Question

The spot price per share is $115 and the risk free rate is 5% per annum on a continuously compounded basis. The annual volatility is 20% and the stock does not pay any dividend. All options have a one-year maturity. In answering the questions below use a binomial tree with three steps. Each step should be one-third of a year.

1)Using the binomial tree, compute the price at time 0 of a one-year European put option on 100 shares of stock with a strike price of $115 per share.

Answer 1

For a Binomial Model, The value of a Put option under the risk neutral probability measure is defined as the Discounted value of the expected payoff.

Or

Put option value Pt= e^-rt * E_Q[Payoff(X_T)]

BINOMIAL PRICING (“TREE") MODEL Risk-neutral probabilities erat – d Up-step probability => , u-d where u zpovar+qar and dze-ovar+qat

Here, u= Up step factor

d= Down step factor

q= up step probability(as defined above)

The following Put price has been computed on excel:

sd	20%		dt=	0.33
Rf	5%		u	1.122401	=EXP(B23*SQRT(E23))					Share Price
Spot Price(St)	115.00		d	0.890947	=1/E24					Payoff
Strike(K)	115		q	0.543777	=(EXP(B24*E23)-E25)/(E24-E25)
			t=0	t=1	t=2	t=3
							Formulas
						162.608	=F35*$E$24
						0	=MAX($B$26-G33,0)
					144.8751		=E37*$E$24
					0		=EXP(-$B$24*$E$23)*(G34*$E$26+G38*(1-$E$26))
				129.0761		129.0761	=F39*$E$24
				2.524719		0	=MAX($B$26-G37,0)
			115.00		115		=E41*$E$24
		Pt=	7.091836		5.626959		=EXP(-$B$24*$E$23)*(G38*$E$26+G42*(1-$E$26))
				102.4589		102.4589	=F43*$E$24
				12.79667		12.54107	=MAX($B$26-G41,0)
					91.28551		=E41*$E$25
					21.81371		=EXP(-$B$24*$E$23)*(G42*$E$26+G46*(1-$E$26))
						81.33057	=F43*$E$25
						33.66943	=MAX($B$26-G45,0)

The price for put on 1 share is Pt= 7.091836.

Therefore price for Put on 100 shares is Pt=709.1836.