Dupire Local Volatility PDE Pricing

Article Quant Q&A · Author: MikeMan

Summary

The document presents an implementation question about numerically solving a Dupire partial differential equation to price vanilla options in log-moneyness and time coordinates. The author uses a fully implicit finite-difference scheme and reports difficulty reproducing market Black prices despite testing a flat local volatility and zero interest rates. The code also interpolates a sample volatility curve, sets boundary values, builds a tridiagonal system, and steps the solution through time.

The central issue raised is whether the chosen boundary conditions are appropriate: the code assigns intrinsic-value expressions at both spatial endpoints. The document does not provide a corrected scheme, a diagnosis, or numerical comparisons that isolate the source of the pricing mismatch. It is useful as an example of implementation concerns in local-volatility PDE pricing, but readers would need to check the PDE’s coordinate convention, time and space step definitions, boundary behavior, and matrix indexing before relying on the results.

Key ideas

The document applies an implicit finite-difference method to the Dupire pricing equation in log-moneyness space.
It compares computed vanilla prices with Black prices and reports that the results do not match.
The author suspects that setting both spatial boundaries to intrinsic value may be unsuitable.
The code’s discretization, coordinate definitions, time stepping, and boundary implementation require scrutiny; no solution is established.

Tags

Options Derivatives pricing Volatility Statistics

Full text

# How do you numerically solve the Dupire Local Volatility PDE in log moneyness-time space?


# How do you numerically solve the Dupire Local Volatility PDE in log moneyness-time space?












I am trying to implement a numerical solution to price vanilla calls. I am using the Dupire equation in log moneyness-time (k = ln(F/T)) space as per below PDE

I have tried solving it using a fully implicit scheme in Matlab and can't match market prices (black prices). I have used a flat local vol of 30% and a zero interest rate in the below matlab code. I suspect my k boundary condition may be the issue as I set it to the intrinsic value along these boundaries?

```
clc; clear;
xmin = -1.0890280712271;
xmax = 0.8471200635454;
T = 0.22739726;

M =97; % Number of space steps
dx = (xmax-xmin)/M;
N = 14; % Number of time steps
dt = (xmax-xmin)/N;

Price = zeros(N+1,M+1);
T_range = 0:dt:T;
x_range = xmin:dx:xmax;

LV = [0.416943422   0.371500369 0.356353595 0.343735066 0.3316299   0.319610453 0.309502636 0.320541186 0.35365883];

X = [-0.475738211   -0.237869105    -0.158579404    -0.079289702    0   0.079289702 0.158579403 0.237869105 0.475738211];

vols = interp1(X,LV,x_range,'linear');
for i = 1:(M+1)
    if (x_range(i)<min(X))
        vols(i) = LV(1);
    else
        if (x_range(i)>max(X))
            vols(i) = LV(end);
        end
    end
end
vols(:) = 0.9;
%% Initial Condition
Price(1,:) = max( 1 - exp(x_range) , 0);

%% Boundary Conditions
% min k boundary
Price(:,1) = max( 1 - exp(xmin) , 0);

% max k boundary
Price(:,M+1) =  max(1 - exp(xmax), 0) ;

%% A matrix
alpha = -(vols.^2)/(2*dx^2);
a = -alpha*dt*(1+dx/2);
b = dt*(2*alpha + 1/dt);
c = alpha*dt*(1-dx/2);
A = zeros(M-1);

for j = 2:M-1
    A(j,j-1) = a(j);
end

for j = 1:M-1
    A(j,j) = b(j);
end

for j = 1:M-2
    A(j,j+1) = c(j);
end

b = zeros(M-1,1);
b(1) = Price(1,1) * a(1);
b(M-1) = Price(N+1,M+1) * c(end);

%% Solve for Y
for i = 2:N+1
    Price(i,2:M) = A \ (Price(i-1,2:M)' - b);
end
```

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.

Numerical Dupire Local Volatility Pricing in Log-Moneyness Space

Summary

Key ideas

Tags