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Calculating Heston Threshold Probabilities with Digital Options

Article Quant Q&A · Author: Jakobovski
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Summary

The document explains how to obtain the probability that a stock price modeled by Heston will finish above a specified threshold at a future date. In options terminology, this probability is the undiscounted value of a digital call on the modeled spot process. The Heston model’s characteristic function can be used with Fourier methods to calculate the result.

It identifies two approaches: an analytic Fourier-transform expression for the digital option and the COS method, which recovers a distribution function from the characteristic function. The answers point to standard references for derivations and mention that implementations of Heston analytics can encounter branch-cut issues. No derivation, parameter choices, numerical example, or comparison of method accuracy is provided, so the document serves mainly as a pointer to the computational framework and its implementation caveat.

Key ideas

  • The probability of finishing above a threshold equals the undiscounted digital-call value for the spot process.
  • Heston threshold probabilities can be calculated using Fourier methods.
  • The COS method recovers a distribution function from the model’s characteristic function.
  • Branch-cut handling can complicate implementations of Heston analytics.
  • The document names methods but does not provide a worked calculation or accuracy comparison.

Tags

OptionsEquitiesVolatilityDerivatives pricing
Full text
# How do I calculate the probability of a stock being above or below a value using the Heston model?


# How do I calculate the probability of a stock being above or below a value using the Heston model?












How can I use the Heston Model to calculate the probability of a stock being above or below a certain value on a given date in the future?

## Answer by q.t.f. (score 2)

https://quant.stackexchange.com/a/19179

In options pricing language, the probability of a spot process being above a given level $K$ at time $T$ is the undiscounted price of a digital call option on that spot process. In the Heston model, there is an analytic expression for this in terms of Fourier transform. You can find this in various standard references, e.g. Alan Lewis's book "Option Valuation Under Stochastic Volatility" or by google search. When I try "digital option in heston model" the top result I find is a student paper http://www.cs.ubbcluj.ro/~studia-m/2003-3/lazar.pdf which at first glance looks basically correct, though I don't see a discussion of the branch cut issue which often trips up unwary implementors of Heston model analytics.

## Answer by Aborna (score 1)

https://quant.stackexchange.com/a/19188

COS method is an efficient way to recover the distribution function from the characteristic function in the Heston model. For other methods, you may refer to "Inverting Analytic Characteristic Functions and Financial Applications".

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.