A European call option has a strike price of $20 and an expiration date in six m
ID: 2758077 • Letter: A
Question
A European call option has a strike price of $20 and an expiration date in six months. The premium for the call option is $5. The current stock price is $25. The risk-free rate is 2% per annum with continuous compounding. What is the payoff to the portfolio, short selling the stock, lending $19.80 and buying a call option? (Hint: fill in the table below.)
Value of ST
Payoff
ST 20
ST > 20
How much do you pay for (or receive with) this portfolio at date 0? Is there an arbitrage opportunity?
If there is an arbitrage opportunity, then answer the following:
What is the minimum profit, expressed as a present value? Will investors trade to exploit the opportunity? If they will trade to exploit the opportunity, explain why security prices change and describe how security prices change. (30 points)
Make sure you answer all parts of this question.
Value of ST
Payoff
ST 20
ST > 20
Explanation / Answer
Payoff to the portfolio at maturity= Payoff on stock short sale+Payoff on call option+Payoff on loan
Payoff on stock short sale=(-ST)
Payoff on call option=max(ST-20,0)
Payoff on loan=19.8*exp(.02*.5) (T=.5 yrs=6 months)
Payoff to the portfolio= (-ST)+max(ST-20,0)+19.8*exp(.01)
i)ST > 20,Payoff to the portfolio= (-ST)+ST-20+19.8*exp(.01)=-20+19.8*exp(.01)= $0
ii)ST 20,Payoff to the portfolio= (-ST)+0+19.8*exp(.01)=-ST+19.8*exp(.01)
Pay to portfolio at date 0=Sell the stock at S0- lending $19.80 -call premium
Pay to portfolio at date 0=25-19.8-5=$0.2 is recieved.Hence there is arbitrage opportunity as there is risk less profit of $0.2 can be made.
The minimum payoff possible is $0 after maturity T=.5 yrs.Therefore the minimum profit possible is $0.2+Present value of $0=$0.2.
Yes investors shall trade to exploit this arbitrage opportunity they shall enter the short position that is they sell the stock this shall cause the price of the stock to decrease as the selling is more or that the supply of stock increases while demand remains the same which shall put downward pressure on the price.
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