Question

A stock price is currently $50. Over each of the next two three-month periods, it is expected to increase by 10% or fall by 10%. Consider a six month American put option with a strike price of $49.5. The risk free rate is 6%. Work out the the two step binomial option pricing fully and fill in the asked questions. (Work out using 4 decimals and then enter your answers rounding to two decimals without $ sign)

a) S0uu= Blank 1

b) ƒuu= Blank 2

c)  S0dd= Blank 3

d) ƒdd= Blank 4

e)  S0u= Blank 5

f) ƒu= Blank 6

g)  S0d= Blank 7

h) ƒd= Blank 8

i) Option price today (ƒ) = Blank 9

j) At which node will American put option exercised early? (Enter A, B, C or None) Blank 10

Answer 1

	Two Step Binomial Tree
	r=	risk free rate	6%
	t=	Length of time of a step=delta t	0.25	=3 months
	S0=	Current Stock Price	50
	f=	Current Price of an Option on the stock.	49.5
	u=	Upward Stock movement , u>1	1.1
	d=	Downward stock movement , d<1	0.9
	Sou=	Stock price after one up step	55	Ans e
	Souu=	Stock price after two up steps	60.5	Ans a
	Sod=	Stock price after one down step	45	Ans g
	Sodd=	Stock price after two down steps	40.5	Ans c
	Sud =	Stock Price after one step up & One step down	49.5
	f=	Option price today	2.882	as per calc =Ans i
	fu=	Payoff from option after one step up	0	as per calc =ans f
	fuu=	Payoff from option after two steps up	0	Ans b -option not exercised
	fd=	Payoff from option after one step down	4.5	=49.5-45 =Ans h
	fdd=	Payoff from option after two steps down	9	=49.5-40.5 Ans d
	fud=	Payoff from option after one step up & one step down	0	=49.5-49.5

	Option Price at Step A			Binomial Tree		Step C
	f=e^-2rt [ p^2*fuu + 2*p(1-p)*fud + (1-p)^2*fdd ]						Suu	60.5
					Step B		fuu	0
	where				Su	55
	p= (e^rt-d)/(u-d)				fu
			Step A				Sud	49.5
			S0	50			fud	0
	Now p=(e^-0.06*0.25-0.9)/(1.1-0.9)		F0	2.882
	or p=0.42556
	So Option price at step A=				Sd	45
		e^-0.06*2*0.25[0.42556^2*0+2*0.425568(1-0.42556)*0+(1-0.42556)^2*9]			fd	4.5
		=0.970446*2.969832 =2.882					Sdd	40.5
							fdd	9
	So f=2.882

	Option Price at Sept B =
	fu= e^-rT[p*fuu +(1-p)*fdd]
	where
	p= (e^rT-d)/(u-d)
	p=e^-0.06*0.25
	p=0.42556
	Now fu=e^-0.06*0.25*[0.42556*0+(1-0.42556)*0]
	or fu=0

So option should be exercised at Step C when the price in down 10% each step ---Ans j