Asian Option Pricing (MATLAB)

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Plot of internal values of asiancontinuous

Contents


Here is the MATLAB implementation of the pricing of Asian options from the paper Unified Asian Pricing by Jan Vecer (2002), Risk, Vol. 15, No. 6, 113-116 (some minor changes have been made by Charles-Albert Lehalle).

[edit] Use of the pricing function

This function is easy to use, it returns the price of the option and the way to plot some internal results.

<<test_asiancontinuous.m>>=
[price, pde_sol] = asiancontinuous(100, 110, .04, .12, .5)
pde_sol.plot()

The result of the last line is quite simple (see figure).

[edit] MATLAB function for pricing

[edit] Main function

<<asiancontinuous.m>>=
function [price, pde_sol] = asiancontinuous(S0,K,r,vol,T)
% ASIANCONTINUOUS - implementation of Vecer's PDE Method for Asion Option
% Pricing, minor modifications by Charles-Albert Lehalle.
% use:
% [price, pde_sol] = asiancontinuous(100, 110, .04, .12, .5)
% pde_sol.plot()

%Implementation of Vecer's PDE Method
%Vecer, J. (2002): "Unified Asian Pricing",  Risk, Vol. 15, No. 6, 113-116 

mesh building

solve pde

price value

fprintf( '\n\nPrice of Asian Option is %8.6f\n\n', price);

description of PDE

initial condition

boundary condition

plot function

end

[edit] PDE Solving with MATLAB

The MATLAB pdepe function is used to solve parabolic-elliptic initial-boundary value problems.

<<solve pde>>=
m = 0;
sol = pdepe(m, @pdef, @pdeic, @pdebc, x, t);
pde_sol = struct('x', x, 't', t, 'u', sol(:,:,1), 'plot', @plot_sol);

[edit] Mesh building

<<mesh building>>=
N = 200;       %number of subintervals in space 
M = 200;       %number of subintervals in time

%more points -> higher precision, but slower

%Xmesh  x
xmin = -1;
xmax = 1;
x = linspace(xmin, xmax, N+1);

%Tspan
t = linspace(0, T, M+1);

[edit] Description of PDE

<<description of PDE>>=
function [c, f, s] = pdef(x, t, u, DuDx)
c = 1;

    f = 0.5*vol^2*( (1-exp(-r*t))/(r*T) - x )^2*DuDx;
    s = vol^2*((1-exp(-r*t))/(r*T) - x)*DuDx;

end

[edit] Initial Condition

<<initial condition>>=
function u0 = pdeic(x)
u0 = max(x, 0);
end

[edit] Boundary Condition

<<boundary condition>>=
function [pl, ql, pr, qr] = pdebc(xl, ul, xr, ur, t)
pl = ul;
ql = 0;
pr = ur - xr;
qr = 0;
end

[edit] Compute price value

The MATLAB pdeval function is used here.

<<price value>>=
%Output of the value of the option
X_0 = (1-exp(-r*T))*S0/r/T - exp(-r*T)*K;
x0 = X_0/S0;

uout = pdeval(m,x,sol(M+1,:),x0);
price = uout*S0;

[edit] Plot function

You can see here how MATLAB embedded functions can be used to implicitly create some internal variables (here the |pde_sol| variable become implicitly stored into the results of the |asiancontinuous.m| function).

<<plot function>>=
function h = plot_sol
figure('Color',[0.9412 0.9412 0.9412 ]);
surf(pde_sol.x, pde_sol.t, pde_sol.u, 'edgecolor', 'none');
axis([min(pde_sol.x) max(pde_sol.x) min(pde_sol.t) max(pde_sol.t) min(min(pde_sol.u)) max(max(pde_sol.u))]);
xlabel('X');ylabel('t');
end
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