymmetric inventory?

With the caveat that I'm not a professional trader and I know as little as I can get away with about market makers, some clarifications, please.

I think given your assumptions the Black-Scholes price for each of a put and a call is the $0.72 you note, so the Black-Scholes price of a straddle would be$1.44. What do you mean, exactly?
You imply that the market maker is strictly trading straddles, but you also write that he is delta hedging. The straddles are pretty nearly delta hedged already, but not quite. How is he supposed to be delta hedging?
You write that with one day left, "The MM is currently long 100 straddles @ 0.65 and short 300 straddles @ 0.80." Do you mean that he bought 100 straddles at $0.65 and sold 300 at$0.80? The Black-Scholes value of a straddle with one day left, and assuming implied volatility is still 20%, is $0.84, or continuing your apparent half valuing,$0.42, so -- ignoring whatever he did to delta hedge -- he's got $350 or$175 given your half valuing in cash from transactions, and is net short 200 straddles worth $168 or$84 half valued.
If the stock rises instantaneously to $104, a call with one day left and still assuming 20% volatility is worth$4.00 and a put only a fraction of a cent, so a straddle is worth $4.00 and the market maker is net short$800 (or $400) on his straddle position, with only$168 (or \$84) in cash to show for it. How do you figure the massive losses that you note? What did he do to "re-hedge his delta?"

Basically, given the information you have given and ignoring whatever other trades were made to delta hedge, the market maker was ignoring gamma and lost money because of it. He ought to have considered this an acceptable cost of how he was trading, and just eaten his loss and carried on; if he can't do that, he should change how he trades and maybe reconsider his chosen profession.
Apologies, I did  not realize I was halving the straddle values. Disregard my above example. Let me use this simple example instead

Let's say the ATM is 100, and the 105 strike call has a theo implied vol of 10% I quote my bid @ 8% vol and my ask @ 12% vol. The call expires in a couple days, today is Monday and it expires on the Friday tenor. A lot of customer are buying from me & selling to me and I'm delta-hedging, the market isn't moving much.

It's Thursday and I now have an asymmetric inventory because more people bought the call from me rather than sold it to me. I end up being a net short 200 calls @ 12% implied vol. Now what happens when the market starts to move quite a bit and I jack up my theo vol estimate to say 15% and I quote a bid @ 13% now and an offer @ 17% vol on the call.

The market's realized vol does indeed move past the 12% implied vol that I quoted and sold the old calls at. And I'm short 200 of them @ 12% vol and have been delta hedging. I'm quoting the calls @ 13% vol-bid and 17% vol-offer. I'm hoping more customers will come in. I'm at a net loss right now because my delta-hedging costs (gamma) have far outpaced the compensation (theta) I've received.

My theo vols were off by X vol points and now even though I jacked up theo vols on the call (single strike), maybe for the rest of Thursday and Friday, no customers buy the call from me, or sell to me. Am I just stuck with my loss? How do I make money in this scenario? And what would I do?

Marsden
Posts: 140
Joined: August 20th, 2001, 5:42 pm
Location: Maryland

### Re: How does an Options Market Maker (OMM) deal with an asymmetric inventory?

I guess I think of market making differently from how you do.

As I see it, market making is essentially an administrative role: the market maker is helping people make trades at reasonable prices while trying to take no net position himself. If he is taking a position, then he has stepped out of the role of market maker and into becoming a speculator.

Toward the end of being a market maker, the ideal situation is that he does not quote prices at all: other people provide bids and asks, and if a market order comes in, he decides either to assign it to the highest bid or the lowest ask, or to take the trade himself somewhere in between.

And what should generally determine his position is whether it will take him closer to having no exposure: he might, near the beginning of trading on a contract, decided to take a trade, making him either one long (or one short) on the contract. And ideally, that's as far out of balance as he ever gets on the contract: if an identical market order comes in, he lets it go to whoever has the best ask (or bid); if an opposite market order comes in, he takes it -- hopefully making a slight profit relative to his earlier trade -- and goes back to a zero position.

And he might follow similar reasoning if a bid or ask comes in between standing offers: if it takes him out of a net position at a profit, he generally takes it; if if moves him off of a zero position, he might take it if he thinks he'll be able to back out of the position profitably.

And he likely will also try to balance things between contracts, going 1 short on a 105 call when he's already 1 long on a 100 call, for example.

But he probably doesn't want to go even 1 long on a 105 call when he's already 1 long on the 100.

So I consider what you describe as speculation rather than market making, and when you speculate and lose, you're stuck with your loss.

Again, I'm not a professional trader and I try to understand all of the mechanics of markets only to the minimum level necessary to keep me out of trouble. But I hope that helps a little.