Hi Adam,
Couple of important points here...
1) Forget about options, let's first chat about the effect of dividends on share prices. Theoretically, a share value is the PV of all the future dividends. So, if a share costs £10, it means that the market perceives that all the future cashflows (let's say that this is £1 every year forever) are worth £10 today. If the company decides to increase the £1 to £1.20, say, it can't just magic this extra 20 pence from nowhere, it simply means that it is going to run out of money to give you - the share price will decrease accordingly in the future. The current value of the share is £10 and this can't change just because the company decides to increase the dividend and give you more money now. The exception to this is what we would call the information content of dividend policy. If a company suddenly announces a higher dividend, people might think "oh, they must have done well, that's good" and the share price might go up. However, some people might think "oh no, why are they giving out more money now rather than holding it back to grow the business etc. they must have run out of ideas" and the share price might go down.
Notice how the EMH is lurking in the background to all of this - you should think about this. The announcement of a higher-than-usual dividend is information that has just gone from insider to public. So, if the share price changes at all, what does this mean?
Most of the time, a change in dividend policy might well drive the share price down temporarily because you'd end up shifting your shareholder base. Investors who like high dividends choose companies that pay high dividends. Investors who prefer low dividends choose companies that pay low dividends (they would rather the company keeps the cash and uses it to grow the business, making more money for them later). So, if you change your dividend rate, it just means all the investors who don't like this will sell and all the investors that do like it will probably then buy - this shift could take time and you might see a temporary dip in share price as the market "gets annoyed with all your messing around", say.
But, theoretically, a change in dividend policy does not affect the share price. The £10 is what it's worth, doesn't matter how the company juggles around getting the money to you - today you've got £10 worth of stuff.
2) Now let's think about options.
If you have a call option, this is a bit like you have cash and are standing around waiting to maybe swap with the share. If dividends go up, this is bad - someone else is getting those extra dividends, not you. So the call option value goes down as q increases. Lambda is negative for a call.
If you have a put option, this is a bit like you have a share and are standing around waiting to maybe swap with cash. If dividends go up, this is good - you can earn extra dividends on the share as you stand there waiting to maybe swap. So the put option value goes down as q increases. Lambda is positive for a put.
True that there may be a secondary effect springing from the information content of the dividend policy but the primary effect is this. Besides, as discussed above, it's not easy to know which way the secondary effect would go!
Good luck!
John
