Question

1. A Simple Two-period Binomial Binary/Digital Option Pricing Riskless bond 16–20-25 225 120 64 < 25

Derivative security (European binary/digital option paying 100 when the stock price is between 50 and 100 and paying zero otherwise). Each step is 12 months

The actual probabilities are 1/3 for up and 2/3 for down at each node.

a. (3 points) What are the risk-neutral probabilities of up and down?

b. (3 points) What is the value of the binary option at each node?

c. (3 points) In the portfolio strategy that replicates the binary option, what are the holdings in the stock and the bond at each node?

Answer 1

a. What are the risk-neutral probabilities of up and down?

The formula for risk-neutral probability of an up move is (r−d)/(u−d).

where

r = 20/16 = 10/8 or 5/4

u = 120/64 = 15/8

d = 40/64 = 5/8

Now, The risk-neutral probability of up is

(r − d)/(u − d) = (10/8 − 5/8)/(15/8 − 5/8) = 1/2

and therefore the risk-neutral probability of down is 1 − 1/2 = 1/2.

Stock Price = (120/2 + 40/2 )/1.25 = 80/ 1.25 = 64

b. What is the value of the binary option at each node?

7 O2 + Tool2 , lo < 1.25 으 = 32 1.25 J 10012 + 이르 = 40 125

c. In the portfolio strategy that replicates the binary option, what are the holdings in the stock and the bond at each node?

Since this is a riskless bond, there is no risk , it is advisable to invest $32 in the bond and nothing( i.e $0 ) in the stock.