Question

The current price of stock XYZ is 100. In one year, the stock price will either be 120 or 80. The annually compounded risk-free interest rate is 10%. i. Calculate the no-arbitrage price of an at-the-money European put option on XYZ expiring in one year. ii. Suppose that an equivalent call option on XYZ is also trading in the market at a price of 10. Determine if there is a mis-pricing. If there is a mis-pricing, demonstrate how you would take advantage of the arbitrage opportunity.

Answer 1

Answer (i)
XYZ Current Price	S	100
Strike Price	E	100
Risk-free Rate	Rf	10%
Up Factor	u	1.2
Down Factor	d	0.8
Up Price	uS	120
down Price	dS	80
R	(1+Rf)^1	1.1
Let Take upside probability = p
Risk Neutral Probability = p = (R - d) / (u-d)
Risk Neutral Probability = p = (1.1-0.80) / (1.20-0.80)
Risk Neutral Probability = p = 0.75
1- p = 0.25
If Stock reach 120, pay off 0 with probability 0.75
If Stock reach 80, pay off 100 -80 = 20 with probability 0.25
Using Binominal function, Put Option Price
= (0*0.75 + 20*0.25) / 1.1 = 4.55
Answer (ii)
Using Bionominal function, no-arbitrage call price
= (20*0.75 + 20*0.25) / 1.1 = 13.63
Yes, there is mis-pricing.