Question

Explain the difference between a call option and a put option. Explain the difference between an American option and European option. Find the value of a call option using the binomial option pricing formula for single period when given the following information: you have an option with 6 months until expiration, the payoff in the up scenario is $12, and the payoff in the down scenario is $0, the risk-free rate is 5%, the weight for the up scenario is 1.1, and the weight for the down scenario is 0.7.

Answer 1

An option is a derivative instrument that gives the holder a right, but not an obligation, to perform. Options are deferred settlement contracts. Usually buyers of stock become part owners of the company on purchase of stocks, but in case of options, the option holder has only the right to buy the stock at a future date on the ocurrence or fulfilment of certain conditions.

An option gives the holder a right to (i) BUY or SELL an asset, (ii) at an agreed price called the exercise price or strike price, (iii) on or before a specified period of time.

Differences between Call option and Put option:

Call Option	Put Option
It is an Option which gives the holder the right to buy an asset.	It is an option which gives the holder the right to sell the asset
A call option shall be exercised only when the Exercise price is lower than the Market Price.	A put option shall be exercised only when the exercise price is higher than the Market Price.
The seller/writer of the option is under an obligation to sell the asset when the option holder i.e., the buyer in a call option exercises his option to buy the shares or stock.	The seller/writer of the option is under an obligation to buy the asset when the option holder i.e., the seller in a put option exercises his option to sell the shares or stock.

Difference between an American option and a European Option:

In an American Put or Call Option, the option holder can exercise his right to buy or sell at any time before the expiry date of the option or on the date of the expiry of the option as well.

In a European Put or Call Option, the option holder can exercise his right to buy or sell only on the expiry date of the option.

Calculations as per the Binomial Options pricing model for obtaining the value of a call:

Sl.No.	Particulars	Notation	Value
1	Value of call or payoff in case of up scenario	Cu	$12
2	Value of call or payoff in case of down scenario	Cd	$0
3	Weight for the up scenario	u	1.1
4	Weight for the down scenario	d	0.7
5	Risk free rate of return	r	0.05
6	Tenor of options contract in months = 6/12 =0.5	t	0.5
7	Future value factor (Continuous Compounding factor) = er * t = e0.05*0.5 = e0.025 = 1.0253 ( Value taken from e tables)	f	1.0253

As per the Binomial Option Pricing formula the value of a call is given by the following formula:

Value of a Call = [C_u [(f-d)/(u-d) ] + C_d [ (u-f)/(u-d) ] ] / f

Therefore applying the values from the table above to the formula we now have:

= [ 12 [ (1.0253 - 0.7)/(1.1 – 0.7) ] + 0 [ (1.1 – 1.0253)/( 1.1 – 0.7) ] ] / 1.0253

= [ 12 [ (0.3253)/( 0.4 ) ] ] / 1.0253

= [12 * 0.81325] / 1.0253

= 9.759 / 1.0253

= 9.5182

Therefore value of a call as per the Binomial Option pricing formula is $ 9.5182.