R19.P1.T4 Hull V3 - Implied volatility

[381447]

Hi there,

Page 21 of the study notes - Implied Volatility example :

For example, assume:
 Stock price (S) is $10
 Strike (K) is $10
 Term (t) is six months (0.5)
 Riskless rate is 5%
 Call price is $1.25
$1.25 = Black-Scholes [$10, $10, t=0.5 years, r = 5%, ]
Solve for the implied volatility:   .405

Can someone explain me how you get the implied volatility please ? It drives me nuts .....

 

[Dr. Jayanthi Sankaran]

Hi @381447

The implied volatility is the value of sigma that when substituted into the Black-Scholes equation gives C = $1.25. Unfortunately, it is not possible to invert the Black-Scholes equation so that sigma is expressed as a function of S, K, r, T and C. However, an iterative search procedure can be used to find the implied sigma.

In page 137 of Chapter 7 of BSM, Hull has an example of a European call on a non-dividend paying stock = 1.875 when S = 21, K = 20, r = 0.1 and T = 0.25. He starts by trying sigma = 0.20. This gives a value of C = $1.76 which is too low. Because C is an increasing function of sigma, a higher value of sigma is required. He then tries with sigma = 0.30. This gives C = $2.10 which is too high. He then tries a value of sigma = 0.25 and so on until the implied volatility = 0.235 or 23.5% per annum.

Hope that helps - it is an iterative search procedure. You have to grin and bear it - sorry

Thanks!

 

[QuantMan2318]

To find the implied volatility, you may also use the DerivaGem software from the Rotman School website that has been provided by Hull in his book

 

[Dr. Jayanthi Sankaran]

Hi @QuantMan2318,

Is the DerivaGem software user friendly? I have not used it so far...

Thanks!

 

[QuantMan2318]

Hi @QuantMan2318,

Is the DerivaGem software user friendly? I have not used it so far...

Thanks!

Its a bit rusty but it does the job well!

 

[Dr. Jayanthi Sankaran]

Thanks a lot - appreciate it

 

[Dr. Jayanthi Sankaran]

Thanks for the star Nicole and David - appreciate it

 

[381447]

ok crystal clear !! Thx for the explanations