Can an option continue to gain value even if the underlying stock does not?

Question

Example. It's 7/1/2008. XYZ stock is selling at $49. I buy a Dec 08 XYZ call strike price $50 for $2. On 7/2 XYZ stock jumps to $51. My option goes to $3. From 7/2 to December, XYZ stock stays at $51. Will my option stay at $3, or increase in value?

Answer 1

The option can definitely change value even if the stock price stays the same.  

There are different components to a call option (as well as a put). Call = Intrinsic Stock Component + Volatility Component + Interest Rate Component.  

The interest rate component is usually a pretty small factor.

Stock can stay the same but volatility (the market's expectations for the stocks future riskiness - defined as standard deviation) can move, and therefore your option value will change.  If you ever heard the term "vega" - that's an options sensitivity to volatility changes.  

So in your example you need more info to answer.  Specifically, has volatility changed?

Let's say it has not changed.  Then your option would decrease a little from $3 with stock still at $51.  This is because the volatility component of the option decays a little bit every day.  Right before expiration, with stock still at $51 and strike = 50, the value of the call would be only the intrinsic value, $1.  This decay towards intrinsic value is called "theta".

One note - This guy below me is incorrect with the idea that volatility movement rarely changes the value of options.  Take one look at the recent performance of the VIX, S&P 500 volatility, and you can see how volatile volatility itself has been over the past 6 months.  Additionally, at the current high levels of volatility, the time value of options will be higher and consequently their sensitivity to changes in volatility will be higher.  So basically movements in volatility are a very real risk in the currrent market environment.

Answer 2

In general, your option will decrease in value as time passes.  The speed at which the value goes down increases as time passes.  In the above example, assuming nothing else changes, after the option goes to $3 to will slowly erode to $1 (the intrinsic value).

The value of the option can increase without changes in the underlying price.  This happens if the perceived riskiness of the underlying increases.  This expected/perceived riskiness is called implied volatility.  This happens rarely.