Question

A stock price is currently $80. It is known that at the end of four months it will be either $75 or $88. The risk free rate is 6 percent per annum with continuous compounding. What is the value of a four–month European put option that is currently $1 out-of-the-money? Use no-arbitrage arguments.

Answer 1

	European Put option is currently $1 Out-of-the money
	that means it is traded at $79 (Spot Price $80 - $1).
	Value of Put Option if after 4 months price will be $75 (Pd)
	= max ((StrikePrice - Price after 4 months),0)
	= max (($79 - $75),0)
	= max ($4,0)
	= $4
	Value of Put Option if after 4 months price will be $88 (Pu)
	= max ((StrikePrice - Price after 4 months),0)
	= max (($79 - $88),0)
	= max (-$9,0)
	= $0
	Let Probability of increase in Price be P
	So,
	P = (R-d) / (u-d)
	Where,
	R = Spot Price after 4 months
	= Spot Price * e^(6%*4months)
	= Spot Price * e^(6%*4/12)
	= $80 * e^(0.02)
	= $80 * 1.0202                     (Where Excel Formula for, e^(0.02) = EXP(0.02))
	= $81.616
	d = Lower Possible Price on Maturity = $75
	u = higher Possible Price on Maturity = $88
	P = (R-d) / (u-d)
	= ($81.616 - $75) / ($88 - $75)
	= $6.616 / $13
	= 0.50892
	Value of Put Option on expiry
	= Probability of Price Increase * Value of option when price increases
	+ (1-Probability of Price Incraeses) * Value of option when price decreases
	= 0.50892 * 0 + (1-0.50892) * 4
	= 0.50892 * 0 + 0.49108 * 4
	= 0 + 1.9643
	= $1.9643
	Value of Put Option today
	= Value of Put Option on Expiry / e^(6%*4months)
	= $1.9643 / e^(6%*4/12)
	= $1.9643 / e^(0.02)
	= $1.9643 / 1.0202                          (Where Excel Formula for, e^(0.02) = EXP(0.02))
	= $1.9254