# Arbitrage beyond Upper and Lower Bounds

There are six factors affecting the value of an option before expiration. They are: the price of the underlying stock, the exercise price of the option, the time remaining until expiration, the risk free rate of interest, the volatility of the underlying assets and dividends expected during the life of the option. These factors set the general boundaries for option prices. If the option price is above the upper bound or below the lower bound, there are profitable opportunities for arbitrageurs.

Upper bounds: A call option gives the holder the right to buy a stock or an index for a certain price. As the option can never be worth more than the stock/index, the stock price/index is an upper bound to the option price.

C≤S, where

C= Price of call option

S = Current stock price/current index level

If the above relationship does not hold true, an arbitrageur cans easily make a risk less profit by buying the stock and selling the call option.

A put option gives the holder the right to sell a stock/index for a price X. No matter how low the stock price/index level becomes, the option can never be worth more than X. Therefore P≤X

If the above relationship does not hold true, the arbitrageur would make profit by writing puts.

Lower Bounds: The lower bound for the price of a call option is S – X (1 + r)T. If the price of a call is not worth at least this much, then it will be possible to make risk less profits.

If the call is available at a premium which is less than the lower bound, that is, if S#X(1 + r)r < C, the arbitrageur can buy the call and short the stock/index and earn risk less profits.

Consider a three month Nifty call option with a strike price of 1,060. The spot index stands at 1,015. The risk free rate of interest is 12 percent per annum. In this case, the lower bound for the option price is 1,150 – 1,060(1 + 0.12) –0.25 = 1,150 – 1,030.40 = Rs 119.60. If the call premium falls below Rs 119.60, arbitrage opportunities exist. Suppose the call is available at Rs 115, an arbitrageur can buy the call and short the index and thereby realize a cash flow of Rs 1,150 – 115 = Rs 1,035. If he invests Rs 1,035 for three months at 12 percent per annum then Rs 1,035 grows to Rs 1,066.05. At the end of the expiry of the options, the index can be either above 1,060 or below 1,060. If the index is above 1,060, the arbitrageur exercises his option and buys back the index at 1,060 making a profit of Rs 1,066.05 – 1,060 = 6.05. If the index is at say 1,050, the arbitrageur buys the index at the market price, that is, 1,050, and makes a higher profit of 1,066.05 – 1,050 = Rs 16.05.

The lower bound for the price of a put option is X(1+ r)-T – S

If the price of a put is not worth at least this, then it will be possible to make risk less profit.

If the put is available at a premium which is less than the lower bound, that is, if X(1+r)*T — S<P, the arbitrageur can buy the put the stock/index by borrowing money and earn risk less profits.

Consider a two month Nifty put option with a strike of 1,260. The spot index stands at 1,185. The risk free rate of interest is 12 percent per annum. The lower bound for the put option is X(1+r) –T – S =
1,260 (1+ 0.12)-0.166 – 1,185 = Rs 51.50. Suppose the put is available at a premium of Rs 40 the arbitrageur can borrow Rs 1,225 for two months to buy both the put the index. At the end of the two months, the arbitrageur will be required to pay at 12 percent per annum, Rs 1,249.50. At the end of the two months, the index can be either below 1,260 or above 1,260. If the index is below 1,260 the arbitrageur will exercise the option of selling the index at 1,260 repaying the loan amount of Rs 1,249.50 and thereby making a profit of Rs 10.50. If the index goes up to 1,270, the arbitrageur will discard the option, sell the index at 1,270 repay the loan amount of Rs 1,249.50 and make a higher profit of Rs 1,270 – Rs 1,249.50 = Rs 20.50.