Question

A cash-or-nothing option is another example of an exotic option in the 'binary options' category. For example, a cash-or-nothing European call pays off nothing if the asset price ends up below the strike price at maturity and pays a fixed amount, Q, if it ends up above the strike. Use the 4-step binomial tree to price the cash-or-nothing European call with the fixed payoff $20. Assume that the spot price is $100. The strike is $95. The time to maturity is 1 year. The risk-free rate is 5%, volatility 20%.

Answer 1

For 4 step tree, each step is 3months or 0.25 years & Assuming continuously compounded risk free rate

u = exp(s*t^0.5) = exp(0.2*0.25^0.5) =exp(0.1) = 1.105171

d = 1/u = 0.904837

So, the tree looks like

t=0	t=3	t=6	t=9	t=12	Value
				149.1825	20
			134.9859
		122.1403		122.1403	20
	110.5171		110.5171
100		100		100	20
	90.48374		90.48374
		81.87308		81.87308	0
			74.08182
				67.032	0

p= (exp(0.05*0.25)-0.904837)/(1.105171-0.904837) = 0.537808

So,

At t=9

Value of option when S=134.9859

=(p*value of option at S=149.1825+(1-p)*value of option at S=122.1403)*exp(-0.05*0.25)

=19.75156

Value of option when S=110.5171

=(p*value of option at S=122.1403+(1-p)*value of option at S=100)*exp(-0.05*0.25)

=19.75156

Value of option when S=90.4837

=(p*value of option at S=100+(1-p)*value of option at S=81.87)*exp(-0.05*0.25)

=10.62255

Value of option when S=74.08182

=(p*value of option at S=81.87+(1-p)*value of option at S=67.03)*exp(-0.05*0.25)

=0

At t=6

Value of option when S=122.1403

=(p*value of option at S=134.9859+(1-p)*value of option at S=110.5171)*exp(-0.05*0.25)

=19.5062

Value of option when S=100

=(p*value of option at S=110.5171+(1-p)*value of option at S=90.48)*exp(-0.05*0.25)

=15.33926

Value of option when S=81.87308

=(p*value of option at S=90.48+(1-p)*value of option at S=74.08)*exp(-0.05*0.25)

=5.641931

At t=3

Value of option when S=110.5171

=(p*value of option at S=122.1403+(1-p)*value of option at S=100)*exp(-0.05*0.25)

=17.36189

Value of option when S=90.48374

=(p*value of option at S=100+(1-p)*value of option at S=81.87308)*exp(-0.05*0.25)

=10.72237

SO, VALUE OF OPTION TODAY

=(p*value of option at S=110.5171+(1-p)*value of option at S=90.48374)*exp(-0.05*0.25)

=14.11561

So, value of this option is $14.12