Upper bound option price in volatility dimension

Question

All,

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?

Answer 1

This does not directly answer your question, but here is a suggestion:

Most options, with the exception of barrier options, tend to behave linearly for extreme values of the model state variable(s). You can use this to program a very generic linear boundary condition that in my experience works fine for most pricings, again with the exception of barriers, for which Dirichlet applies.

Let $x$ be a state variable in your model (in your case Finite Difference method in Matlab for SABR volatility model fails to provide correct option values the forward $F$ or the stochastic volatility $\alpha$), then linearity in $x$ on the boundary $x_{\text{min}}$ or $x_{\text{max}}$ means the condition is $\frac{\partial^2V}{\partial x^2} = 0$ on the boundary. Plug that in your pricing PDE and you are left with a PDE on the boundary that has only 1st order derivatives in $x$, which you then approximate using the non centered finite difference $(V_{1,...} - V_{0,...})/\delta x$ or $(V_{i_{\text{max}},...} - V_{i_{\text{max}}-1,...} )/\delta x$. These give you the additional equations that you need to fill in the matrix that represent your discrete linear operator.

I have used this generic condition successfully in my implementation of many models, including SABR, for pricing all kinds of products.

Also as a general remark your should remember that a finite difference scheme is an approximation trough discretization of a continuous problem. For practical purpose it generally does not matter if the option value is slightly off near the boundary because the probability of getting there is very small (again with the exception of barriers).