Updated: Jan 18
Option pricing and volatility? What a serious matter! Here you will be described the 6 factors affecting the theoretical price of options and, hopefully, all your questions options premium related will be answered.
This article approaches the option pricing mainly from an economical and conceptual point of you. It is not my purpose investigating the mathematical implications standing behind the Black-Scholes model and, to say the truth, traders do not really need mathematical skills in order to understand the variables making the model.
Options Theoretical Price Calculation
When you trade financial instruments on the stock exchange there must always be somebody on the other side who is willing to trade with you at the agreed price. On the opposite side of the trade there are always market makers, which must sell you one or more contracts in order to get your orders filled.
As a result, the premium paid to buy stock options is directly related to a series of risks that market makers face. In fact, market makers determine the option theoretical price by organizing a package of asset into a risk-free position and then finding the current value of that package based on the current interest rate.
The 6 elements Impacting on Options Premium
In order to understand how this process works you should be aware of all the components making up the option premium. The Black-Scholes Model is the theoretical price model mainly used to perform such calculations.
Actually, there are many pricing models out there all bringing to similar results, but the Black-Scholes model is the original and the most well-known. In order to understand the option pricing, you should be aware of the meaning of the following elements:
Price of the underlying stock or financial instrument;
Option exercise price or strike price;
Options time value;
Options implied volatility;
Risk-free interest rate;
Dividends paid over the life of options.
Stock price and options strike prices
The option premium is the price traders accept to pay to hold call options or put options for a certain period of time and have the chance to exercise the rights included within their contracts. The option premium is made up of intrinsic and extrinsic value.
P (option premium) = Vi (option intrinsic value) + Ve (option extrinsic value)
P = Vi + Ve
The intrinsic value (Vi) is the real value carried by options and works differently for calls and puts. The intrinsic value of calls is the difference between the value of the underlying security and the selected option strike.
Call (intrinsic value) = value of the underlying security - call option strike
The intrinsic value of puts is the difference between the option strike and the value of the underlying security.
Put (intrinsic value) = put option strike - value of the underlying security
Options carrying intrinsic value are in-the-money. Options not carrying intrinsic value are out-of-the-money. Options whose strike price is very close to the price of the underlying security are at-the-money.
The extrinsic value (Ve) is the value carried by options related to time value and implied volatility. The premium paid to purchase out-of-money options is completely made up of extrinsic value. Such premium is higher for longer-term options, because being longer the time before expiration, will be higher too the probability for this financial instrument to go in-the-money by expiration.
Ve (extrinsic value) = IV (option implied volatility) *t (option time value)
Option Time Value
A substantial part of the option extrinsic value is given by the time remaining until expiration. The further is the expiration date, the more time value will be included in the price of options. All options are subject to time decay, meaning that options lose value as time goes through.
Time decay is not a linear value. The less time an option has until expiration, the faster that option is going to lose its value. The effect of time on the option premium is measured by the Greek Theta.
Option implied volatility
A substantial part of the option extrinsic value is given by implied volatility. It is related to investor’s expectations and is a gauge of uncertainty within the market. The understanding of its value is very important in today’s volatile financial markets because it may influence option premiums to such an extent to make options underpriced or over-priced in different market conditions. The effect of implied volatility on the option premium is measured by the Greek Vega.
The impact of implied volatility on the options price is higher as they approach to expiration. This statement is true if you consider IV on its own. However, the impact of implied volatility is directly proportional to the amount of time left on your options (as you can see from the formula). It means that IV has a stronger effect on the price of longer-term options.
Below the complete option premium formula:
P = Vi + (IV*t)
P = Option premium
Vi = Option intrinsic value
IV = Implied volatility
T = Current time until expiration
The above mentioned formula is explained to make you understand the main components of an option pricing model. It is very intuitive and easy to understand, but is not exhaustive. In fact, it does not consider the effects that interest rates and dividends can have on options.
A minor part of the option extrinsic value is given by the impact of interest rates. When interest rates are changed the price of options is affected. Definitely, interest rates are considered by traders the least important of the variables affecting the option premium because they are changed not too often and the impact of such changes on the option theoretical price is quite insignificant. The effect of interest rates on the option premium is measured by the Greek Rho.
The option value is influenced by the release of cash dividends. Dividends affect option prices by impacting directly on the price of the underlying security. As the stock price is expected to decrease by the amount of the dividend on the ex-dividend date, high dividends release involve lower call premiums and higher put premiums. Options traders should take into consideration the dividends paid when calculating the option theoretical price and be aware of their impact on the options’ risk profile.
The Black-Scholes Model
For purpose of completeness, I am going to report below the Black-Scholes Model. Even though it is presented here in a simplified version, the model may appear hard to most of you and so it does for me. The good news is that to become a professional trader you do not really need to master the mathematical calculations within it.
What you need is to deeply understand how all the elements in this formula work and can affect your position. For this reason, it is on those elements that this article was focused on. Anyway, to learn more about the original mathematical Black-Scholes Model, I recommend visiting other sources.
Price of a call option in the Black-Scholes model:
C = S N (d1) - X e-rTN (d2)
Price of a put option in the Black-Scholes model:
P = Xe-rT N (-d2) - S N(-d1)
Put-call parity in the Black-Scholes model:
P = C - S + Xe-rT
C = call option price
P = put option price
S = price of the underlying stock
X = exercise price of an option
r = risk-free interest rate
T = current time until expiration
N() = area under the normal curve
d1 = [ ln (S/X) + (r + σ2/2) T ] / σ T1/2
d2 = d1 - σ T1/2
The Black-Scholes formula is based on the following assumptions:
Options can be exercise only at expiration. Consequently, Black and Scholes refer only to European style options rather than American style options.
The volatility of the underlying security remains constant over the period of calculation.
The risk-free interest rate remains constant over the period of calculation.
The underlying stock does not pay dividends over the period of calculation.