The price of an option.
After the Options general introduction in the first post, we have seen how call options and put options work. Now, in order to get the whole picture, its the moment to understand a bit better how to determine their price.
I’m telling you in advance: if you are looking for an endless sequence of mathematical formulas, you will be disappointed. Please, consider this is an introductory guide and not an academic course, but I want to help anyway. If you are looking for the theoretical demonstrations go here or here and enjoy them.
Having said this, we can start with a simple observation. A financial instrument quoted 10 right now is more likely to rise to 11 than to 15 and, considering decreasing trends, it works exactly the other way round. Starting from 10, our security is more likely to fall to 9 rather than to 5.
With the options it’s the same. Let’s consider options written on any security now quoted 10 $. A 11 $ call option is more risky for the seller then a safer 15 $ strike one, this means that it simply will cost more. Concerning put options instead, things work the other way round: a 9 $ exercise price option will cost more than a 5 one. The point here, like for anytime you take some risk in change of money, is the probability to make profits or losses.
Now you might point out: “it costs more, it’s obvious and I agree, but how much exactly ..?” Or if you are an analytical person you might ask yourself which is the “right” price of the option. Well, that’s a good guess, this is the critical point and things could go in a rather complicated and longish direction.
But since we are smart and don’t want to get mad with useless calculations, we are going to choose our direction wisely. My assumption here is that you are interested in trading and not in philosophy or better in stochastic differential equations.
This means that I’m pretty sure you will appreciate if we see the basic formula just to have an idea and then we analyze a bit better the variables that are important in a trading framework. Nevertheless, just to know what are we speaking about, a bit of formulas are necessary, so let’s see…
The options classic formula.
The price of an option, as you can imagine, is calculated using mathematical and statistical formulas. There are several models, which is the name given by economists for a given set of assumptions and hypotesis, but probably the most famous is the Black & Scholes model. Below you can see the model resulting formula:
Where
and
Basically what does it say ..? It says something quite logical if you watch the question from the seller perspective. Since he doesn’t want to loose money he will simply multiply any possible loss for the probability that it can happen. In this way he find the correct value of the option.
Actually the path is a bit more complicated because they assume to buy and sell constantly the underlying creating in this way a covered position. However, in general, give it up, trust me. You will never use that method when trading. What you will use is…
The options price expressed by the market.
On the market, as you can imagine, there are professional trading companies who use any kind of software to calculate the theoretical price. Usually they are also dealers on that option’s market and dealers, in most cases, are obliged to quote a bid / ask price.
Since the arbitrage mentioned above is possible and anyone would like to eat for free, the correct price will be in some point between the bid / ask spread. After all, what is software for if not making life easier…? So in place of stochastic differential equations and lognormal distribution we can give a look to our option book and see:
Going deeper and deeper in the mathematical science we can state that the correct value will be: (324 + 334) / 2 = 329 approx and this way we solve the price question. Now we can start to see the things you will keep an eye on when trading, that is to say…
The elements that determine the option’s price.
The distance between the underlying value and the strike price.
We have already discussed this aspect at the beginning of the post and it’s useless to repeat it once again. Let’s say, in few words, that an option very close to the market price is more likely to give some return. A more probable return simply means it will cost more. If you imagine a market price of the underlying around 20,000, on your trading platform you would see for example quotes like this:
What we have not considered before is the relation between the price and the strike in relative terms, in other words what happens to close options and distant ones when there is a strong (but even a week) movement of the underlying price.
We could use some models to describe this aspect but, again, I think we are better explaining it with a little of common sense. When there is an earthquake, if you are one mile far from the epicenter it’s a problem while if you are on the other side of the earth it is not at all. The same with options. The strikes closer to the market price, other conditions being equal, will react more and faster to any changes in the underlying price.
Since we are here, let’s see also some terminological aspects, just to make you ready when you will meet them. The relation between the variation of the option price and the corresponding variation of the underlying value is called Delta by economists. With a formula that should not scare you, the mirrored 6 trivially means “how much varies” V (that is the market value of the option) when there is a variation of S (that is the value of the underlying):
This relation is used to create some advanced strategies that (maybe) we will see later on, while for now we’ll get on with the second element, that is to say…
The time missing to the expiration date of the option.
Referring to our previous example “which is the difference if you have to stay in a place where there can be some earthquake for one month or for a year..?” Easy: in one year you run many more risks than in a month. Again, with options the game is the same.
The more is the time left to the expiration date, the more is the probability that the underlying trend can be influenced by some relevant event. Option’s value will reflect this statistical evidence and in absolute terms, all other conditions being equal, the greater will be the time to maturity the higher will be the price and vice versa.
Like we have done before, we can also consider the relation between prices and time in relative terms, that is to say how the price of the option varies as time goes on and the expiration date gets nearer.
Well, you will understand that the probability framework is the same as described above, it’s obvious: more time means more risks and vice versa. So imagine a situation in which the underlying doesn’t move significantly and you have bought or sold an option. Well, its value will have the following dynamic:
Terminology again. This situation is described in finance with an indicator called Theta, which expresses what mentioned above: how the price of the option (V) varies compared to the variation of the expiring date time (T).
Now that we have understood the importance of time in the option pricing, we can go on with our analysis considering a less obvious element although still important, that is…
The market volatility.
Back to our earthquake example: given a 5 degree one, is it more risky if it is undulatory or vibrating..? You know perfectly that vibrating ones are worse, they are usually more likely to produce big damages to buildings. So in the option framework an historically “vibrating” underlying will be more dangerous than an “ondulatory” one. Let’s illustrate the concept with a chart, have a look at the following two securities trends:
Well, even at a glance, you will see clearly that the blue is more risky than the red one, price movements are much more nervous and broad. This variability, on a visual level, is called volatility in finance.
In more formal terms, volatility is an index that measures the risk of a financial instrument. There are several ways to calculate it, some are simple and others more complicated. None of them is infallible because all always rely on a set of simplifying assumptions.
Perhaps the most intuitive one is the standard deviation, that is: the average deviation of the security from its average in a given period of time. Confused..? Don’t worry, forget about it.
Quantifying the volatility is like the getting the price ”right” issue. Any trading platform offers many computational tools that do the job for you, among these there is the standard deviation as well as other risk indicators. In the worst case, if you really want to do it manually, I suggest you to use an Excel spreadsheet that already incorporates the function to calculate this kind of indicator.
Coming back to the relation between the volatility and the options price you will have guessed that, since its more risky, an option written on a very volatile security will cost more than one written on a stock that moves more smoothly. Even in this case, its fairly logical.
In relative terms if you consider how the option value changes when the underlying volatility rises or decreases, you can get what in finance is called Vega, in a formula:
At this point, having seen the Vega and the other indicators above, I guess you will be a bit tired but there is a good news for you: only one element is missing and we will go through it quickly because it is not so important. So with the last effort let’s see…
The interest rate.
Just to be sure we speak the same language: the interest rate is a variable which measures the economic cost of money. You can see this when you need a loan, if you go to the bank and ask for money they will ask you back more that means it has a cost. The cost is expressed by the interest rate. Like all other economic variables, the interest rate trend is subject to fluctuations determined by the demand and the supply.
How does the interest rate deal with options..? Well, let’s avoid both to get bored, me in writing and you in reading. Let’s say that the interest rate is used to calculate the theoretical price. As I told you at the beginning, in Black & Scholes model (and in other models too), it’s assumed to borrow securities to create a covered position and, thanks to this, you can then give a value to the options.
In short, we can say that since it’s an element that contributes to the price determination, any variation of it will have some impact on the price. In general you can expect that the higher the interest rate is, the lower will be the premiums of any option.
This is valid for put as well as call options in any expiration date and any strike price. In this case too we have a relative relation that expresses the situation described above, that in finance is called Rho. Like in all the cases we have analyzed before, V represents the option value while r is the interest rate:
Last but not least, since we have seen many things, we can make a small summary to have all the elements under our eyes at a glance. So following you can find…
A simple summary about option pricing in a trading framework.
- For what concerns the determination of the theoretical price don’t bother with mathematical and statistical formulas. The best thing you can do is trust the values expressed by the market (if and only if the market is fairly liquid, otherwise the best thing to do is to trade on another market).
- The nearer the exercise price is or gets to the market price of the underlying, the greater will be the option value.
- The price of the put and call options, other factors being equal, tends to decrease as the expiration gets nearer.
- If the volatility of the market increases, you will see that also the option’s value will increase.
- Vice versa if the interest rates rise, the prices of the options tend to decrease.
These are the important variables you will need to take a look at when you open a position and begin trading on any market. Obviously, as said in our previous posts, these elements are all important but the main game is played by your expectation on the underlying movement.
If your expectations are wrong you will be likely to loose some money and, obvious, if they are correct there are good probabilities to get some profit. The so called greeks most of the cases can increase or decrease their entity. This observation is valid with put and call in long as well as in short positions. That’s it.
