Question

Suppose a stock was selling today at $40 and the interest rate was 5%. You found that a 1 year European Call on this stock with EX=23 was selling for $26 and a 1 year European Put on this stock with EX=23 was selling for $10.

Describe a way to construct a risk-free arbitrage opportunity for yourself.

Answer 1

Now,
Present Value of Exercise Price assuming Rate of Interest Compounded Continuously
= Exercise Price / e^rt
Where,
Exercise Price = $23
r = Rate of Interset = 5% = 0.05
t = Period = 1 Year
So,
e^rt
= e^0.05*1
= e^0.05
Value of e^0.05 using excel Formula
=EXP(0.05)
= 1.05127
So,
Present Value of Exercise Price using Rate of Interest Compounded Continuously
= Exercise Price / e^rt
= $23 / 1.05127
= $21.88
Value of Stock & Put Option
= $40 + $10
= $50
Value of Call and Present Value of Exercise Price
= $26 + $21.88
= 47.88
There is violation of Put-Call Parity Theorem, as Value of Stock and Put Option
is not equal to Value of Call and Present Value of Exercise Price.
Call plus risk free investments are underpriced and stock plus put are overpriced.
So, Arbitrage Opportunity will involve the following -
1) Buy Call Option and Risk Free Investments
2) Short sell of Stock and Sell Put options
Arbitrage Gain
Sale of Stock	$ 40.00
Sale of Put Option	$ 10.00
Buy a Call Option	$(26.00)
Buy a Risk Free Investments	$(21.88)
(Present Value of Exercise Price)
Total Arbitrage Gain	$    2.12