Calculation of asian options for the Black–Scholes model

The paper deals with one of fundamental problems of financial mathematics, namely, allocation of resources between financial assets to ensure sufficient payments.
When constructing mathematical models of the dynamics of financial indicators, various classes of random processes with discrete and continuous time are used. Therefore, the theory of martingales is a natural and useful mathematical tool in financial mathematics and engineering. In this paper, the Black–Scholes model is considered in continuous time with two financial assets
$$ \begin{cases} \qquad \quad B_t=1, &\\ dS_t=\sigma S_tdW_t, &S_0>0, \end{cases} $$
The representation Theorem 1 of square integrable martingales is studied to calculate coefficients of the martingale representation. These coefficients allow further redistribution of the securities portfolio to obtain the greatest profit.
Theorem 1. Let $X=(x_t, F_t)_{0\leqslant t\leqslant T}\in\mathrm{M}_t$ and $W=(W_t, F_t)_{0\leqslant t\leqslant T}$ be a Wiener process with respect to the natural filtration. Assume that a family of $\sigma$-algebras $(F_t)_{0\leqslant t\leqslant T}$ is right continuous. Then there exits a stochastic process $(\alpha(t,\omega), F_t)_{0\leqslant t\leqslant T}$ with $\mathrm{E}\int_0^T\alpha^2(t,\omega)dt<\infty$ such that for all $0\leqslant t\leqslant T$,

\begin{eqnarray} x_t=x_0+\int_0^t\alpha(s,\omega)dW_s, \\ \langle x, W\rangle_t=\int_0^t\alpha(s,\omega)ds. \end{eqnarray}

Here, $\langle \bullet, \bullet\rangle_t$ is a mutual quadratic characteristic of processes.
The practical result of the research is the solution of the problem of constructing a hedging strategy. The option was used as the main financial instrument.
To construct a hedging strategy in the case of the model under consideration, we apply Theorem 1 to the martingale
$$ M_t=\mathrm{E}(\mathrm{f}_T|F_t), $$
where $f_t=\left(\frac1T\int_0^T S_tdt-K\right)_+$ is the payment function.
We found a quadratically integrable process $(\alpha_t)_{0\leqslant t\leqslant T}$ adapted with the filtration $(F_t)_{0\leqslant t\leqslant T}$ such that for all $t\in[0, T]$
$$ M_t=M_0+\int_0^t\alpha_s dW_s. $$
The strategy $\Pi=(\beta_t,\gamma_t)$ is calculated by the formulas
$$ \beta_t=\mathrm{E}f_t+\int_0^t\alpha_sdW_s-\gamma_tS_t, \quad \gamma_t=\alpha_t/\sigma S_t. $$