This exam consists in 4 problems. Please read carefully before starting and just

ID: 2782030 • Letter: T

Question

This exam consists in 4 problems. Please read carefully before starting and justify clearly all your results and statements. Problem 1 (20 points). Consider the ninomial Asset Pricing Model, with u-2--1/2, r 1/4 and Se-8. Consider a European Call Option with strike price K r= 0, maturing at time N = 3. (1) State the non arbitrage condition and compute the Risk-neutral probabilities. (2) Using a hackward recursion,find the arbitrage free price of the option at tine 0, i.e. Vo under the risk-neutral probability

Explanation / Answer

Binomial asset pricing model works on the non abritrage condition that all risk free investments atleast earn risk free rate of return. The binomial is used to calculate stock price or interest rate based on interest rate paths.

To compute the risk neutral probabilities i.e probability of up move (suppose we denote it as Pu) = 1+r-d / u-d
and the probability of down move (suppose we denote it as Pd) = 1-Pu

Given in the question u=2, d=1/2=0.5, r=1/4=0.25%
So Pu = 1+r-d / u-d
=1+0.0025-0.5 / 2-0.5
=0.5025 / 1.5
=0.335
and Pd=1-Pu= 1-0.335 = 0.665

Year 0

Year 1

Year 2

Year 3

So*u^3

So*u^2

So*u

So*u^2*d

So*u*d

So*d

So*u*d^2

So*d^2

So*d^3

Here with K=9 the value of call will be max(year3So-K, 0), and then reverse calculate using formula = (C*Pu)+(C*Pd) / (1+Rf)

55.00

21.70

8.36

5.00

3.16

1.67

0.56

0.00