Why Option Pricing Reverses Finite-Difference Time Steps
Summary
The document clarifies why finite-difference descriptions for option pricing may appear to reverse the usual pairing of explicit schemes with forward time differences and implicit schemes with backward differences. It explains that explicit and implicit refer to whether each time step can be computed directly or requires solving a system of equations. The choice of time-difference direction depends on how the problem is expressed and whether the PDE has an initial or terminal condition.
For Black-Scholes pricing, values are propagated backward from a terminal payoff, which changes how time steps are labeled relative to a standard initial-value problem. The discussion is conceptual and does not derive a particular discretization, compare numerical stability, or give an implementation. It helps resolve terminology, but a reader still needs the specific PDE formulation and scheme to determine which equations must be solved at each step.
Key ideas
- Explicit and implicit schemes are distinguished by whether each time step requires solving a system of equations.
- A forward or backward time difference alone does not determine whether a scheme is explicit or implicit.
- The direction of time propagation depends on the PDE formulation and its initial or terminal condition.
- Option pricing commonly starts from a terminal payoff and works backward in time.
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Full text
# Confusion about terminology : Finite difference for option pricing
# Confusion about terminology : Finite difference for option pricing
Consider the following initial-boundary value problem for $u = u(x,t),$
$$u_t - a u _{xx} = f(x,t) \text { for } 0 < x < L \text { and } 0 < t< T$$ along with bunch of initial and boundary conditions.
If we want to employ finite difference method to solve the above parabolic problem in 1D, then under explicit Euler method, we use forward difference approximation in time and under implicit Euler method, we use backward difference approximation in time.
I'm slighty confused about the terminology for finite difference method for option valuation. I was reading Paolo Brandimarte's Numerical Methods in Finance and in chapter 9 Option Pricing by finite difference method, to solve Black-Scholes PDE, under explicit scheme, he approximates the derivative with respect to time by a backward difference and under implicit scheme, he uses forward difference approximation in time. Isn't that the exact opposite of the previous definition?
Does it have something to do with terminal conditions in Black-Scholes PDE rather than initial condition and we go backward in time ?
## Answer by Quantuple (score 4, accepted)
https://quant.stackexchange.com/a/60527
An explicit (resp. implicit) finite difference scheme means you do not need to (resp. have to) solve a linear system of equations to find the solution at each intermediate time step. Whence their names!
Consequently, it really depends on if you have an initial, versus a terminal condition in time, and in what 'axes' you express and how you discretise the PDE you are solving.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.