to need to bound $\Omega$. Since this equation describes prices, and it doesn't
make sense to talk about prices being negative, we will set the lower bound of
$\Omega$ to be 0. Then, for an upper bound, we will choose a very large number,
-one that $S$ is not very likely to ever get to. We will call this $S_\text{max}$
-. So, $\Omega=[0,S_\text{max}]$.
+one that $S$ is not very likely to ever get to. We will call this $S_\text{max}$.
+So, $\Omega=[0,S_\text{max}]$.
Second, after truncating the domain, we need to ask what boundary values we
should pose at this now finite boundary. To take care of this, we use "put-call"
This means that this is not an equation that is posed going forward in
time, but in fact going *backward* in time. Thus it makes sense to solve this
problem in reverse by making the change of variables $\tau=T-t$ where now $\tau$
-denotes "time before the strike time $T$".
+denotes the time before the strike time $T$.
With all of this, we finally end up with the following problem:
@f{align*}{
@f}
But, because the matrix $\textbf{B}$ involves an advective term, we will choose
$\theta=0$ there -- in other words, we use an explicit Euler method to treat
-advection. Conversely, since the matrix $\textbf{D}$ involves the diffusive term
-, we will choose $\theta=1/2$ there -- i.e., we treat diffusion using the second
+advection. Conversely, since the matrix $\textbf{D}$ involves the diffusive term,
+we will choose $\theta=1/2$ there -- i.e., we treat diffusion using the second
order Crank-Nicolson method.
So, we arrive at the following:
@endcode
What is more interesting is the output of the convergence tables. They are
-outputted into the console, as well into a LaTex file. The convergence tables
+outputted into the console, as well into a LaTeX file. The convergence tables
are shown above. Here, you can see that the the solution has a convergence rate
of $\mathcal{O}(h)$ with respect to the $H^1$-norm, and the solution has a convergence rate
of $\mathcal{O}(h^2)$ with respect to the $L^2$-norm.
// when the option expires.\n
// - <code>asset_volatility</code>: The volatility of the stock price.\n
// - <code>interest_rate</code>: The risk free interest rate.\n
- // - <code>strike_price</code>: The aggreed upon price that the buyer will
+ // - <code>strike_price</code>: The agreed upon price that the buyer will
// have the option of purchasing the stocks at the expiration time.
//
// Some slight differences between this program and step-26 are the creation
// they are fairly normal values for these parameters. Although the stock
// price has no upper bound in reality (it is in fact infinite), we impose
// an upper bound that is twice the strike price. This is a somewhat arbitrary
- // choice to be twice the strike price, but it is large enought to see the
+ // choice to be twice the strike price, but it is large enough to see the
// interesting parts of the solution.
template <int dim>
BlackScholes<dim>::BlackScholes()
// @sect4{<code>BlackScholes::add_results_for_output</code>}
- // This is simply the function to stitch the solution peices together. For
+ // This is simply the function to stitch the solution pieces together. For
// this, we create a new layer at each time, and then add the solution vector
// for that timestep. The function then stitches this together with the old
// solutions using 'build_patches'.
convergence_table.write_tex(error_table_file);
// Next, we will make the convergence table. We will again write this to
- // the console and to the convergence LaTex file.
+ // the console and to the convergence LaTeX file.
convergence_table.add_column_to_supercolumn("cells", "n cells");
std::vector<std::string> new_order;
new_order.emplace_back("n cells");
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