Question

A stock selling at $50 will either go up 20% or go down 10% each month for the next 3 months. The risk-free rate is 12% per annum with continuous compounding. Assume that a European put option is available for a strike price of $55 and a maturity of 3 months.

a. Use a 3-step binomial model to calculate the price of the put option.

Answer 1

q=e(-rt)-d/u-d

Here u=1.2 , d=0.9 ,x= 50 r=.12 t=.25(3/12) strike price=55

so q=e(-(.12*.25)-.9/12-.9

=e(-.03)-.9/.3

=.97044-.9/.3

=.2348

Value of put option at point 2

p2=e(-rt)*(p*Pupup + (1-q)Pupdwn)

p= price of put option

At Pupup:
50*1.2*1.2=72 so Pupup=0(more than strike price)

At Pupdwn:
50*1.2*.9=54 so Pupdwn=1(55-54)

At Pdwndwn:
50*.9*.9=40.5 so Pdwndwn=14.5(55-40.5)

So
p2=.97044*(.2348*0+(1-.2348)*1)

= 0.74258
p3=.97044*(.2348*1+(1-.2348)*14.5)

=10.995

So value of put option is :
=.97044*(.2348*0.74258+(1-.2348)*10.995)
=.97044(0.1743+8.413374)
=.97044*8.5877
=8.3338
=8.34