Question

A. σ = 0.4, r = 0.1,

Δt = 1/52, S0 = 40. Construct a binomial tree with three periods (i.e., periods 0,1 and 2. Assume that each time period is one week.) [Answer with two decimal points! i.e., 40.36]

B. Price a two week European Put option with a 41 strike by hand

Answer 1

Given that, S₀=40, K=41, r=0.10, T= 2 weeks = 0.03846 years, n=2

Volatility, σ = 0.40

Now, u = e^{σ x sqrt(t),} where t= T/n = 0.03846/2 = 0.01923

          = e^{0.40 x sqrt(0.01923)}

          = 1.0570

Similarly, d = 1/u = 1/1.0570 = 0.9460

Also, a = e^{r x t} = e ^{0.10x 0.01923} = 1.0019

p = (a-d) / (u-d) = (1.0019-0.9460)/(1.0570-0.9460)

             = 0.0559 / 0.1110

             = 0.5035

1-p = 1-0.5035 = 0.4965

Refer the attached diagram above of two-Step binomial tree, we need to calculate the value of put at various nodes.

For this, we need to first evaluate the put option prices starting at Node D, E, F respectively.

With Strike price as 41:

Value of Put at Node D, f_u = max{(41 - 44.69),0} = 0.00

Value of Put at Node E, f_d = max{(41 - 40.00),0} = 1.00

Value of Put at Node B,        

f = e^-rt * [p * f_u + (1-p) * f_d]

           = e^{-0.10 x 0.01923}*[0.5035*0.00 + 0.4965*1.00]

           =0.9981*0.4965

           = 0.4956

Thus, Put Option price at Node B = 0.4956

Similarly, we calculate value of put at Node C

Value of Put at Node E, f_u = max{(41 - 40.00),0} = 1.00

Value of Put at Node F, f_d = max{(41 - 35.80),0} = 5.20

Value of Put at Node C,        

f = e^-rt * [p * f_u + (1-p) * f_d]

           = e^{-0.10 x 0.01923}*[0.5035*1.00 + 0.4965*5.20]

           =0.9981*3.0855

           = 3.0796

Thus, Put Option price at Node C= 3.0796

Similarly, we calculate value of Put at Node A using the values at Node B and Node C

Value of Put at Node B, f_u = 0.4956

Value of Put at Node C, f_d = 3.0796

Value of Put at Node A,        

f = e^-rt * [p * f_u + (1-p) * f_d]

           = e^{-0.10 x 0.01923}*[0.5035*0.4956 + 0.4965*3.0796]

           = e^{-0.10 x 0.01923}*[0.2495 + 1.5290]

           =0.9981*1.7786

           = 1.7752

Thus, Put Option price at Node A= 1.7752