a) A 6 month European call option on dividend paying stock is currently selling for $5.
The stock price is $50, the strike price is $45, and a dividend of $0.9 is expected
in every 2 months. The risk free interest rate is 12% per annum for all maturities. Are
little
there any arbitrage opportunities? Explain.
explanation
Answers
Answer:
Options Arbitrage
As derivative securities, options differ from futures in a very important respect. They represent rights rather than obligations � calls gives you the right to buy and puts gives you the right to sell. Consequently, a key feature of options is that the losses on an option position are limited to what you paid for the option, if you are a buyer. Since there is usually an underlying asset that is traded, you can, as with futures, construct positions that essentially are riskfree by combining options with the underlying asset.
Exercise Arbitrage
The easiest arbitrage opportunities in the option market exist when options violate simple pricing bounds. No option, for instance, should sell for less than its exercise value. With a call option: Value of call > Value of Underlying Asset � Strike Price
With a put option: Value of put > Strike Price � Value of Underlying Asset
For instance, a call option with a strike price of $ 30 on a stock that is currently trading at $ 40 should never sell for less than $ 10. It it did, you could make an immediate profit by buying the call for less than $ 10 and exercising right away to make $ 10.
In fact, you can tighten these bounds for call options, if you are willing to create a portfolio of the underlying asset and the option and hold it through the option�s expiration. The bounds then become:
With a call option: Value of call > Value of Underlying Asset � Present value of Strike Price
With a put option: Value of put > Present value of Strike Price � Value of Underlying Asset
Too see why, consider the call option in the previous example. Assume that you have one year to expiration and that the riskless interest rate is 10%.
Present value of Strike Price = $ 30/1.10 = $27.27
Lower Bound on call value = $ 40 - $27.27 = $12.73
The call has to trade for more than $12.73. What would happen if it traded for less, say $ 12? You would buy the call for $ 12, sell short a share of stock for $ 40 and invest the net proceeds of $ 28 ($40 � 12) at the riskless rate of 10%. Consider what happens a year from now:
If the stock price > 30: You first collect the proceeds from the riskless investment ($28(1.10) =$30.80), exercise the option (buy the share at $ 30) and cover your short sale. You will then get to keep the difference of $0.80.