European Option (Definition, Examples) | Pricing Formula with Calculations

What is the European Option?

A European option can be defined as a type of options contract (call or put option) that restricts its execution until the expiration date. In a layman’s terms, after an investor has purchased a European option, even if the price of the underlying security moves in a favorable direction i.e. increase in the price of the stock for call options and decrease in the price of the stock for put options, the investor cannot take advantage by exercising the option early.

Types of European Option

Below are the two types of European options.

#1 – European Call Option

Holders of such contracts have the ability to buy a predetermined quantity of the underlying at the expiration date at a predetermined price, also known as the strike price. The investor is bullish on the market.

#2 – European Put Option

Investors can sell a predetermined quantity of the underlying at the expiration date at the strike price. Investor opinion is bearish.

Formula of European Option

Black Scholes Merton Model or BSM model is more suited for pricing of European options since one of the assumptions that this model rests on is that the options aren’t exercised early.

Pricing a European Call Option Formula

Price _Call= P₀N(d₁) – Xe^-rtN(d₂)

Where,

d1 = [ln(P₀/X) + (r+v²/2)t]/v √t and d₂= d₁– v √t
P₀= Price of the underlying security
X= Strike price
N= standard normal cumulative distribution function
r = risk-free rate
v= volatility
t= time until expiry

Pricing a European Put Option Formula

Price _Put= Xe^-rt *(1-N(d₂)) – P₀*(1-N(d₁))

Where d₁and d₂can be calculated in the same way as in the pricing of call option explained above.

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Practical Example of European Option

Stock XYZ is trading for $60. The strike price is $60. Volatility is 10% and the risk-free rate is 5%.

You can download this European Option Excel Template here – European Option Excel Template

Calculate the value of both 1-year call and put options written on it using the BSM model.

Calculation of d1

d1 = [ln(P₀/X) + (r+v²/2)t]/v √t
= LN(60/60)+(5+10^2/2*1)/(10*SQRT(1))
=0.55

Calculation of d2

d₂= d₁– v √t
= 0.55-60*SQRT(1)
= 0.45

We got the following values from normalization table

So the calculation of the price of the call option using the above table –

Price of call= $4.08

calculation of the price of the put option using the above table –

Price of put= $1.16

Prices are also calculated in the solved example excel sheet.

Before we move forward to its advantages and disadvantages, let’s talk a little about the upper and lower price bounds for European call and put options.

Lowest bound for a Call =0 (Option price can never fall below zero)
Highest bound for a Call= P₀(Current value of the underlying)
Lowest bound for a Put = 0
Highest bound for a Put = Xe^-rt(Present value of strike price)

How European Options Differs from American Options?

European Options differ from American options in the sense that American option holders have the liberty of exercising the option anytime, be it on or before the expiration date. Investors of such options can, however, choose to sell their holdings in the market before expiration date approaches. Under such circumstances, profit is essentially the difference between the premium earned and the premium paid.

Also, European and American options are mutually exclusive in the market. Both aren’t offered at the same time, so investors don’t have a choice.

Both these options basically differ on three grounds:

European options	American options
Can be exercised only at its expiration date	Can be exercised anytime during the life of the option
Traded over the counter	Traded only on exchanges
Generally, have lower upfront cost i.e. premium	Have higher premiums
Can have stocks and foreign currencies as the underlying	Can have stocks, bonds, commodities and derivatives as the underlying security

Advantages

There is a pre-determined time of expiration of the contract that gives the investor some certainty.
These options are less expensive than American options. Due to the added flexibility of any-time exercise in American options, the upfront cost tends to be higher than European options.
This tends to be less risky and the pricing complexity is also less due to the availability of limited options of contract execution.

Disadvantages

European options though less risky than the American options aren’t devoid of risks. They can be subject to other types of unique risks. The meticulous approach has to be adopted in order to avoid such risks.
One such risk is the trading lapse risk. Trading of European options closes on the end of the business day on a Thursday that falls before the third Friday of the expiration month. This can lead to an unexpected change in the price of the underlying.
The settlement price might be a little tricky to determine due to the trading lapse risk.
Investors don’t have the ability to exercise their options for taking advantage of a favorable price move.
The majority of such options are traded over the counter, so there isn’t much regulation involved, adding another degree of risk to it.
BSM model used for pricing might not be the most accurate model due to some unreal assumptions involved in the calculation.

Limitation

European options aren’t much accessible due to it being traded over the counter.
Pricing model takes certain assumptions of dividend, volatility and risk-free rate being constant during the entire duration which is unreal and hence prices might be different from the real world.

Conclusion

The naming of European and American options has nothing to do with the respective geographical locations. It simply gives an idea of when an option can be exercised. A majority of options traded in the US are known to be European options. Due to the difference in the features of the two options, investor expectations also vary. Like, an American option holder would expect the prices to move favorably before the expiration date arrives. However, the same isn’t true for European option holders who hope that such changes happen only at expiration.