Question

A call option on a non-dividend-paying stock has a market price of $2. The stock price is $15, the exercise price is $13, the time to maturity is three months, and the risk-free interest rate is 5% per annum. What is the implied volatility?

Answer 1

Call Price = SN (d₁) – N (d₂) Ke ^-rt

Call Price = $ 2

S = $ 15 (Stock Price)

K = $ 13 (Exercise Price)

r = 5%

i.e, 2 = 15 * N (d₁) - N (d₂) * 13 * e^-5%*90/365

Using iterative and trial and error method, we can try calculating at Implied Volatility say at 0.61 where the value shall be 2.96 and at 0.62 the value shall be 2.99, hence the vol lies in between 61% and 62%.

1st at 61%
Inputs:
Stock Price (S)	$15.00
Strike Price (X)	$13.00
Volatility (s)	61.00%
Risk-Free Rate	5.00%
Time to Expiration (T)	0.2466
Dividend Yield	0.00%
Output:
D1	0.66458
D2	0.36168
N(D1)	0.74684
N(D2)	0.64120
Call Price	$2.96910

• =15*e^{(-0.00%*0.0329)})*0.68028-$13*e^{(-0.50%*0.0329)}*0.66655
• =$2.96910

Trial and Error Method – Call Price at 62%

2nd at 62%
Inputs:
Stock Price (S)	$15.00
Strike Price (X)	$13.00
Volatility (s)	62.00%
Risk-Free Rate	5.00%
Time to Expiration (T)	0.2466
Dividend Yield	0.00%
Output:
D1	0.65879
D2	0.35092
N(D1)	0.74498
N(D2)	0.63718
Call Price	$2.99298

• =15.00*e^{(-0.00%*0.0329)})*0.67327-$13*e^{(-0.50%*0.0329)}*0.65876
• =$2.99298

Particulars	Amount
Call Option Value	2.00
Stock Price	15.00
Strike Price	13.00
Risk Free Rate	5.00%
Time to Expire	90.00
Call Price at 21%	$2.96910
Call Price at 22%	$2.99298
Volatility (s) at 21%	61.00%
Volatility (s) at 22%	62.00%
Implied Volatility	20.406%

• = 61% + (3. – 2.99298) /(2.96910 – 2.99298)x (62% – 61%)
• =20.406%

Therefore, the implied Vol shall be 20.41%