Lower Bound for Call Options

The lower bound for any non-dividend paying call option is:

This means that the lower bound is equal to the current stock price minus the options strike price multiplied by the natural e, to the power of negative risk-free interest rate multiplied by the options time to expiry.

Example:

Say that the current stock price = $20 (So)
And that the strike price = $18 (K)
The risk-free interest rate = 10% (r)

And the time to time to expiry = 1 year (T)

If you have done the calculations correctly then according to this rule the lower bound for this call option on a non-dividend paying stock is $3.71.

Let’s say that the market is quoting the European call option at $3.00. Such a price is less than the lower bound or “theoretical minimum”. What would happen is that an arbitrageur would short the stock (i.e. sell some of the shares underwritten this stock to buy them back at a future date), and then buy the call option.

This will provided the arbitrageur with a cash inflow of $20.00 - $3.00 = $17.00.

If this amount is then invested for 1 year at the market risk-free interest rate of 10% per annum, then the $17.00 will grow to 17е0.1 = $18.79. Therefore at the end of the year, when the option will expire if the stock price is greater than $18.00 (the strike price), then the arbitrageur will be able to exercise the option for $18.00. By doing this, it will enable them to close out the short position to make a profit of $18.79 - $18.00 = $0.79.

This means that the arbitrageur will buy back the stock at a cheaper rate than what he/she originally shorted (sold stock he/she didn’t own) and then invested that money at the risk-free rate, and then finally met the shorting obligations by buying the stock back with the aid of the option at the strike price.

What happens if the stock price is less than the strike price after 1 year (at time of option expiry?)

Then the arbitrageur will make an even GREATER profit. This is because he/she will be able tot buy back the stock at the market price which will be even cheaper than that of the strike price to close out the shorted stock position.

Say for example the stock price after 1 year = $17.00

The arbitrageur’s profit will then = $18.79 - $17.00 = $1.79

Which is $1 more than the previous example where the stock is greater than the strike price.

The additional difference in profit is the difference between the strike price and the stock price.

In a more formal approach this investment is demonstrated under 2 portfolios

• Portfolio A: 1 European call option plus an amount of cash equal to Ke-rT
• Portfolio B: 1 Share

Under portfolio A, the cash will be invested at the risk-free interest rate, and will then grow to the K in time T. If ST > K, then the call option will be exercised at the maturity of the option and as a consequence portfolio A will be worth ST.

However if on the other hand ST < K, then the call option will not be exercised at the maturity of the option, but instead the option will become worthless. The portfolio as a consequence will be worth K.

Therefore at time T, the portfolio A is worth:

Max (ST, K)

Under portfolio B on the other hand, it will be worth ST at the time T. Therefore because of this, portfolio A is always worth as much as, and can be worth more than, portfolio B at the time the option matures (expires). Hence, the worse thing that can happen to a call option is that it expires worthless, the value cannot be negative.