Week 2 - Boundary and Initial Conditions

(1). vanilla option

`(delv)/(delt)+1/2sigma^2s^2(del^2v)/(dels^2 )+(r-q)s(delv)/(dels)-rv=0`

0<s<+`oo`,t`in`[0,T),

`V(S,T)={(S-X)^+,For call,`

{(X-S)^+,For put,

(2). Barrier options.

`(delv)/(delt)+1/2sigma^2s^2(del^2v)/(dels^2 )+(r-q)s(delv)/(dels)-rv=0,`

`0<S<H,tin[O,T)`

`V(S,T)=(S-H)^+`

`V(H,T)=0,`

(3).Arthimatic average options.

`(delv)/(delt)+(S-V)/t+(delv)/(delA)+1/2sigma^2s^2(del^2v)/(dels^2)+(r-q)s(delv)/(dels)-rv=0`

(4).Lookback options.

`(delv)/(delt)+1/2sigma^2s^2(del^2v)/(dels^2)+(r-q)s(delv)/(dels)-rv=0`

`(S,M,T)is given,0<S<M<+oo,tin[o,t)`

(5).CIR Model.

``(delv)/(delt)+1/2prop^2r(del^2v)/(delr^2)+(eta-gammar)(delv)/(delr)-rv=0`

`0<r<+oo,tin[O,T)`

`where eta=1/2 sigma^`

(6).Vasicek Model.

`(delv)/(delt)+1/2prop^2r(del^2v)/(delr^2)+(eta-gammar)(delv)/(delr)-rv=0`

`-oo<r<oo,tin[O,T)`