I'm working with natural cubic splines and local volatility. I'm reading and using the techniques in Martin Hanke's paper [L=Computation of Local Volatilities from Regularized Dupire Equations]http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.69.9928&rep=rep1&type=pdf[/L]
With zero dividend we have that the Dupire equation can be written like
(1)
where
u = cubic spline
uT = first derivative of u w.r.t. maturity
uX = first derivative of u w.r.t. strike
uXX = second derivative of u w.r.t. strike
X = strike prices
r = risk free interest rate
sigma(T,X) = local volatility
I have done the Reinsch algorithm etc. and derived everything mentioned above. Then he goes ahead and defines the system
(2)
where
D is a diagonal matrix with the denominator of (1) as its elements
z is the variance
b is a vector of the enumerator of (1)
Hence D should be a diagonal-matrix 100x100
z should be a 100x1 vector.
b should be a 100x1 vector.
This system doesn't guarantee a postivei solution so one needs to regularize it. That system looks like this:
(3)
I have 10 strike prices per maturity (10 maturities). In the picture above m = n = 10.
What I don't get is how the left hand side in (3) can be correct w.r.t. dimensions. As mentioned before we have that D^2 is a 100x100 matrix. But L'L is 110x110 (won't show all the steps but if you use that m = n = 10 you will end up with L'L being 110x110).
Would really appreciate some help, I have written everything in matlab and all that is remaining is this last step of the problem.
With zero dividend we have that the Dupire equation can be written like
where
u = cubic spline
uT = first derivative of u w.r.t. maturity
uX = first derivative of u w.r.t. strike
uXX = second derivative of u w.r.t. strike
X = strike prices
r = risk free interest rate
sigma(T,X) = local volatility
I have done the Reinsch algorithm etc. and derived everything mentioned above. Then he goes ahead and defines the system
where
D is a diagonal matrix with the denominator of (1) as its elements
z is the variance
b is a vector of the enumerator of (1)
Hence D should be a diagonal-matrix 100x100
z should be a 100x1 vector.
b should be a 100x1 vector.
This system doesn't guarantee a postivei solution so one needs to regularize it. That system looks like this:
I have 10 strike prices per maturity (10 maturities). In the picture above m = n = 10.
What I don't get is how the left hand side in (3) can be correct w.r.t. dimensions. As mentioned before we have that D^2 is a 100x100 matrix. But L'L is 110x110 (won't show all the steps but if you use that m = n = 10 you will end up with L'L being 110x110).
Would really appreciate some help, I have written everything in matlab and all that is remaining is this last step of the problem.
