Question

Suppose that the annual volatility (σ) of spot silver is 20% and that we are trying to price a European call option written on silver. The spot price is $17 per ounce, the exercise price of the option is $18, and it has exactly 12 months till expiration. Silver pays no dividend and the continuously compounded risk-free rate of interest is 2% per year.

a) Using time intervals of six months ( t = 1/2 years), construct the binomial price tree for silver for twelve months. Also calculate the risk-neutral probabilities.

b) Determine the call price today using binomial pricing. Make sure that you construct the call price tree fully. What is the risk-neutral probability, calculated at t=0, that this call option will be in the money at expiration?

Answer 1

Two Step Binomial Tree
r=	risk free rate	2%
t=	Length of time of a step=delta t	0.5	Half year in each step
Sigma=	Price volatility=	20%
S0=	Current Stock Price	17
K=	strike price	18
f=	Current Price of an Option on the stock.
u=	Upward Stock movement , u>1
d=	Downward stock movement , d<1
Sou=	Stock price after one up step
Souu=	Stock price after two up steps
Sod=	Stock price after one down step
Sodd=	Stock price after two down steps
Sud =	Stock Price after one step up & One step down
f=	Option price today
fu=	Payoff from option after one step up
fuu=	Payoff from option after two steps up
fd=	Payoff from option after one step down
fdd=	Payoff from option after two steps down
fud=	Payoff from option after one step up & one step down
Option Price at Step A
f=e^-2rt [ p^2*fuu + 2*p(1-p)*fud + (1-p)^2*fdd ]

Findin the value of a
where a=	e^r*delta t
so a=e^0.02*0.5
a =1.01005
u= e^sigma*Sq rt of delta t
so u=e^0.20*Sqrt0.5
or u=1.1519
d= e^-sigma*Sq rt of delta t =1/u
d=1/1.1519 =0.8681
p= (a-d)/(u-d)
p=(1.01005-0.8681)/(1.1519-0.8681)=
p=0.50018
1-p=0.4998
So the risk neutral probability of price going up=0.50018 and going down =0.4998	Ans a

Option Price at step A
f=e^-2rt [ p^2*fuu + 2*p(1-p)*fud + (1-p)^2*fdd ]			Binomial Tree-Ans a		Step C
f0=e^-2*0.02*0.5*[0.50018^2*4.557+2*0.50018*(1-0.50018)*0+(1-0.50018)*0]			delta t=0.5 years each	Suu	22.557
f0= 1.117				Step B		fuu	4.557
				Su	19.582
So value of option today =$1.117 per ounce ( Ans b)			fu	2.257
Option Price at Sept B =
fu= e^-rT[p*fuu +(1-p)*fdd]		Step A				Sud	16.999
=e^-0.02*0.5*[0.50018*4.557+(1-0.50018)*0 ]	S0	17			fud	0
fu=2.257		F0	1.117
Risk neutral probability of the option being in money at t=0 is =p=0.50018
				Sd	14.758
				fd	0
						Sdd	12.811
						fdd	0