Question

A stock index is currently trading at 50. Paul Tripp, CFA, wants to value two-year indexoptions using the binomial model. In any year, the stock will either increase in value by20% or fall in value by 20%. The annual risk-free interest rate is 6%. No dividends arepaid on any of the underlying securities in the index.

a. Construct a two-period binomial tree for the value of the stock index.

b. Calculate the value of a European call option on the index with an exercise price of 60.

c. Calculate the value of a European put option on the index with an exercise price of 60.

d. Confirm that your solutions for the values of the call and the put satisfy put-call parity

Short answer

a. As below.

b. As below

c. As below

d. relationship exists with minor variation

Step-by-step solution

Construction of two-period binomial tree ‘a’

2 period = 1 period of 365 days x 2 = 730 days

U = eσ√Δt

U = 1.2214

D = 1 / U

D = 1 / 1.2214 (Where D = 0.8187)

The binomial parameters:

U = 1 + percentage increase in a period, if there is increase in stock price= 1.2214

D = 1 + percentage decrease in a period, if there is decrease in stock price= 0.8187

R = 1 + Risk free rate = 1.06184

Period 1:

Up: $50 x 1.2214 = $61.07

Down: $50 x 0.8187 = $40.94

Period 2:

Up: $61.07 x 1.2214 = $74.59

Down: $61.07 x 0.8187 = $50.00

Up: $40.94 x 1.2214 = $50.00

Down: $40.94 x 0.8187 = $33.52

Calculation of value of European call option ‘b, c’

When no dividend δ = 0.

else it would be subtracted from the risk free rate.

Probability of upward stock price movement =

πu = e (r-δ)Δt – D / U – D

0.6038 = 1.06184 – 0.8187 /(1.2214 -0.8187)

Probability of downward stock price movement = 1 – 0.6038 = 0.3962

Value of call expiration = max (O, S – X)

Period 2:

Up-Up: max (O, S – X) = max (0, $74.59 - $60.00) = max (0, $14.59) = $14.59

Up-Down: max (O, S – X) = max (0, $50.00 - $60.00) = max (0, - $10.00) = $0.00

Down-Up: max (O, S – X) = max (0, $50.00 - $60.00) = max (0, - $10.00) = $0.00

Down-Down: max (O, S – X) = max (0, $33.52 - $60.00) = max (0, - $26.48) = $0.00

Period 1:

$14.59 x 0.6038 =(probability of upward stock movement) = $8.81

$0.00 x 0.3962 =(probability of downward stock movement) = $0.00

$0.00 x 0.6038 =(probability of upward stock movement) = $0.00

$0.00 x 0.3962 =(probability of downward stock movement) = $0.00

Hence the value of the call in period 1 (with discount factor 0.94176) =

($0.00 x 0.94176) + ($8.81 x 0.94176) + ($0.00 x 0.94176) + ($0.00 x 0.94176) =$8.30

Period 0:

$8.30 x 0.6038 =(probability of upward stock movement) = $5.01

$0.00 x 0.3962 =(probability of downward stock movement) = $0.00

Hence the value of the call in period 0 (with discount factor 0.94176) =

($5.01 x 0.94176) + ($0.00 x 0.94176) = $4.72

Validity of call put parity formula‘d’

The put-call parity formula = C- P = S0 – PV(X)

Substituting the value= $4.5123 - $7.9108 = -$3.3985

$50.00 - $60(1.06)2 = -$3.3998

Hence the call put parity exists with just minor deviation.