Classifying Black–Scholes PDEs by Dependence on the Solution

Summary

The document clarifies that a partial differential equation’s linearity depends on how it involves the unknown function and its derivatives, not simply on whether volatility is constant. In the heat-equation form presented, the unknown and its time and spatial derivatives enter linearly, with coefficients independent of the unknown. The equation therefore remains linear when volatility varies as a function of time or space, though it then has variable coefficients rather than constant coefficients.

Nonlinearity arises if volatility or another coefficient depends on the solution itself or its derivatives, or if nonlinear terms and cross-products of the unknown and its derivatives appear. This is a conceptual classification rather than a derivation of option prices; the document does not explore boundary conditions, solution methods, or particular stochastic-volatility models.

Key ideas

• Linearity is determined by dependence on the unknown function and its derivatives.
• Volatility that varies only with independent variables can produce a linear equation with variable coefficients.
• Dependence of volatility on the solution or its derivatives can make the equation nonlinear.
• Constant versus variable coefficients is a separate distinction from linear versus nonlinear.

Tags

# Linear Or nonlinear Black Scholes Equation


# Linear Or nonlinear Black Scholes Equation












I have been going through the analytical solutions of black scholes equation which transforms it to a heat equation. $$u_{t}=\frac{1}{2}\sigma^{2}u_{xx}$$ Now if the volatility is constant , then its the linear form. and if the volatility is variable, then its the nonlinear form ? Please give reference too with the answer if possible.

## Answer by Magic is in the chain (score 3, accepted)

https://quant.stackexchange.com/a/49171

The linear/non-linear classification is concerned about the dependent variables, and its derivatives. To verify whether the equation is linear, you should be checking that the equation is linear in each of these variables, and the coefficients of these are functions of the independent variables (t and x in your example).

In your example, the dependent variables and its derivative are $u$, $u_t$ and $u_{xx}$. As the equation is linear in all of them, the coefficient don't depend on u or its derivatives, and there are no cross terms (e.g., $u \, u_x$), so the equation is linear. The criteria you have listed - constant vs variable volatility - will help differentiate whether the equation is constant-coefficent. On the other hand, if volatility depends on u or its derivatives, then that would make it non-linear.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.