As one time step be equal to 4 months

With u=1.11 and d =0.9 the stock lattice, value of call option at t=2  is given below

		32.0346	4.0346	0.0000
	28.86	25.9740	0.0000	2.0260
26.00	23.40	21.0600	0.0000	6.9400
t=0	t=1	t=2	Value of Call option at t=2	Value of Put option at t=2

For European Call option

Under No arbitrage approach,

From t=1 to t=2 when stock price is $28.86

Let X shares be purchased and one call option be shorted to create the no arbitrage portfolio

So, X*32.0346- 4.0346 = X*25.974 -0

=> X = 0.6657

(At this node the Riskless portfolio consists of Long position in 0.6657 Stocks and Short position in 1 Call option)

So, Value of option(C1h) at t=1 when stock price is $28.86 is given by

0.6657*28.86 - C1h = 0.6657*25.974/exp(0.06*4/12)

=> C1h= $2.2636

Similarly From t=1 to t=2 when stock price is $23.4

X*25.974- 0 = X*21.06 -0

=> X = 0

(At this node the Riskless portfolio consists of Long position in 0 Stocks and Short position in 1 Call option)

So, Value of option(C1L) at t=1 when stock price is $23.40 is given by

0*23.4 - C1L= 0*21.06/exp(0.06*4/12)

=> C1L= 0

and  From t=0 to t=1when stock price is $26

X*28.86- 2.2636 = X*23.40 -0

=> X = 0.4146

(At this node the Riskless portfolio consists of Long position in 0.4146 Stocks and Short position in 1 Call option)

So, Value of option(C) at t=0 when stock price is $26 is given by

0.4146*26 - C= 0.4146*23.4/exp(0.06*4/12)

=> C= $1.27 (Value of 8 month European Call option)

For European Put option

Under No arbitrage approach,

From t=1 to t=2 when stock price is $28.86

Let X shares be purchased and one put option be purchased to create the no arbitrage portfolio

So, X*32.0346 + 0 = X*25.974 +2.026

=> X = 0.3343

(At this node the Riskless portfolio consists of Long position in 0.3343 Stocks and Long position in 1 Put option)

So, Value of option(P1h) at t=1 when stock price is $28.86 is given by

0.3343*28.86 + P1h = 0.3343*32.0346/exp(0.06*4/12)

P1h = 0.8492

Similarly From t=1 to t=2 when stock price is $23.40

X*25.974+2.026= X*21.06 +6.94

=> X = 1

(At this node the Riskless portfolio consists of Long position in 1 Stock and Long position in 1 Put option)

So, Value of option(P1L) at t=1 when stock price is $23.4 is given by

1*23.40 + P1L= (1*25.974+2.026)/exp(0.06*4/12)

=> P1L= 4.0456

and  From t=0 to t=1when stock price is $26

X*28.86 + 0.8492 = X*23.40 + 4.0456

=> X = 0.5854

(At this node the Riskless portfolio consists of Long position in 0.5854 Stocks and Long position in 1 Put option)

So, Value of option(P) at t=0 when stock price is $26 is given by

0.5854*26 +P= (0.5854*28.86+0.8492)/exp(0.06*4/12)

=> P= $2.17

From put call parity

C+K*exp(-rt) = P +S

LHS = 1.27+ 28*exp(-0.06*8/12) = $28.17

RHS = 2.17+26 = $28.17

As LHS = RHS , the Put call parity holds

Under risk neutral valuation ,the risk neutral probability is given by

p = (exp(0.06*4/12)- 0.9)/(1.11-0.9) = 0.5724

So, Value of European call option

= (p^2*value of option when stock is $32.0346 + 2*p*(1-p)*value of option when stock is $25.974 + (1-p)^2*value of option when stock is $21.06) / exp(0.06*8/12)

= (0.5724^2*4.0346)/exp(0.06*8/12) = $1.27

Under risk neutral valuation

Value of European put option

= (p^2*value of option when stock is $32.0346 + 2*p*(1-p)*value of option when stock is $25.974 + (1-p)^2*value of option when stock is $21.06) / exp(0.06*8/12)

= (2*0.5724*0.4276*2.026+0.4276^2*6.94)/exp(0.06*8/12)

=$2.17