Asymptotic behavior of European options In some of the numerical work on Black-Scholes generalized models, the boundary conditions on the truncated domain taken from the asymptotic behaviors of European call options, which are given by
$$\lim_{x \to -\infty} u(\tau,x)=0,~\text{and}~\lim_{x \to \infty}u(\tau,x)-K(e^{x-\int_0^\tau D(s)ds}-e^{-\int_0^\tau r(s)ds}),$$
where, $u(\tau,x)$ is the option's value depending on the price variable $x$ and time variable $\tau$ with the strike price $K$. The parameters $r$ and $D$ represents the interest rate and dividend yield, respectively.
My questions are:

*

*What precisely the financial importance of these asymptotic behaviors?


*What type of hedging strategy is being processed at the truncated boundaries in numerical approaches?

*

*I guess $x$ is the log moneyness $\log(S_0/K)$ where $S_0$ is today's stock price. Then $\lim\limits_{x\to-\infty}$ corresponds to $\lim\limits_{S_0\to 0}$ and
$\lim\limits_{x\to+\infty}$ corresponds to $\lim\limits_{S_0\to+\infty}\,.$ It is clear that in any model for the dynamics of $S_t$ we must have (abusing notation)
$$
\lim_{S_0\to 0}u(\tau,S_0)=0\,
$$
because a call option on a stock must be worth zero if the stock goes to zero.
This is the first boundary condition. For very large $S_0$ (or $x$) we must have in any model
\begin{align}
u(\tau,S_0)&\approx S_0\,e^{-\int_0^\tau D(s)\,ds}-Ke^{-\int_0^\tau r(s)\,ds}\\
&=K\left(e^{x-\int_0^\tau D(s)\,ds}-e^{-\int_0^\tau r(s)\,ds} \right)\,,
\end{align}
because for a very large stock price the call option must approach that difference regardless of the stock volatility, or the dynamics of $S_t$.
This is the second boundary condition.


*In a tree or PDE approach you choose a large enough upper boundary at which you apply the second boundary condition. You ignore all values of $S_t$ that are above that boundary. The hedging strategy was only a no arbitrage argument used to derive the pricing equation that you are solving. I don't think you need to worry about the hedging strategy at those boundaries, unless I misunderstood question 2.
