Question

An asset’s price is $68.14 and its volatility is 30% per year. A European put on the asset has three months until expiration, a strike price of $70.00, and the risk-free interest rate is 2.2% per year. According to a one-period binomial option pricing model, what is the option’s value?

Answer 1

P0 = Spot Price = $68.14

Volatility = 30% Per year means 7.5% Quarterly

r = Risk free interest rate = 2.2% p.a. means 0.55% Quarterly

X = Strike Price = $70.00

Up Price possible = $68.14 x (1+0.075) = $73.25

Down Price possible = $68.14 x (1-0.075) = $63.03

Simple method:

We create portfolio by buying h number of shares and to protect the same we sell 1 call.

The value of portfolio today = $68.14h - C (Call)

After 3-Month, Price is $73.25 then value of protfolio = $73.25h - $3.25

After 3-Month, Price is $63.03 then value of protfolio = $63.03h - $0 (Call option not exercise by buyer) = $63.03h

Risk less fortfolio so $73.25h-$3.25 = $63.03h - $0

h = 0.318

if h=0.318 then value of portfolio on expiry = $63.03h - $0 = ($63.03 x 0.318) - $0 = $20.04

Therefore value of portfolio today is PV of $20.04

=$20.04 x 1/1.0055 (Simple Interest taken)

=$19.93

But we know that value of portfolio today is $68.14h - C

So,

$68.14h - C = $19.93

$68.14(0.318) - C = $19.93

C = $1.73852 Round off $1.74 (Value of call)

For calculate put option we use put-call parity.

C + PV(x) = P + S

where:

C = price of the European call option

PV(x) = the present value of the strike price (x), discounted from the value on the expiration date at the risk-free rate

P = price of the European put

S = spot price or the current market value of the underlying asset

$1.74 + ($70/1.0055) = P + $68.14

$1.74 + $69.62 = P +$68.14

P = $3.22 (Put option value)

Summery option value at strike price $70 for 3-Month expire option as below

Call Option Value = $1.74

Put Option Value = $3.22

This method easy to understand and remember.