Question

Suppose the current stock price is $120 and the stock price in a year can be either $150 or $100. The risk-free rate is 2% per year, compounded annually. Compute the price of a European put option that expires in a year. The strike price is K=$130 (Hint: This is a put option case, not a call option. Be careful when you compute the cash-flow at expiration date. All other calculations should be the same as call option case.)

Answer 1

Current Stock Price (So) = $ 120

Risk free Rate (r) = 2% per annum compounded annually

Expected Price in a year

S(upward) = $ 150

S(downward) = $ 100

Strike Price (K) = $ 130

Risk Neutralisation Model:

Fair Future Price = So * (1+i)^n

Fair Future Price = 120* (1+0.02)^1

Fair Future Price = 120* (1.02)

Fair Future Price = 122.40

Let the Probability of attaining Upward price at the time of Expiry = "P"

Then,

($ 150 * P) + ($ 100 * (1 - P)) = $ 122.40

($150 - $100) P = $ 122.40 - $ 100

$40 P = $ 22.40

P(Upward) = 0.56

Therefore P(Downward) = 1- 0.56

P(Downward) = 0.44

Therefore, Price of Put Option =

= [(0.56 * 0) + (0.44 * (130 - 100)] / (1+0.02)^1

= [(0.56 * 0) + (0.44 * 30)] / (1.02)

= $13.2 / 1.02

Fair Price of Put Option = $ 12.94

Note:

Expiry Price	Probability	Moneyness	Exercise Price	Profit at Expiry date
$        150	0.56	Out of Money	Not Exercised	$                -
$        100	0.44	In the Money	$     130	$               30

Probable profit = (0.56 * 0) + (0.44 * $30)

Probable profit on Expiry date= $13.2

PV of Profit = $ 13.2/1.02

Value of Put Option = $12.94