Question

Consider a two-period binomial model in which a stock currently trades at a price of K65. The stock price can go up 20 percent or down 17 percent each period. The risk-free rate is 5 percent. Calculate the price of a put option expiring in two periods with exercise price of K60.

Answer 1

Given Data Point:

Current Stock Price = S = K65

Strike Price = X = K60

Risk Free Rate = r = 5%

Up Move = 20% ; Hence, every period price can increase by a factor of (1 + 20%) = u = 1.2

Down Move = 17% ; Hence, every period price can also decrease by a factor of (1 - 17%) = d = 0.83

A two period binomial tree can be constructed as given below:

Here, uS = 1.2 x 65 | dS = 0.83 x 65 | uuS = 1.2 x 1.2 x 65 | udS=duS = 0.87 x 1.2 x 65 | ddS = 0.83 x 0.83 x 65

Understanding of the Concept:

In a multi-stage binomial option pricing model, the option is valued at each and every stage, starting from the last stage to arrive at the final value of the option. The stepwise method is given below:

Step 1: Pay-off from the options are calculated at the last stage (here Stage 2).

Payoff P++ = Max (X - uus , 0) ...... where, X = strike price & uuS = Stock price @ stage 2 post consecutive up-moves

Step 2: Using a weighted average probability and the present value measure, the value of option is calculated at the preceding stage (here Stage 1). For Ex: Value of P+ will as given below

Option (t-1) = P+ = [ (pi x P++) + ((1 - pi) x P+-) ] / (1 + r) ..... where, pi = Up-Move Probability

Step 3: Repeat Step 2 till you reach Stage 0 to get the current price of the option.

Now let us calculate the Value of Put Option for this question using the abovementioned steps

Step 1: Pay-off from the options are calculated at the last stage

P++ = Max (60 - 93.60 , 0) = 0

P+- = P-+ = Max (60 - 64.74 , 0) = 0

P-- = Max (60 - 44.78 , 0) = 15.22

The binomial tree now looks like:

Step 2: We need to calculate P+ and P- using the formula mentioned in Step 2 above.

But, to use that formula, first, we need to find the Up-move Probability = pi

pi = (1 + r - d) / (u - d)

pi = ( 1 + 5% - 0.83) / (1.2 - 0.83)

pi = 59.45%

P+ = [ (pi x P++) + ((1 - pi) x P+-) ] / (1 + r)

P+ = [ (59.45% x 0) + ((1 - 59.45%) x 0)] / 1.05

P+ = 0

P- = [ (pi x P-+) + ((1 - pi) x P--) ] / (1 + r)

P- = [ (59.45% x 0) + ((1 - 59.45%) x 15.22) ] / 1.05

P- = 5.87

The binomial tree now looks like:

Step 3: Repeating Step 2, till we arrive at Stage 0 i.e. P.

In this case, using the formula in Step 2 with P+ & P- values, we can calculate P.

P = [ (pi x P+) + ((1 - pi) x P-) ] / (1 + r)

P = [ (59.45% x 0) + ((1-59.45%) x 5.88) ] / 1.05

P = 2.27 ...... Value of Put Option .... Answer

The final binomial tree now looks like:

Hence, the price of Put Option expiring in two periods is K2.27.