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

Under Black-Scholes:

Code: Select all

```
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.