Question

Alpha Tunis LLC, a newly publicly listed company in San Diego, is currently valued at $100. Over the next two six-month periods, analysts forecast it will go up by 10% or down by 10%. The current risk free rate is 8% per year. Given that the current strike price is $100, based on the Binomial Option Pricing model,

(i) Estimate the value of one-year European call option. Clearly show the binomial trees as part of your calculations.

(ii) Estimate the value of one-year European put option. Clearly show the binomial trees as part of your calculations.

(iii) Demonstrate if the Put-Call parity hypothesis holds. Clearly show your workings.

Please use 4 decimal places in your workings.  

Answer 1

Solution:

Given in the example is

- Current stock price is $100 i.e. S0 is 100.

- Two six months period. Do T for each period is 0.5

- Price will go up 10%. ie u is 1.1

- Price will do down by 10%. ie d is 0.9

- Risk Free rate is 8% p.a. ie  r is 0.08

- Stike Price is $100. So K is $100

We shall use the following two functions to solve the binomial tree:

A] The binomial tree for the European Call Option is as under:

Stock Value at Node B is calculated as 100*u i.e. 100*1.1. All up arrows have Previous Node Stock Value * 1.1; All Down arrows have Previous Node Stock Value * 0.9

Calculations at Node D,E,F,C for Option Value are pretty clear.

Now lets focus on Node B. Here the option value is calculated using above two formulas.

Lets calculate the p value.

p = (e^(0.08*0.5)-0.9)/(1.1-0.9) = 0.70405 <<-- we will be using this across the example

Now lets calculate the value of option at NOde B.

f = e^(-1*0.08*0.5)*(0.70405*21+(1-0.70405)*0) = 14.2054 <- Option value at NOde B

Now , moving to Node A.

f = e^(-1*0.08*0.5)*(0.70405*14.2054 +(1-0.70405)*0) = 9.6092 <<- Option value at Node A

So, Call option value at Node A = 9.6092

B] A] The binomial tree for the European PUT Option is as under:

Calculations at Node D,E,F are pretty clear.

Calculations at Node B:

f = e^(-1*0.08*0.5)*(0.70405*0+(1-0.70405)*1) = 0.284342

Calculations at Node C:

f = e^(-1*0.08*0.5)*(0.70405*1+(1-0.70405)*19) = 6.078944

Calculation at Node A:

f = e^(-1*0.08*0.5)*(0.70405*0.284342+(1-0.70405)*6.078944) = 1.920841

So, PUT option value at Node A = 1.920841

C] Put Call Parity

Put call parity hypothesis is :

We know from given data and our computations:

c = 9.6092

k = 100

e^-rt = 0.923116

p = 1.920841

S0 = 100

Putting in the above function:

LHS = 101.9208

RHS = 101.9208

Hence, Put Call Parity Hypothesis holds.

I hope you undetstand the solution. Thanks.

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