Estimate a future option price given greeks and a 1$ move in underlying

Question

Let us say underlying is at 50$, and the 50 call is at 0.50.

delta=0.5, gamma =0.5, theta = .02, vega = .10.
I apologize if these numbers are not realistic, I am very poor at Greeks.

My question is: If tomorrow underlying moves 1$ to 51$. Is it possible to do a rough estimate of the option price based on the Greeks? The option will have 1$ intrinsic value.

And extrinsic value will be .50 - .02(theta decay) = .48? So it will cost 1.48?

Another way to think is: Price = current price (.50) + delta effect (0.50) - theta effect (.02) = 1.48 Will the other Greeks come into play too? How?

Answer 1

It's not that straightforward, even though your gamma will change your delta on the fly, you likely won't see the full $.48 after such a small move.

If the vega drops due to lack of volatility while the stock is moving up, those few percentage points up might help your delta (2% gain $50 to $51 in your example) but will be partially negated by volatility going down.

I mean, don't be surprised to see it at closer to $1.33 or something. The market is out to make money, not to make you money.

Answer 2

The delta-gamma approximation would be:

(IOP) + (DELTA)(UPC) + (.50)(GAMMA)*(UPC)^2

where:

IOP = Initial Option Price

UPC = Underlying Price Change

So the expected new price would be:

(.50) + (.50)(51-50) + (.50)(.50)(51-50)(51-50) which equals $1.25

Note that this is just an approximation. With the stats that you offered, it's going to be close to expiration. There's an interplay between higher implied volatility and days until expiration (DTE).

For the stats you provided, you could have a higher IV and lower DTE or a lower IV and a higher DTE. In each case, the $1 overnight change will alter the option price by different amounts.