Hi, Just wondering if the additional pages referenced by Graham Aylott are available to view as per the thread posted previously: Deriving vega for call option | Actuarial Education (acted.co.uk) If still avialble that would be great as I am looking for guidance on differencing the PDE w.r.t vega. Many thanks, Darragh
Hi Steve, Thanks for posting this. I've been looking at the question but still a bit stuck. So the PDE = def,f/def,r +rSt*del,f/del,s + 0.5*sigma^2*St^2*del^2,f/del,s^2 = rf (taken from page 11 of chap 15). If I do as per the question and take the first derivative above with respect to sigma as per the question, I get the following: del^2,f/del,t*del,sigma + rSt*del^2,f/del,s*del,sigma + 0.5*sigma^2*St^2*del^3,f/del^2,s*del,sigma + sigma*St^2*del^2,f/del,s^2 = r*del,f/del,sigma ? I used the product rule on the term 0.5*sigma^T. I'm not sure where to go next - do I need to plug in vega derived in the previous q and plug in and simplify? Thanks, Darragh
Every time you find a del,f/del,sigma in your equation you should replace it with "V" for vega, this will give you: del,V/del,t + rSt*del,V/del,s + 0.5*sigma^2*St^2*del^2,V/del^2,s + sigma*St^2*del^2,f/del,s^2 = r*V The last term in the LHS is zero because gamma = del^2,f/del,s^2 = 0.
Thanks Steve, I think I get it now, was very rusty on the multivariable calculus. So just to check I fully understand, taking first term which is del^2,f/del,t*del,sigma I use fact that the order of the differentiation in multivariable calculus doesn't matter so I write and del,*defl,f/del,t*del,sigma and then finally del(vega)/del,t? Same approach for rest of the variables noting that all of the greeks for share price will be zero (except for delta), which will collapse a few terms. Cheers, Darragh
