Estimating constant Parameters for function with known Output and two variable inputs

Hey,

i am currently working on the estimation of SVI(stochastic volatility implied) parameters for implied volatilites, which i got from a set of option prices from 1996-2020. I have filtered thus far, that i now have maturities from 8-120 days, for each time-to-maturity i have a log-moneyness roughly between -0.5 and 0.5 and for each combination of those two i have an implied volatility.

What i now need is an SVI paramertization, which gives me an implied volatility for continous log_moneyness and time-to-maturitities pairs.

The original SVI function looks like this:

total_implied_variance(log_moneyness, tau) = a + b ( rho(log_moneyness - m) + sqrt((log_moneyness - m)^2 + sigma^2));

but in order to be able to interpolate in time-to-maturity as well i need to define:

tau = unique(maturity);

a = a(0)+a(1)*tau;

b = b(0)+b(1)*tau;

rho = rho(0)+rho(1)*tau;

m = m(0)+m(1)*tau;

sigma = sigma(0)+sigma(1)*tau;

so that my paramaters have a linear function implemented to them.

I have attached a picture of what my data looks like. First column is tau(=time-to-maturity), Second column describes the log-moneyness and the third describes my obtained implied_volatilites.

So in the end i require a set of 10 parameters (a0,a1,b0,b1,rho0,rho1,m0,m1,sigma0,sigma1), which enable me to calculate every point on the surface with inputs log_moneyness and tau with the minimized error of estimation.

As i am fairly new to matlab, i was wondering if one of you might be able to help. I have tried to implement fmincon and others, but until now with no success. Please let me know, if there is any more information you need.

All the best, Kai.

[Kai Koslowsky]

I wanted to share what i got so far and how i did it:

%generate matrix of log_moneyness and ttm

A = [log_moneyness maturity];

xdata = A;

ydata = implied_volatility;

%create function

fun = @(x,xdata) (x(1)+x(2).*xdata(:,2)) + (x(3)+x(4).*xdata(:,2)).*((x(5)+x(6).*xdata(:,2)).* (xdata(:,1) -(x(7)+x(8).*xdata(:,2))) + ... sqrt((xdata(:,1) - (x(7)+x(8).*xdata(:,2)).^2 + (x(9)+x(10).*xdata(:,2)).^2)));

%Fit the model using the starting point from former parameter estimation

%via SVI

x0 = [0.000202758890760171 0.000202758890760171 -0.249601998773279 -0.249601998773279 ...

0.5 0.5 0.0114 0.0114 1.73380716030294e-05 1.73380716030294e-05];

opt = optimoptions (@lsqcurvefit, 'StepTolerance', 1e-10);

x = lsqcurvefit (fun,x0,xdata,ydata,[],[],opt);

This gives me my 10 parameters. However, the estimates are not that close to what i hoped they would be. Is there a way of fitting them even closer to my observations? I would like to use a function like the 'Argmin' function, but have not had any luck yet. All the best to you.