Hello,
I am playing with different local vol models and try to calibrate them using forward Dupire PDE.
Let F denotes a forward rate
$$dF_t=\sigma(t,F_t)dW_t$$, and c as $$c(t,K)=E[(F_t-K)^{+}]$$
The Dupire PDE gives :
$$\partial_{t}c(t,k)=\frac{1}{2}\sigma(t,F_t)^2\partial_{kk}c(t,k)$$
If I choose $$\sigma(t,F_t)=60bps$$, and a short maturity : 3months, by solving the pde , I should expect the implied vols of c(0.25,k) for differents strikes k to be 60bps.
I do not know where I am missing something , I cannot get what I expect : That is how I solve the pde :
For a given set of strikes $$\{k_0,...,k_0+n*\delta k\}$$
I have equations :
$$c(0.25,k_l)-\frac{1}{2}*0.25*(60bps)^2*\frac{c(0.25,k_{l+1})-2c(0.25,k_l)+c(0.25,k_{l-1})}{\delta k^2}=max(F_0-k_l,0) \\ with 1\leq l \leq n-1$$
I assume $$\partial_{kk}c(t,k_0)=\partial_{kk}c(t,k_n)=0$$
The left and right diagonals are $$\{0,-m,...,-m,0\}$$
the second(middle) diagonal is $$\{1,1+2m,...,1+2m,1\}$$
with $$m= \frac{1}{2 \delta_k^2}*0.25*(60bps)^2$$
Finally, to get the prices, I just use https://en.wikipedia.org/wiki/Tridiagon ... _algorithm
Any clue? thanks