Options: How they're priced



Page Summary:
Options, whether on Stocks or Commodities are priced using mathematical models. Find out how these work and what the 6 main factors are. 
Intrinsic and Time value 
The price of an option consists of two parts 
 Intrinsic value, and
 Time value



What is Intrinsic Value 
Intrinsic value is the amount of money an option is worth if it were exercised and turned into the underlying. As the majority of readers are interested in the Stockmarket the underlying in these pages is always going to be shares.
However, the intrinsic value of many options is zero. For example 

 If Tesco's shares are trading at £3.50 the £4.00 call option would have zero intrinsic value
 Because the £4.00 call gives the owner the right but not obligation to buy shares at £4.00
 Why would anyone elect to exercise the option (and buy shares at £4.00) when they could buy them for £3.50 in the cash market
 So intrinsic value = zero


The option however would not be worthless because it still has time till expiry so its value would be made up entirely of time value. 
 But if Tesco's shares are trading at £3.50 then the £3.00 call would have an intrinsic value of exactly £0.50
 The owner of the call could exercise it, buy shares at £3.00 and then immediately sell them at £3.50 for a profit of £0.50 (the intrinsic value)
 But the option would be worth more than £0.50 (depending on how long till expiry) because it is possible for Tesco's share price to rise further?
 The £3.50 call would therefore be worth £0.50 + time value


To summarise  only inthemoney options (both calls and puts) have an intrinsic value. Atthemoney calls/puts and outofthemoney calls/puts only have time value. 
What is Time Value 
Time value can be summarised by this simple equation  
Time value = the price of an option less its intrinsic value

So if the option is outofthemoney its price will consist entirely of time value. For example 
 Tesco's shares are trading at £3.50 and the £4.00 call is worth £0.20
 The entire £0.20 is time value
But if Tesco is trading at £3.50 the £3.00 would be worth £0.70 and that is made up of both intrinsic value + time value as is shown below 
Time value = an option's price (£0.70)  intrinsic value (£0.50) = £0.20

The 6 factors that determine an options price 
There are six inputs that determine the price of a stock option and they vary in importance.

 Price of the underlying security
 The option's strike price
 Time remaining until expiration
 Volatility
 Dividends (obviously only relevant for dividend paying securities)
 Interest rates


1. The price of the underlying security 
The primary influence of an options premium is the price of the underlying security. If it is rising then call options will generally rise in price and put options will generally fall in price.
If the share is falling in value then put options will generally rise in price and calls will fall.
The further inthemoney an option is the more it will be worth and vice versa with outofthemoney options.

2. The option's strike price 
The strike price determines whether an option has any intrinsic value or not.
For call options, the lower the strike price the more value a call option will have, and vice versa.
For example 

 If a share is trading at £4.00 then £3.00 calls will be worth more than the £3.25 calls which in turn will be worth more than the £3.50 calls etc
 It would be the same for out of the money calls
 If the shares are trading at £4.00 then the £4.25 call would be worth more than the £4.50 calls and so on


For put options, the higher the strike price the more value a put option will have and vice versa. For example  
 If a share is trading at £10.00 the £14.00 puts would be worth more than the £13.00 puts which in turn would be worth more than the £11.00 ones
 It would be similar for outofthemoney puts
 If the shares are at £10.00 then the £9.00 put would be worth more than the £8.00 put and in turn more than the £7.00 put


3. Time remaining until expiration 
Options will always expire at a known date in the future and the further this date is from expiration the more value an option will have. For example (if the current month is January) the March £1.20 Vodafone call might be priced at £0.10 but the December £1.20 Call is at £0.35.
This is because Vodafone's share price has more potential to move higher (or lower) over a longer period of time.
Time is especially important when pricing options and theoretically an option will lose some time value everyday. How much it loses will depend on how much time is left till expiry.
As expiration approaches, the speed which time value decays speeds up, ie the rate of decline is not linear. The diagram below highlights this.

Chart showing time decay versus time to option expiry

An option with 6 months+ till expiry loses a fraction of time value everyday but in the final 2 weeks value is almost in freefall.
So what does this mean for the buyer of options? Common sense suggests that buying shortdated options (less than 1 month till expiry) will be a losing trade more often than not, unless your timing is perfect or you're incredibly lucky (the two are often related).
The increase in the rate of decline in an option's value as it gets nearer to expiry makes a lot of sense. This point was discussed in the introduction to Options 

 What is the chance of Tesco's shares rallying 25% over the next 6 months  Very possible
 And the chance of them rallying 25% in 3 months  Possible
 And 25% in 1 month  Not that possible
 And 25% in 5 days  virtually impossible
 And 25% in 1 day  99% impossible unless there's an act of God or surprise takeover


4. Volatility 
Volatility is the bogeyman when it comes to pricing options, and it's something that many new to options have no clue about. Disgracefully many of the leaflets or media articles explaining the benefits of options don't even mention volatility.
But if you fail to understand option volatility and how it affects prices then you'll have no chance of making any money. And you could even lose large amounts of cash even if you're right about the future direction of a share or market.
Option volatility is the measure of the risk of the possible amplitude of an option's price. It does therefore not imply a bullish or bearish trend in stock price movement  rather it is the expected fluctuation in price, either up or down.
As volatility moves higher both calls and puts will generally move up in value as well. Conversely, when volatility moves lower both call and put prices will generally lose value.
A beginner to options, without knowing anything about volatility, will therefore expect call prices to move higher if the underlying moves higher. The calls will also lose value as the stock moves lower.
But the combination of sharp volatility rises and falling stock prices can cause call options often to dramatically rise in price. The beginner is therefore totally confused as the call options seem to be doing the complete opposite of the dictionary definition, ie call options are rising in value as the market slumps.

5. Dividends 
Dividends have a mild effect on an option's price. Personally I don't worry about them as all known information about the stock price will already be factored into the option prices.

6. Interest Rates 
Interest rates are a factor in the pricing of an option but they are minor compared to say volatility or the underlying share price.
As interest rates rise, call options will rise but put options will fall. Conversely, as interest rates fall call prices will fall and put prices will rise.
I wouldn't worry too much about the effect of interest rates on the price of options as they'll have a negligible effect. Plus, interest rates don't change that much during the year and when they do the announcement has normally always been factored in beforehand.

The role of Mathematics in Options 
Maths hardly plays an important role in the stockmarket, however it is important with options. In fact, one could easily argue that options are mathematical products.
The pricing of options is always based on a series of calculations using option valuation methods such as the Black Scholes model Wikipedia link. But is it necessary to get bogged down in all the analysis especially if Maths has never been your strong point?
There is of course no correct answer; it all depends on how you view your own trading. People who want to use options occasionally only really need to understand the mathematical basics. Conversely, the more serious you are about options the more mathematical type analysis you'll have to use.
If you do want a more analytical approach then you'll need the help of some dedicated options software. I use the excellent Hoadley Options package which is an excel addon  See FAQ section  What is Hoadley Options software.
Ultimately though, what will make or lose the average retail trader money in options is not how much theory they know or how intelligent they are. The decider will be how good their view is on the market and that's where most traders and investors should spend at least 80% of their research time.

It's not all about the Maths  Supply and demand is also important 
An option market maker when asked to quote a price for a stock option or strategy won't start thinking about the prospects of the company or the state of the economy  these facts are probably totally irrelevant to him.
Instead, he will rely on his computer models to quote the price.
But if the computer spits out a price of say 48  50 and the whole market wants to buy the market marker will add in some common sense and raise his quote until there is some sort of equilibrium in the marketplace among both buyers and sellers.

The 'Greeks' 
Option prices are determined by mathematical models (plus some natural supply and demand in the market place) and these in turn kick out a number of output values which are known as the 'Greeks'.
These Greeks are useful tools to anticipate the behaviour of option prices and can help traders to determine which options or strategies are the best to use.
There are 5 Greeks 

 Delta  the expected change in an option's price with a change in the underlying price
 Gamma  the expected change in the rate of the delta's change
 Theta  the expected change in an option's price with the passage of time
 Vega  expected change in an option's price with a move in volatility
 Rho  the expected change in an option's price with a change in interest rates (Rho is therefore the least important of the Greeks)


The quandary of the Greeks 
Remember the iceberg analogy that I made when introducing options? Understanding an option is like understanding an iceberg  there's so much more below the surface than above.
Having a good grasp of options and how to use them in their basic form is the 10%  15% that's above the waterline. But to really understand options and use semi to complex strategies you're going to have to dive deep beneath the surface and do a great deal of work. And I mean a great deal of work.
This is the quandary of explaining the Greeks in detail. It simply cannot be done over a few pages. Check out the following Wikipedia link on the Option Greeks  this is about 3,000 words and yet it only scratches the surface. However, for most people the information, and especially the maths, will be too hardcore.
So I have decided to not to go into any detail on the Greeks apart from Delta, which is the most important one and also the easiest to understand.
If you do want more information then I suggest you look at the following book  which should be required reading for anyone wanting to get a better options education.

What is Options Delta and how to use it 
The delta measures the change in an option's price relative to the underlying security. For example  
 If the underlying share rises by 10p the call option with the delta of 0.5 will rise by 5p
 If the underlying share falls by 15p the call option will fall by 7.5p


Deltas for calls are always positive, deltas for puts are always negative and they work in the same way. So if a put option has a delta of 0.75 it will move 75% of the value of the underlying share, for example  
 If the share falls £1.00 the put option that has a delta of 75% will gain £0.75
 If the share rises by £0.50 the put option will lose £0.375


Note that deltas are not constant (none of the Greeks are). They'll be moving all the time as the underlying share price changes. If an option opens with a delta of 0.7 in the morning, that might fall to 0.59 by the close of business that day.
The further inthemoney an option is the higher its delta will be and vice versa. If BT is trading at £1.50 the £1.00 call will probably have a delta of 90%+. But the £2.00 call's delta will be nearer 20%.
Delta is the most important of the Greeks but Gamma is of note as well. Gamma is the expected change in the rate of the delta's change. All very interesting but how and where do you find details of an options Greeks? Simple, use software and the one of the best free packages around is Hoadley options.

Summary  How are options priced 
If you have a basic grasp of how an option works it's not that hard to make an accurate estimation of their prices in your head. You only need a few inputs 

 What is the stock's current price
 What is the strike of the option (and whether it's a call or put)
 When does the option expire


With just that information you'll easily be able to work out if the option has any intrinsic value or not and then add in time value. But to get an accurate price you need more information with the most important input being volatility which is discussed in depth on its own page.
Finally, if you want to use options occasionally in your trading, and using them as part of your financial toolbox, then I wouldn't worry too much about the complex mathematical tools used to price them up.
This is because you can be sure that any option you see quoted on a large Exchange, such as Euronext for UK stock options, is priced to almost perfection by the professional market makers. In effect they have done a lot of the work for you. Where you will make or lose money won't relate too much to your skill at using options, rather whether your view on the underlying market is correct or not.
But if you want to extensively use options as part of your trading then remember this  the most successful option traders are normally the most educated (in options).

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