Can the Delta be used to calculate the option premium given a certain target?

Question

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,

Answer 1

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.

Answer 2

One thing I would like to clear up here is that Black Scholes is just a model that makes some assumptions about the dynamics of the underlying + a few other things and with some rather complicated math, out pops the Black Scholes formula. Black Scholes gives you the "real" price under the assumptions of the model. Your definition of what a "real" price entails will depend on what assumptions you make. With that being said, Black Scholes is popular for pricing European options because of the simplicity and speed of using an analytic formula as opposed to having a more complex model that can only be evaluated using a numerical method, as DumbCoder mentioned (should note that, for many other types of derivative contracts, e.g. American or Bermudan style exercise, the Black Scholes analytic formula is not appropriate). The other important thing to note here is that the market does not necessarily need to agree with the assumptions made in the Black Scholes model (and they most certainly do not) to use it. If you look at implied vols for a set of options which have the same expiration but differing strike prices, you may find that the implied vols for each contract differ and this information is telling you to what degree the traders in the market for those contracts disagree with the lognormal distribution assumption made by Black Scholes. Implied vol is generally the thing to look at when determining cheapness/expensiveness of an option contract.

With all that being said, what I'm assuming you are interested in is either called a "delta-gamma approximation" or more generally "Greek/sensitivities based profit and loss attribution" (in case you wanted to Google some more about it). Here is an example that is relevant to your question. Let's say we had the following European call contract:

• Stock price of $50
• Exercise price of $64
• Time to expiry of 2 years
• Vol of 25%
• Interest rate of 5%
• (Assume no dividends)

Popping this in to BS formula gives you a premium of $4.01, delta of 0.3891 and gamma of 0.0217. Let's say you bought it, and the price of the stock immediately moves to 55 and nothing else changes, re-evaluating with the BS formula gives ~6.23. Whereas using a delta-gamma approximation gives:

• 4.01 + (0.3891)(55 - 50) + 0.5(0.0217)*(55-50)^2 ~= 6.23

The actual math doesn't work out exactly and that is due to the fact that there are higher order Greeks than gamma but as you can see here clearly they do not have much of an impact considering a 10% move in the underlying is almost entirely explained by delta and gamma.