How can I calculate how much an option would be worth after X days if the underlying stock changed by +/- $Y?

Question

I don't really know how to anticipate the changes in an option's price.

If I have an call set to expire the following Friday at $10 above the current share price, I know in the absence of other changes it loses value with each day that passes getting closer to its expiration. (Talking about American options.) But it gains value as the share price gets closer to the strike price.

So, if two days pass but the stock goes up $3, would the option have gained value or lost value? Is there any way to calculate this?

Answer 1

Delta and theta are useful for estimating how much an option's price will change with for a one point change in the price in the underlying in one day. But they are impractical for determining a multi-point change over a longer period of time.

As an example, XYZ is $50 and the ATM trading for $2.10 has a delta of .52 . That means that if everything else is unchanged, at $51 the call should be worth $2.62 -ish. At $55 it will be worth $4.70 ($2.10 + 5 x $0.52). But in reality, at $55, the call will be worth $5.60 . Why the difference? Because delta is non linear and increases as the underlying increases (at $55, the delta is close to .85) and therefore, you can't multiply the delta of current price times a multi-point change in price and come up with anything resembling accuracy.

Making things even more complicated, implied volatility affects delta and therefore price. And none of this has even addressed theta decay.

The only way to adequately anticipate the change in an option's price from the passage of time and/or underlying price change and/or change in implied volatility is to use an option pricing calculator of which many are available online. Vary whatever pricing parameters amuse you and the numbers will be much more accurate.

If you want to be even more accurate, use the current bid of a long option to project the sale price at the later date (you STC at the bid) or conversely, use the current ask to project the purchase price for a short (BTC). This is only relevant where the B/A of the option is wide.

Answer 2

You'd need to know the delta and the theta of the option. You can either calculate them yourself using a model like Black-Scholes (assuming you have a market price and can imply a volatility, and know the other factors that go into the model) or, you can see if your broker quotes "greeks" as well (mine does).

The delta is the sensitivity (rate of change in value) to the underlying stock price, and the theta is the sensitivity to time passing (usually expressed in $/day). So if your option has a delta of .5 and a theta of -.04, when one day passes and the underlying stock goes up $3, the option will gain roughly $1.50 due to the underlying stock price and lose $0.04 due to time passing.