The value of an option before expiration depends on six factors:
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The level of the underlying index
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The exercise price of the option
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The time to expiration
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The volatility of the index
The risk-free rate of interest
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Dividends expected during the life of the option
These factors set general boundaries for possible option prices. If the option price is above the upper bound or below the lower bound, there are profitable arbitrage opportunities. We shall try to get an intuitive understanding about these bounds.
Upper bounds for calls and puts
A call option gives the holder the right to buy the index for a certain price. No matter what happens, the option can never be worth more than the index. Hence the index level is an upper bound to the option price.
C (less than equal to) I
If this relationship is not true, an arbitrageur can easily make a riskless profit by buying the index and selling the call option.
As we know a put option gives the holder the right to sell the index for X. No matter how low the index becomes, the option can never be worth more than X. Hence,
P (less than equal to) X
If this is not true, an arbitrageur would make profit by writing puts.
Lower bounds for calls and puts
The lower bound for the price of a call option is given by S-X (1+r) –T The price of a call
must be worth at least this much else, it will be possible to make risk less profit
S-X (1+r) –T < C
Consider an example. Suppose the exercise price for a three-month Nifty call option is 1260. The spot index stands at 1386 and the risk-free rate of interest is 12% per annum.
In this case, he lower bound for the option price is 1386-1260 (1+1.2) –0.25 i.e. 161.20 Suppose the call is available at a premium of Rs.150 which is less than the theoretical minimum of Rs. 163.20. An arbitrageur can buy a call and short the index. This provides a cashflow of 1386-150 = 1236.
If invested for three months at 12% per annum, the Rs.1236 grows to Rs.1273. At the end of three months, the option expires. At this point, the following could happen:
1. The index is above 1260, in which case the arbitrageur exercises his option and buys back the index at 1260 making a profit of Rs.1273 - 1260 = Rs.13.
2. The index is below 1260 at say 1235, in which case the arbitrageur buys back the index at the market price. He makes an even greater profit of 1273 - 1235 = Rs.38.
The lower bound for the price of a put option is given by X (1+r) -T –S. The price of a put must be worth at least this much else, it will be possible to make riskless profits.
X (1+r) -T –S. < P
Consider an Example. Suppose exercise price for three- month Nifty put option is 1260. The spot index stands at 1165 and the risk-free rate of interest is 12% per annum. In this case the lower bound for the option price is Rs.59.80.
Suppose the put is available at a premium of Rs.45 which is less than the theoretical minimum of Rs.59.80. An arbitrageur can borrow Rs.1210 for three months to buy both the put and the index. At the end of the three months, the arbitrageur will be required to pay Rs.1246.3. Three months later the option expires. At this point, the following could happen:
1. The index is below 1260, in which case the arbitrageur exercises his option, sells the index at Rs.1260, repays the loan amount of Rs.1246.3 and makes a profit of Rs.13.7.
2. The index is above 1260 at say 1275, in which case the arbitrageur discards the option, sells the index at 1275, repays the loan amount of Rs.1246.3 and makes an even greater profit of 1275 - 1246.3 = Rs.28.7.
¦
The level of the underlying index
¦
The exercise price of the option
¦
The time to expiration
¦
The volatility of the index
The risk-free rate of interest
¦
Dividends expected during the life of the option
These factors set general boundaries for possible option prices. If the option price is above the upper bound or below the lower bound, there are profitable arbitrage opportunities. We shall try to get an intuitive understanding about these bounds.
Upper bounds for calls and puts
A call option gives the holder the right to buy the index for a certain price. No matter what happens, the option can never be worth more than the index. Hence the index level is an upper bound to the option price.
C (less than equal to) I
If this relationship is not true, an arbitrageur can easily make a riskless profit by buying the index and selling the call option.
As we know a put option gives the holder the right to sell the index for X. No matter how low the index becomes, the option can never be worth more than X. Hence,
P (less than equal to) X
If this is not true, an arbitrageur would make profit by writing puts.
Lower bounds for calls and puts
The lower bound for the price of a call option is given by S-X (1+r) –T The price of a call
must be worth at least this much else, it will be possible to make risk less profit
S-X (1+r) –T < C
Consider an example. Suppose the exercise price for a three-month Nifty call option is 1260. The spot index stands at 1386 and the risk-free rate of interest is 12% per annum.
In this case, he lower bound for the option price is 1386-1260 (1+1.2) –0.25 i.e. 161.20 Suppose the call is available at a premium of Rs.150 which is less than the theoretical minimum of Rs. 163.20. An arbitrageur can buy a call and short the index. This provides a cashflow of 1386-150 = 1236.
If invested for three months at 12% per annum, the Rs.1236 grows to Rs.1273. At the end of three months, the option expires. At this point, the following could happen:
1. The index is above 1260, in which case the arbitrageur exercises his option and buys back the index at 1260 making a profit of Rs.1273 - 1260 = Rs.13.
2. The index is below 1260 at say 1235, in which case the arbitrageur buys back the index at the market price. He makes an even greater profit of 1273 - 1235 = Rs.38.
The lower bound for the price of a put option is given by X (1+r) -T –S. The price of a put must be worth at least this much else, it will be possible to make riskless profits.
X (1+r) -T –S. < P
Consider an Example. Suppose exercise price for three- month Nifty put option is 1260. The spot index stands at 1165 and the risk-free rate of interest is 12% per annum. In this case the lower bound for the option price is Rs.59.80.
Suppose the put is available at a premium of Rs.45 which is less than the theoretical minimum of Rs.59.80. An arbitrageur can borrow Rs.1210 for three months to buy both the put and the index. At the end of the three months, the arbitrageur will be required to pay Rs.1246.3. Three months later the option expires. At this point, the following could happen:
1. The index is below 1260, in which case the arbitrageur exercises his option, sells the index at Rs.1260, repays the loan amount of Rs.1246.3 and makes a profit of Rs.13.7.
2. The index is above 1260 at say 1275, in which case the arbitrageur discards the option, sells the index at 1275, repays the loan amount of Rs.1246.3 and makes an even greater profit of 1275 - 1246.3 = Rs.28.7.