Question

Consider a 2-period binomial non-recombining model. Let r = 0.05 be the discrete interest rate for each of the periods, S(0) = 100 and suppose that the stock price follow four possible scenarios:

Scenario	S(1)	S(2)
?1	Su	Suu
?2	Su	103
?3	90	Sdu
?4	90	80

It is known that the risk-neutral probability for each scenario ?i , i = 1, 2, 3, 4 satisty P ? (?1) = 0.2, P? (?2) = 0.4, P? (?3) = 0.3, P? (?4) = 0.1.

(a) Draw the tree of the stock prices using S^u , S^uu, and S^du, and find the risk-neutral probability for each route on the tree.

(b) Find the prices S^u , S^uu, and S^du

(c) Find today’s price of a European call option with strike price K = $100 maturing after two steps.

(d) Find today’s price of an American put option with strike price $110 and expiration date in two steps. Should the American option be exercised early? If so, when?

Answer 1

First we draw the binomial tree with the given data.

The Stock prices in next period are calculated as product of the up-move/down-move factor and the current stock price.

Here, for period 0 to 1,

The up-move factor can be calculated from d1 as below.

So, Su = u1 * S(0) = 1.11*100 = 111 (Note: The value of u1 is rounded to two decimal places)

For period 1 to 2 in down-move section i.e. when stock price is 90

Down-move factor, dd2 = 80/90 = 0.89

=> Up-move factor, du2 = 1/dd2 = 1.12

So, Sdu = 90 * 1.12 = 100.8

For period 1 to 2 in up-move section i.e. when stock price is Su

Down-move factor, ud2 = 103/Suu = 103/111 = 0.93

Up-move factor, uu2 = 1/ud2 = 1.08

So, Suu = 111*1.08 = 119.88

The completed tree with stock prices looks as below

Risk Neutral probability:

Consider that the risk neutral probability at a node (where stock price is known or has been calculated) is "p" for an up-move, the risk-neutral probability for down-move will be "1-p".

The risk neutral probability at a point is given as below

In this question, r = 0.05, =1 (since every period is considered single unit),

Values of u & d will depend on the path selected and can be taken from the completed tree above.

By substituting the values and calculating, we get the below results

p1 = 0.72

p2 = 0.81

p3 = 0.70

Therefore the tree with the risk neutral probability looks as below.

(c) A European option can only be exercised at the time of maturity and not before. We exercise the call option only if the stock price is more than the strike price (in this case $100). The value of the option at time of exercising becomes the difference between the stock price and the strike price at that instant. In case the option is not exercised, the option value becomes 0.

Then we calculate the current price of the European option by multiplying them with the risk neutral probabilty and discounting it.

At period 2,

Option value at scenario 1, V1 = $(119.88-100) = $ 19.88

Option Value at scenario 2, V2 = $(103-100) = $3

Option value at scenario 3, V3 = $(100.8-100) = $0.80

Option value at scenario 4, V4 = $ 0 since the stock price is less than strike price, we do not exercise the option

At period 1,

Option value in Up-move (i.e. Stock price = $111), Vu = (p2*V1 + (1-p2)*V2)*(1/1.05) = $15.88

Also at the same point, Vu = $(100-111) = $11

Since the exercising of option in period 2 gives higher value we take $15.88 for calculating the final option value

Option value in Down-move (i.e. Stock price = $90), Vd = (0.7*0.8 + 0.3*0)*(1/1.05) = $0.53

In down-move we do not exercise the option at period 1 since the value is less than $100. So the option value at period 1 remains $0.53

At period 0,

Value of option, V0 = (0.72*15.88 + 0.28*0.53) * (1/1.05) = $ 11.03

(d) In an American option, we can exercise the option any time before the expiration. We exercise the put option if the strike price is higher than the stock price at the instant of exercising it. So we need to check the stock price at every point. In case of points like the $90 stock price at period 1 above, there will be one option value from period 2 and one option value if exercising at period 1. We consider the value whichever is higher.

In the tree above, we see that the stock price at time 2 is less than $100 in scenario 4.

The value of option = 100 - 80 = $20

the value of option at period 1 = $20 * 0.3 * (1/1.05) = $5.71

Also at period 1, value of option = $(100-90) = $10 Since this is higher value, we exercise it here and this value will be considered to calculate option value at period 0.

the value of option at period 0 = $10 * 0.28 * (1/1.05) = $2.67