1
$\begingroup$

I am trying to obtain the Theta from Closed Formula by using Finite Difference methods and I observe some discrepancies. For instance, here with the following parameters:

Spot:50, Strike:50, Rate: 0.12 Time To maturity: 0.25, Volatility: 0.3

BS Closed Form: -8.528247797676

BS Forward FD: -5.8503941392455516

I applied a change of -1/365 in T to compute the BS Forward FD.

Please note that I am perfectly able to match Delta, Gamma and Vega. I don’t know what is wrong with Theta. Any idea?

$\endgroup$
1
$\begingroup$

First and foremost I do not agree with you Closed Form value. I get $\Theta=-8.963$. There are various of BS calculator you can use the check your results and in general you should do that. Here is one: https://goodcalculators.com/black-scholes-calculator/

Have in mind that maturity T is fixed then your forward FD problem should look like this: $$ \Theta(T-t_0) \approx \frac{C(....,T-t_0+h)-C(....,T-t_0)}{h} $$ $C(...,T-t)$ denotes the BS call price for time To maturity $T-t$. Choose a small value of $h$ say $h=1/100000$ and let the other parameters be those your mentioned in your post then at time 0 and maturity $T=1/4$ your FD problem will return: $$ \Theta(T-0)=\Theta(T) \approx \frac{C(....,T+h)-C(....,T)}{h}=-8.963 $$ Even for a much bigger value of $h$ namely $h=1/365$ the result is $\Theta(T)=-8.946$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.