Question

On April 22 2015, at market close, the price of BP (symbol BP) was 40.50. The prices of various call options with expiration on July 22 2015 is set out below. Given this, derive the smile for BP. Assume the annual interest rate is 3.0%.

Strike Price	Price
$27.50	$14.17
$30.00	$11.26
$32.00	$9.30
$35.00	$6.50
$38.00	$4.00
$40.00	$2.50
$42.00	$1.61
$45.00	$0.67
$48.00	$0.27
$50.00	$0.14
$52.50	$0.10
$55.00	$0.05
$57.50	$0.06

Answer 1

Solution :

Find the implied volatility at all the given strike price using black Scholes formula

In order to find the volatility, you have to use goal seek function in excel and then it would be easier otherwise you have to do trial and error manually and put some values of volatility in the formula and then see if it gives the Call premium value correctly

In the below given excel sheet, I have used goal seek function and make call premium equals to given value in the question by changing the volatility cell and found the values of implied volatility and then plotted the smiling curve of volatility against the strike price