

Option Pricing  Part II
In the previous week's article, we discussed some of the basic tenets of options. These included how they operate, and the concepts of intrinsic and time value. Today, we will continue this discussion further, looking at how option prices change when other variables change (i.e. the Greeks). Let's have a look at this hypothetical table below:
The market is crude oil. Assume the current spot price is $80/barrel. The options below represent one month call options on crude oil at varying strikes.
Strike 
$78 
$79 
$80 
$81 
$82 
Price 
$3 
$2.50 
$2 
$1.50 
$1 
Intrinsic Value 
$2 
$1 
$0 
$0 
$0 
Time Value 
$1 
$1.50 
$2 
$1.50 
$1 
Delta 
0.7 
0.6 
0.5 
0.4 
0.3 
Gamma 
0.08 
0.09 
0.1 
0.09 
0.08 
Vega 
0.05 
0.06 
0.07 
0.06 
0.05 
Theta 
0.05 
0.06 
0.07 
0.06 
0.05 
Rho 
0.02 
0.017 
0.015 
0.013 
0.01 
Last week, we discussed intrinsic and time value. The only additional point I'd like to mention is that time value is highest for ATM options. The further ITM or OTM we go, the less the time value will be.
Delta is a measure of how the option price moves with the underlying asset moves. It measures the percentage change in the option price when the underlying asset moves by 1%. In the example above, the $80 call has a delta of 0.5. (ATM options will generally have a delta of 0.5) This means that if crude oil rises by 1%, the call option will rise by 0.5%. As we move in the money, the options have a higher delta, meaning they are more sensitive to changes in the underlying asset. The $78 call has a delta of 0.7, meaning it will move more than the $80 call if crude oil rises by 1%. The option delta approaches 1.0 the further in the money we go. A similar logic applies to the OTM options. They are less sensitive to the underlying that ATM or ITM options and they have deltas less than 0.5. Again, as we go further OTM, the delta will approach 0.
While call options have a positive delta, put options have a negative delta. This means that they move in opposite directions to the underlying asset. So if crude oil rises, a crude oil put will fall in value. An ATM put option will have a delta of 0.5. An ITM put will have a delta between 1 and 0.5 (the further ITM the option is, the closer delta will be to 1). Lastly, an OTM put will have a delta between 0 and 0.5.
Moving on, the gamma of an option is a measure of how the delta changes as the underlying changes. Taking the example of $80 call, it has a delta of 0.5 (because it is ATM) and a gamma of 0.1. If the price of crude oil rises by 1%, the delta will increase by 0.1 (from 0.5 to 0.6). Once crude oil has risen, the option is now ITM. The increase in delta makes sense as ITM options have higher deltas relative to ATM options. Gamma is positive for both put and call options, and is highest when the option is ATM. The further ITM or OTM we go, the lower gamma will be.
Vega refers to volatility. The vega of an option represents how much the option will move when volatility increases. Vega is positive for both call and put options, and like gamma, is highest when the option is ATM, and decreases as we go further ITM or OTM. Vega also falls as the option nears expiry. Thus it is highest at the beginning of an option's life, and decreases continuously over time.
Volatility can be a difficult concept to grasp. In the case of option pricing, volatility is measured using standard deviation. All options have a positive vega, meaning that they are positively correlated to volatility. So the more volatile an asset is, the higher option prices for that asset will be. A simple example should help to illustrate why this is the case:
Let's assume the fair value of an option represents the expected value of the option. Asset 1 has a current value of 100. Tomorrow, it will either fall to 90 or rise to 110 (with a 50% probability of a fall and 50% probability of a rise). What is the expected value of an ATM call option that expires tomorrow? An ATM call option with a strike of 100 will have a payoff of 0 if the asset's price falls to 90 and a payoff of 10 if the asset's price rises to 110. The expected payoff is 5, so the fair value of the option is 5.
Assume asset 2 also has a current value of 100, but it is more volatile than asset 1. Tomorrow, asset 2 will either fall to 80 or rise to 120 (with a 50% probability of a fall and a 50% probability of a rise). An ATM call option with a strike of 100 will have a payoff of 0 if the asset's price falls to 80 and a payoff of 20 if the asset's price rises to 120. The expected payoff is 10, so the fair value of the option is 10. As we can see, the more volatile asset 2 has a higher option value than the less volatile asset 1. This shows us how option prices are positively correlated with volatility.
Back to the Greeks, the next one to discuss is theta. Theta represents how much the option will change in value due to the passage of time. More specifically, it is how much the option will fall in a single day due to the fact that we are one day closer to expiry. As I discussed last week, the greatest enemy of the option buyer is time. Upon the expiry of an option, the option will have zero time value (and will have only intrinsic value). Theta represents the fall in the time value component due to the passage of time. Note that time value can rise due to other factors such as gamma or vega, but over the lifespan of the option, the time value must fall to zero.
Lastly, we have rho. Rho represents the change in option prices due to changes in interest rates. Calls have a positive rho and puts have a negative rho. Rho is usually small for options, and given that interest rates tend to stable over short periods, most investors don't use rho as part of their analysis when making investment decisions. It is more important when considering very long term options – as interest rates can change significantly over a long period of time.
Hopefully by now we all understand how options are priced, and what factors can affect option prices. There is a wealth of academic literature on option pricing (and it is fascinating), and I recommend those who are interested to pursue it further. Over the next two weeks, we'll discuss some trading strategies for options, and the risks and rewards associated with them.
This column, A Fresh Perspective, is authored by Asad Dossani. Asad is a financial analyst and columnist. He actively trades his own and others' funds, investing primarily in currency, commodity, and stock index derivative products. Prior to this, he worked at Deutsche Bank as an analyst in the FX derivatives team. He is a graduate of the London School of Economics. Asad is a keen observer of macroeconomic trends and their effects on global financial markets. He is deeply passionate about educating investors, and encouraging individuals to take part in and profit from financial markets. To put it colloquially, he wishes to take Wall Street products and turn them into Main Street profits!
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