Question

PROBLEM 2. Consider a two-step Binomial model. In Figure 1 you are given an incomplete pricing tree, which corresponds to a European put option with strike price K = 65. (a) (5 Points) Compute the per period interest rate r and the risk-neutral probability p*. (b) (10 Points) Find the price of the put option at t = 0. Moreover, determine the complete binomial tree for the stock price.

Answer 1

(a)

We can write following two equation for the prices of the put at the two nodes at the end of the first period.

0xp + 14.6(1-p) = 2.6545(1+r) and 14.6p + 35.06(1-p) = 17.09(1+r)

from the first equation 1+r = 5.500094(1-p) , substituting in the second equation we have

14.6p + 35.06(1-p) = 17.09x5.500094(1-p) or 14.6p =58.93661(1-p)

or p = 0.801459 , substituting in the first equation r = 0.09199 or 9.199 %

(b)

The price of the put option at t=0 is

p_E(0) = (1+r)^-1{2.6545p + 17.09(1-p)}

= 1.09199^-1.{2.6545x0.801459 + 17.09x(1- 0.801459)}

= 5.055

The payoff of the put at expiry(after two periods) is K - S₂ if S₂ < K, or 0 if S₂ >or= K

Therefore at the uppermost node at expiry S₂> 65 and the option is out of the money

At the remaining two nodes the option is in the money and must have

65 - S₂= 14.6 or S₂= 50.4 at the middle node and

65 - S₂= 35.06 or S₂= 29.94 at the bottom node.

In a risk-neutral world the stock price grows at the risk-free rate therefore we must have at t = 1

S₁ = (1+r)^-1.{50.4p + 29.94(1-p)} at the bottom node

or S₁ = 1.09199^-1.{50.4x0.801459 + 29.94x(1-0.801459)}

or S₁ = 42.4343

Therefore up movement ratio u = 50.4/42.4342 = 1.18772

and down movement ratio d = 29.94/42.43423= 0.705561

Thus   S₁ = 50.54/d = 71.6309 at the upper node

and S₂= 71.609u = 85.076 at the uppermost node

and finally

S₀ = 71.6309/u = 63.31

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