a. There are two types of options, the American option (which can be exercised anytime during the period of the contract) and the European contract (which can only be exercised on the expiry of the contract). Both the options can, however, be traded anytime during the contract period.
b. So some of the important notations while valuing/pricing these options are:
c_{0 } = Value of European call at time 0.
c_{T } = Value of European call at time T.
C_{0 } = Value of American call at time 0.
C_{T } = Value of American call at time T.
p_{0 } = Value of European put at time 0.
p_{T } = Value of European put at time T.
P_{0 } = Value of American put at time 0.
P_{T } = Value of American put at time T.
c. The options are said to be in-the-money if it is beneficial for the holder/buyer of the option to exercise the same. On the other hand, if it is not beneficial for the holder of the option to exercise the same, it is called out-of-money. If the spot price equals the exercise price at the expiration of the contract, it is called at-the-money (ATM)
d. The value of an option is the sum of its intrinsic value and the time value of money.
V = IV + TV |
1.1. Call Options
a. If we recall the definition of a call option, under an option, the buyer has the right to buy and the seller has an obligation to sell a specific underlying asset at a pre-decided strike price by certain expiration date.
b. Suppose if S_{T }is the spot price of the asset and X is the exercise price of the underlying asset then there would be the following payoffs:
Pay-Offs |
S_{T }> X |
S_{T }< X |
Long Call |
S_{T }– X |
0 |
Short Call |
-(S_{T }– X) |
0 |
In the above cases, the profits would be as follows:
Profits |
S_{T }> X |
S_{T }< X |
Long Call |
(S_{T }– X) – C_{0} |
-C_{0} |
Short Call |
C_{0 }– (S_{T }– X) |
C_{0} |
1.2. Put Options
a. If we recall the definition of a put option, under such an option, the buyer has the right to sell and the seller has an obligation to buy a specific underlying asset at a pre-decided strike price by certain expiration date.
b. Suppose, if S_{T }is the spot price of the asset and X is the exercise price of the underlying asset then there would be the following pay-offs in the case of a put option:
Pay-Offs |
S_{T }< X |
S_{T }> X |
Long Call |
X – S_{T} |
0 |
Short Call |
-(X – S_{T}) |
0 |
In the above cases, the profits would be as follows:
Profits |
S_{T }< X |
S_{T }> X |
Long Call |
(X – S_{T}) – P_{0} |
-P_{0} |
Short Call |
P_{0 }– (X – S_{T}) |
P_{0} |