An American and European put both have strike price $25, the same underlying ass

Question

An American and European put both have strike price $25, the same underlying asset and the same time to expiry. Consider a three-step binomial model. The underlying asset price is, in crr notation, S = $24 , u = 1.15 and d = 1/u . The return over each time step is R = 1.05 .

(a) Construct a three-step binomial pricing tree for the European put and find the premium then constuct a three-step binomial pricing tree for the American put and find the premium.

(b) If the European put is a down-and-out barrier option with barrier at the asset price $20. Construct a three-step binomial pricing tree for this barrier European put and find the premium.

(c) If the American put is a down-and-out barrier option with barrier at the asset price $20. Construct a three-step binomial pricing tree for this barrier American put and find the premium.

Explanation / Answer

Soln : (a) Step 1: Given parameters, Strike price, K = $25, Current Price S = $24 , upward movement = 1.15, downward d=1/u, Let p be the probability of price go up and 1-p for price go down

r = 5% as it is given that ineach step return = 1.05, we can say that rt = 0.05

As per binomial tree equation, p = (ert -d)/(u-d)  = (e0.05 - 0.87)/(1.15-0.87) = 0.65

1-p = 0.35

Binomial tree :

(a) Step 2: Strike price = 25, so at each node at t=3 we will calculate the value of option and using probability of up and down and discount rate, we will calculate the values at each node like at t =2 , 2nd value of 1.375 = (0.65*0 +0.35*4.13)/e0.05 and accordingly we calculate at each node.

Please note this calculation is for only European call option.

We can see here that European put option premium = 1.19

For american put option , we need to calculate the option value if exercised at that point :

Now, we compare these values from the discountng values at each node , so that execution value at that node can be calculated. like in year 2 3rd value of 1 will be replaced by the value of 1.375 that comes from (0.35*4.13 +0)/ert = 1.375, similarly at each node:

So, we can say that premium for American option put = 1,375

(b) If Europena is down and out barrier put option , barrier at $20 Down and out barrier option means that option is active from spot only when it is above the barrier price and move down to go for null and void.

As we have calculated in part similarly here we will calculate the table of premium at each node, only difference will be that wherever the value of underlying goes below 20, we will consider premium as 0, please refer table, compare it with 1st table:

We can see here that value of this european put option is 0.566

c) Again in the same manner comparison to be done for American option as done in part(A) and values at each node will be calculated with taking barrier at $20, please refer table calculated for premium :

Again here we can see that value of american down and out barrier put option not changed from american put option = 1.375.

0 1 2 3 36.50 31.74 27.6 27.6 24 24 20.87 20.87 18.15 15.78

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