Why Option Pricing Uses Approximation Methods

Summary

The document explains why option pricing sometimes relies on numerical approximations even though European options have a closed-form solution in the Black–Scholes setting. That formula depends on specific assumptions about market dynamics, including geometric Brownian motion for the underlying price. When the model uses dynamics for which a closed-form price is unavailable or unknown, such as a Lévy process, an approximate method may be needed.

It identifies Fourier methods, including the fast Fourier transform, as tools used to obtain approximate prices under more general dynamics and for more complex products. The explanation is conceptual rather than a worked derivation: it offers no comparison of numerical accuracy, computational costs, or implementation choices. It also does not describe particular models or products in detail. The central takeaway is that the need for approximation comes from expanding beyond restrictive assumptions and from practical efficiency or generalization needs, rather than from an inability to evaluate the normal distribution in the standard European case.

Key ideas

• Closed-form option prices are available only under particular model assumptions.
• Black–Scholes prices European options when the underlying follows geometric Brownian motion.
• Lévy-process models may lack a known closed-form option price.
• Fourier methods can approximate prices under more general dynamics and for more complex products.
• The document gives a high-level explanation without numerical evidence or implementation detail.

Tags

# Why do we need approximation in option pricing?


# Why do we need approximation in option pricing?












We know that we can get a closed form for European option price. And we can calculate directly the normal distribution accumulation. But I saw that people use many approximation methods such as Fourier transform ... What are the reasons? Please explain to me in detail, I do not have an full understanding but can obtain interpretation from basic ideas.

## Answer by Ivan (score 1)

https://quant.stackexchange.com/a/41985

We can only get closed-form solutions under certain assumptions about the market dynamics, e.g. in the Black-Scholes framework (share prices follows a GBM), the European option can be valued with the well-known Black-Scholes formula.

For other assumptions where no closed-form solution exists or is known (e.g. share price is a Levy process), FFT methods are used to arrive at approximate solutions.

## Answer by ir7 (score 0)

https://quant.stackexchange.com/a/41975

In short, FFT methods are used for efficiency and need for generalization to more complex dynamics and products. Reference below should help, but there must be many more publicly available.

https://www.ricam.oeaw.ac.at/specsem/sef/events/program/slides/oosterlee_cm_sm.pdf

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.