Nonlinear Black–Scholes Pricing and Comparisons with the Linear Model

Summary

The document discusses Black–Scholes partial differential equations modified to account for transaction costs or discrete hedging. It reports that exact solutions are generally unavailable for the nonlinear equation, so numerical methods are typically used. The discussion points readers toward a book covering nonlinear option pricing, but gives no details about a particular numerical scheme or the book’s results.

It also explains why researchers compare prices from nonlinear and standard linear models: comparisons can benchmark the new model, show where their prices diverge, or support an evaluation of the proposed model. The post does not present a worked example, empirical evidence, or criteria for deciding which model is superior. Its brief answers are general guidance, so conclusions for a particular equation depend on its assumptions and setup.

Key ideas

• Nonlinear Black–Scholes equations can include effects such as transaction costs or discrete hedging.
• Exact solutions are generally unavailable, making numerical methods a common approach.
• Comparing nonlinear and linear model prices can reveal where their outputs differ.
• A comparison may serve as a benchmark or as part of an assessment of a new model.
• The post gives general guidance rather than equation-specific evidence or a numerical example.

Tags

# Nonlinear Black-Scholes model Vs linear Black-Scholes


# Nonlinear Black-Scholes model Vs linear Black-Scholes












I am working on a project related to Nonlinear BS partial differential equation, with terms for transaction costs and/or discrete hedging. I have two questions:

- Is there any exact solution to the Nonlinear BS equation?

- I have read a paper which numerically solved a Nonlinear BS and compared results with Linear BS. Nonlinear BS is supposed to be giving different option price than Linear one. Why should we compare them ? the paper

Lastly, as I am working on a project , it will be extremely helpful if you can provide some references that backs your statements.

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

https://quant.stackexchange.com/a/50380

In general you have don't have an exact solution for the non linear equation - so you have to use numerical methods. Have you seen this Non linear option pricing book. It covers the topics that you mentioned.

People generally do compare the prices produced by non traditional models to the traditional ones - the purpose could be to benchmark the model output, understand/locate the regions where the prices differ, or to demonstrate the superiority of the new model etc.

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