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Crank–Nicolson for Option Sensitivities: A Source Reference

Article Quant Q&A · Author: MXR879
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Summary

The document raises a question about approximating option sensitivities, using rho under the Black–Scholes partial differential equation as a proof of concept. The author proposes differentiating the PDE with respect to the interest rate to obtain an equation for the sensitivity, then asks whether a central difference approximation can be used with the Crank–Nicolson scheme.

The included answer does not explain the derivation or provide a numerical recipe; it points to a treatment in a derivative-pricing textbook. As a result, the document identifies a numerical method and a sensitivity problem but does not supply enough detail to implement or assess the proposed approach. It gives no examples, error analysis, or comparison with alternative ways to calculate rho.

Key ideas

  • The question considers computing rho by differentiating the Black–Scholes PDE with respect to the interest rate.
  • It asks about combining a central difference approximation with the Crank–Nicolson scheme.
  • The response refers readers to a textbook and supplies no derivation or implementation details.

Tags

OptionsDerivatives pricingStatistics
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# Sensitivity Approximation - Crank Nicolson


# Sensitivity Approximation - Crank Nicolson












I am looking into a new method of calculating sensitivities starting off with a proof of concept with Black Scholes PDE. Suppose I want to calculate Rho and take the derivative of the PDE (heresy!!) and end up with a new PDE wrt to the interest rate.

I wish to approximate the above PDE using Crank Nicolson. However I would like to know whether it is possible to create a central difference approximation of:

Any ideas would be great. Thanks!

## Answer by Magic is in the chain (score 1)

https://quant.stackexchange.com/a/46795

Please see below:

Copied from Quantitative Methods in Derivative Pricing by Tavella.

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.