# Stability Analysis Finite Difference Methods Black-Scholes PDE

$$\mathbf{Background}$$

Let $$u = u(x,t)$$ be the solution of the following forward parabolic initial boundary-value problem in one space dimension, $$u_t - a u_{xx} = f(x,t) \text { for } 0 < x < L \text { and } 0 < t

with the boundary conditions \begin{align*} u &= \gamma_0(t) \text { at } x = 0, \text { for } 0 < t < T,\\ u &= \gamma_L(t) \text { at } x = L, \text { for } 0 < t < T, \end{align*} and the initial condition $$u = u_0(x) \text { for } \text { at } t = 0, \text { for } 0 < x < L.$$

The problem is a model of heat conduction in one space dimension and $$a > 0$$ is the thermal conductivity.

Define a finite difference grid on the rectangle $$[0,L] \times [0,T],$$ $$(x_p,t_n) = (p\Delta x,n\Delta t) \text { for } 0 \leq p \leq P \text { and } 0 \leq n \leq N , \text { with } \Delta x = \frac{L}{P}, \text { and } \Delta t = \frac{T}{N}.$$

Suppose $$U_p^{n} \approx u(x_p,t_n),\, f_p^{n} = f(x_p,t_n),\, \gamma_0^n = \gamma_0(t_n),\, \gamma_L^n = \gamma_L(t_n) \text { and } u_{0p} = u_0(x_p).$$

Put $$\rho = a\Delta t / \Delta x^2.$$ The explicit Euler method for the problem is: $$$$U_p^{n+1} = f_p^n\Delta t + \rho U_{p-1}^n + (1-2\rho)U_p^n + \rho U_{p+1}^n \text { for } 1 \leq p \leq P-1 \text { and } 0 \leq n \leq N-1.$$$$

If $$\rho \leq 1/2,$$ or equivalently if $$2a\Delta t \leq \Delta x^2,$$ then the explicit Euler method is stable: $$\begin{multline} \label{stability theorem explicit} || \mathbf{U}_{0:P}^n ||_{\infty} \leq \max \big\{||(\mathbf{u_0})_{1:P-1}||_{\infty}, ||\gamma_0^{0:n}||_\infty, ||\gamma_L^{0:n} ||_\infty \big\} + \sum_{j=0}^{n-1}|| \mathbf{f}_{1:P-1}^{j} ||_\infty \Delta t \,\, \text { for } 0 \leq n \leq N. \end{multline}$$

$$\mathbf{My\,\,Question}$$

I'm computing numerical solution of Black-Scholes PDE using explicit Euler method. I want to undertake stability analysis similar to what I have indicated for the numerical solution of heat equation using explicit Euler. We know that explicit Euler is conditionally stable. However, unlike heat equation, for Black-Scholes PDE, I can't find relevant literature that derives bounds on the discretization steps. I have been reading some textbooks that simply say that $$\Delta t /\Delta x^2$$ should be below a certain threshold but do not indicate that threshold explicitly.

Could someone suggest useful references (journal articles/books/research papers) that undertake stability analysis of this kind for computing numerical solution to Black-Scholes PDE using finite difference methods (explicit/implicit/Crank-Nicolson)?

• It seems your best bet would be to deal with the constant coefficient'' form. Page 6 of <people.math.gatech.edu/~meyer/MA6635/chap4.pdf> shows a change of variable $y = \log S / K$ and function redefinition $P$ that reduce the partial differential equation to $P_t + (\sigma^2/2)P_{yy} + (r - \sigma^2/2)P_y - rP = 0$ which is your heat conduction equation, for which you have a stability analysis already. Now it's a matter of translating the $\Delta t / (\Delta x)^2$ condition to $\Delta t / (\Delta y)^2$. Commented Jan 21, 2021 at 1:11
• So maybe something like $\Delta t / (\Delta S)^2 \leq (\sigma KS)^{-2}$? Commented Jan 21, 2021 at 1:30
• @A rural reader very useful reference. Thanks for that. Do you mind elaborating how you got this bound ?
– user715112
Commented Jan 21, 2021 at 9:42
• I will, below. I think the change of variable is actually $y = \log(S/K)$ so that'll change things a little, but not materially. Commented Jan 21, 2021 at 22:41

The recommendation above is to make the change of variable $$y := \log(S/K)$$ and the function redefinition referenced in the notes. Those two let you transform the original problem $$V_t(t, S) + (\sigma^2/2)S^2 V_{SS}(t, S) + rSV_S(t, S) - rV(t, S) = 0,$$ with appropriate initial condition, e.g. $$V(0, S) = \phi(S)$$, into the equivalent system $$P_t(t, y) + (\sigma^2/2)P_{yy}(t, y) + (r - \sigma^2/2)P_y(t, y) - rP(t, y) = 0$$ with initial condition $$P(0, y) = \gamma(y)$$.
I point this out because you already have a stability analysis for the latter, and if you have stably generated its solution you'll likewise have the solution for the former. You know from your stability analysis that the explicit scheme needs $$(\sigma^2/2)\Delta t / (\Delta y)^2 \leq 1/2$$, that is $$\sigma^2 \Delta t / (\Delta y)^2 \leq 1.$$ With $$y = \log(S/K)$$ I simply computed the differential $$\Delta y = (1/S)\Delta S$$ and substituted to get $$\sigma^2 S^2 \Delta t / (\Delta S)^2 \leq 1.$$ In practice this might mean the explicit scheme for the standard Black-Scholes system requires dynamic step-sizing.
• After making necessary transformations , Black-Scholes system reduces to $u_{\tau}=\frac{1}{2}\sigma ^2 u_{yy}$ where I have made the substitution $\tau = (T - t).$ This is now our heat equation and we can use stability analysis. So, instead of $\Delta t$, I think it should be $\Delta \tau$ in the final answer . Moreover if we see page 13 of the notes (fourth line from last), we have $u_{t}=\frac{1}{2}\sigma ^2 u_{yy}$. I think this $t$ represents $\tau$ because otherwise it should be $u_{t}= - \frac{1}{2}\sigma ^2 u_{yy}$ as on line 2 of page 7 where we have initial condition as $u(y,T)$.