Question

Using a binomial tree, what is the price of a $40 strike 6-month put op-
tion, using 3-month intervals as the time period? Assume the following
data: S = 37:90; r = 0:05; = 32:1%
(a) 3.52
(b) 3.66
(c) 3.84
(d) 3.91

Answer 1

D is right answer

value given in question

Strike price = K = $40

current price =S0= $37.90

price at down node = Sd = $32.1

here we have to use binomial tree so we need value of u and value of d

value of d = Sd/S0 =$32.1/ $37.9 = 0.847

value of u = 1/d = 1/0.847 =1.1806

now find the probability of up (P) = [(1+r) t- d )] / (u - d) = [(1+(0.05*0.5)) - 0.847)] / (1.1806 - 0.847 ) = [(1.0250 - 0.847)] / (1.1806 - 0.847 ) = 0.5335

where r = risk free rate and t is maturity time of option

probability of down (Q) = 1 - P = 1 -0.5335=0.4665

Su = share value at up node = S0 * u = $37.5 * 1.1806 = $44.272

Suu = value at up and up node Su * 1.1806 = $44.272 * 1.1806 = $52.268

Sud = Value st up and down node = Su* d = $44.272 * 0.847 = $37.5

Sdd = value at down and down = Sd * 0.847 = $32.1 * 0.847 = $27.1887

Puu = put value at up up node = Max(0, X - Suu) = Max (0, $40 - $52.268 ) = 0

Pud = put value at up and down node =Max(0, X - Sud) = Max (0, $40 - $37.5) = $2.5

Pdd = value of put option at down and down node = Max(0, X - Suu) = Max (0, $40 -$27.1887) = $12.8113

Pu = put value at up node = [(Puu * P) + (Pud*Q)] / 1+rt=[(0 * 0.5335) + ($2.5 * 0.4655)] / 1+(0.05*0.25)= $1.1493

Pd = put value at up node = [(Pud * P) + (Pdd*Q)] / 1+rt=[($2.5 * 0.5335) + ($12.8113 * 0.4655)] / 1+(0.05*0.25)= $7.2073

P0 = current value of put = [(Puu * P) + (Pud*Q)] / 1+rt=[($1.1493* 0.535) + ($7.2073 * 0.4655)] / 1+(0.05*0.25)= $3.92

where p = probability of up

Q= probability of down

t = interval between node

hence option d is right