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I’m struggling for a while now with a question about options, namely 'which is the best option to buy?'. I have various books on options, but I’m not an mathematician and don’t have (yet) any extensive hands-on experience with options.

According to Cohen (Options Made Easy, 2nd Edition), the Delta of an option is the “change in option price relative to the change in underlying asset price”. He goes on to give an example of an option with an Delta of 0.5 which moves $1, in which case the premium of the option will increase with 0.50 (call) or decrease with 0.50 (put).

Even though Delta’s of options are changing with each change to the various components which make up an option premium, I’m wondering if a Delta can be used to determine the premium of an option given a certain target.

For example, let’s say stock XYZ trades at 50 dollar and we have an price target of +10% (so the share price of XYZ increases to 55 dollar; +$5). Let’s say an option’s current premium is 2.00, with an Delta of 0.40. Can the option premium at the target of 55 dollar be calculated with the following formula?

Current option premium + ( (share price target - current share price) * current delta of the option) = Approximated option premium at the price target

So, with the example figures this option will be worth.. 2.00 + ( (55 – 50) * 0.20) = 3.00 …at the price target?

Besides this, I’m wondering:

  • Isn’t the gamma (that is, the change in delta relative to the change in de underlying asset) needed for such an calculation?
  • If we have a time period in which to achieve this price target of $55, can the Theta (time decay) be incorporated in the calculation of the approximate value at the price target?
  • And, above all, requires this really so much calculation or can the approximated value be more easily and better be derived from something else? (like, say, the same strike of the option at and different expiration month, correcting for time value?)

Edit: My original angle to my question was more in wondering if there was a sort of ‘rule of thumb’ which an investor could use, in trying to choose between different strike of options. The underlying idea to my question was that, if somehow the option premiums could be guessed given the target for the stock, then the investor would be able to select the ‘best’ option for his outlook (i.e. the one with the highest potential return). With the same ‘rule of thumb’ an investor could calculated the potential downside, given his stoploss on the stock.

I agree with DumbCoder that an option model (like the Black and Scholes model; see https://secure.wikimedia.org/wikipedia/en/wiki/Black%E2%80%93Scholes#Mathematical_model) has the potential to answer this question, even though I don't (yet) understand this model.

Any more insights would be highly welcomed,

Regards,

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1 Answer

up vote 6 down vote accepted

In a simple world yes, but not in the real world. Option pricing isn't that simplistic in real life. Generally option pricing uses a Monte Carlo simulation of the Black Scholes formula/binomial and then plot them nomally to decide the optimum price of the option. Primarily multiple scenarios are generated and under that specific scenario the option is priced and then a price is derived for the option in real life, using the prices which were predicted in the scenarios.

So you don't generate a single price for an option, because you have to look into the future to see how the price of the option would behave, under the real elements of the market. So what you price is an assumption that this is the most likely value under my scenarios, which I predicted into the future. Because of the market, if you price an option higher/lower than another competitor you introduce an option for arbitrage by others. So you try to be as close to the real value of the option, which your competitor also does. The more closer your option value is to the real price the better it is for all.

Did you try the book from Hull ?

EDIT: While pricing you generally take variables which would affect the price of your option. The more variables you take(more nearer you are to the real situation) the more realistic your price will be and you would converge on the real price faster. So simple formula is an option, but the deviations maybe large from the real value. And you would end up loosing money, most of the time. So the complicated formula is there for getting a more accurate price, not to confuse people. You can use your formula, but there will be odds stacked against you to loose money, from the onset, because you didn't consider the variables which might/would affect the price of your option.

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Thanks for responding DumbCoder. Yes, you're right and pricing of an option requires more than just an 'simple' formula than the one provided in my question. My intention is not to use options as complex as your answer assumes (sorry for that lack of clarification). –  Jura25 Dec 7 '10 at 15:19
    
). I indeed have the book of Hull, but haven't started reading it (I'm in the beginning of learning options), and even though the formulas look quite complex (at first sight), I guess your suggestion is a good one (thanks). (Btw, I provided some further clarification in the question) –  Jura25 Dec 7 '10 at 15:27
    
@Jura25 - Updated with some more explaination. –  DumbCoder Dec 8 '10 at 12:59
    
Thanks for your further explanation. Well, I know enough: it's time to dive into Hull and get calculating. :) You're right, better to spend time studing complex formula's than the odds (heavily) against me. –  Jura25 Dec 8 '10 at 13:44
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