Why would this integral function be explicit?

I would like to solve the gamma*, other parameters will be given in the codes I wrote.

T = 1;
sigma = 0.3;
r = 0.09;
a = r - sigma^2/2;
syms t gamma_s;
eqa1 = int(exp(-3*t^2*gamma_s*sigma/2/T^2 + a*t + sigma^2*(t - 3*t^4/4/T^3)/2), t, 0, T);
eqa2 = T*exp(a*t - 3*gamma_s*sigma/2 + T*sigma^2/8);

Gamma_s = solve(eqa1 == eqa2,gamma_s);

And it returns

Unable to find explicit solution. For options

[Walter Roberson]

There isn't even a closed form formula for let alone your more complicated formula.

Answer 1

[Star Strider]

I would solve it numerically:

T = 1;
sigma = 0.3;
r = 0.09;
a = r - sigma^2/2;
K = 100;
S_0 = 100;
% syms t gamma_s;
eqa1 = @(gamma_s) integral(@(t)exp(-3*t^2*gamma_s*sigma/2/T^2 + a*t + sigma^2*(t - 3*t^4/4/T^3)/2), 0, T, 'ArrayValued',1);
eqa2 = @(gamma_s) T*exp(a*T - 3*gamma_s*sigma/2 + T*sigma^2/8);

Gamma_s = fsolve(@(x) eqa1(x) - eqa2(x), 1)

Gs = logspace(-4, 8, 1E+5);
figure
semilogx(Gs, eqa1(Gs) - eqa2(Gs))
hold on
plot(Gamma_s, 0, 'p')
hold off
grid

producing:

Gamma_s =
   0.059750259559572

and the plot.

In ‘eqa2’ I substituted ‘T’ for ’t’ because the first function, ‘eqa1’ is not a function of ‘t’, only ‘gamma_s’. That may also be the reason that the symbolic solution failed.

The plot demonstrates that there is only one point where the two functions are equal.

[Ziyi Gao]

Thanks, let me have a try

[Star Strider]

My pleasure!

If my Answer helped you solve your problem, please Accept it!

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