I have a theoretical question about the value of an option when spot price goes to infinity as a function of volatility going to infinity.
I know that for a call option:
The option value equals the discounted payoff when there is zero volatility,
The option value at infinite volatility is equal to the spot. One can derive this fact by evaluating what happens when $\sigma \to \infty$ in the Black-Scholes PDE.
Now assume we add a dimension of, besides volatility, the spot price.
My question is, given "infinite" spot price, how will the option value approach the spot price by moving 'up' in the volatility dimension?
For example, a (EU) call with (A) strike $100$, spot $100000$, $\tau=1$ year, lets say $r=0$%, and volatility $200$% has value $V=99900.2846$, which is almost equal to payoff. when (B) vola $500$%, $V=99981.6396 \approx$ spot. And for (C) vola $1000$%, $99999.9985$, which is even closer to the exact spot. Now lets say in want to truncate my spot (and volatility) domain and need a boundary condition at the upper bound of this (truncated) spot domain. Is there a way to know how the option value on this bound (spot $\to \infty$, vola $\in (0,\to\infty)$) behaves?