Question

Solve the below problems on Excel. 1. The current price of Bank of America (BAC) is $10.The annual stan- dard deviation is 12%. The continuously compounded risk-free rate is 5% per year. Assume BAC pays no dividends. (a) Compute the value of a 1 year European call option with a strike price of $7 using a one-step (At = 1) binomial model. (b) Compute the value of a 1 year European call option with a strike price of $7 using a two-step (At = 0.5) binomial model. (c) Compute the value of a 1 year European call option with a strike price of $7 using a three-step (At = 0.25 ) binomial model. (d) Compute the value of a 1 year European call option with a strike price of $7 using a four-step (At = 0.125 ) binomial model. (e) Compare the option prices from (a), (b), (c), and (d). What are your findings?

Answer 1

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

Or

Call option value Ct= 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)

a)

sd	12%
Rf	5%
BAC(St)	10.00
Strike(K)	7
a	When	dt=	1
		u	1.127497	=EXP(B1*SQRT(D7))					Share Price
		d	0.88692	=1/D8					Payoff
		q	0.683154	=(EXP(B2*D7)-D9)/(D8-D9)
			t=0	t=1
				11.27497	=D17*D8
				4.274969	=MAX(E15-$B$4,0)
			10.00	=B3
		Ct=	3.341394	=EXP(-$B$2*$D$7)*(E16*$D$10+E20*(1-$D$10))
				8.869204	=D17*D9
				1.869204	=MAX(E19-$B$4,0)

b) (continuing the valuation of the tree in the same manner)

b	When	dt=	0.5
		u	1.088557	=EXP(B23*SQRT(D29))
		d	0.918647	=1/D30
		q	0.627791	=(EXP(B24*D29)-D31)/(D30-D31)
			t=0	t=1	t=2
					11.84956
					4.84956
				10.88557
				4.058399
			10.00		10
		Ct=	3.341394		3
				9.186475
				2.359305
					8.439132
					1.439132

c)(continuing the valuation of the tree in the same manner)

sd	12%		dt=	0.25
Rf	5%		u	1.061837	=EXP(B23*SQRT(E23))
BAC(St)	10.00		d	0.941765	=1/E24
Strike(K)	7		q	0.589762	=(EXP(B24*E23)-E25)/(E24-E25)
			t=0	t=1	t=2	t=3
						11.97217
						4.972174
					11.27497
					4.361924
				10.61837		10.61837
				3.791196		3.618365
			10.00		10
		Ct=	3.257639		3.086955
				9.417645		9.417645
				2.590476		2.417645
					8.869204
					1.95616
						8.352702
						1.352702

d)(continuing the valuation of the tree in the same manner)

sd 0.125 1.043339 =EXP(B23 SQRT(E23)) 0.958461 = 1/E24 0.56326 =(EXP(B24E23)-E25)/(E24-E25) 6 300 Share Price Payoff BAC(St) Strike(K) t=0 t=1 t=2 t=3 t=4 11.84956 4.84956 11.35734 4.400955 10.88557 3.972524 10.88557 3.885568 10.43339 3.56342 10.43339 3.477006 10.00 3.172831 3.086955 9.58461 2.714637 9.58461 2.628224 9.186475 2 27343 9.186475 2.186475 8.804878 1.848491 8.439132 1.439132

Note: I'm not answering the last part, as the policy asks to answer only the 1st 4 sub parts.