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Esscher Transforms, Entropy, and Risk-Neutral Pricing

Article Quant Q&A · Author: user53249
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Summary

The document asks why the Esscher transform can produce Black–Scholes pricing when a model driven by a Wiener process appears to allow many equivalent martingale measures. Its answer points to work on pricing claims under Lévy processes and identifies the Esscher measure with a measure that minimizes relative entropy. This offers an interpretation of the transform as a principled selection rule among candidate pricing measures, rather than as an arbitrary choice.

The discussion is brief and provides no derivation, model assumptions, or comparison with other measure-selection criteria. It also does not explain the Brownian Black–Scholes case in detail, so the reader should treat the cited connection as a pointer for further study rather than a complete proof that the transform is uniquely appropriate in every setting.

Key ideas

  • The Esscher measure is presented as a minimum relative entropy measure.
  • The transform is discussed as a way to select a pricing measure in models with multiple candidates.
  • The cited reference concerns contingent claims driven by Lévy processes.
  • The short answer does not establish that the Esscher transform is universally unique or appropriate.

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Derivatives pricingStatistics
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# Why the Esscher transform is the right transform for pricing formula?


# Why the Esscher transform is the right transform for pricing formula?












A Wiener process has infinitely many states of the world at any time step. Does that not mean that there are infinitely many EMM's for any model that uses the Wiener process?

But then if there is only one EMM for this model, how is it possible that the Esscher transform can be exactly the right transform, out of all possible transforms, to get the exact Black-scholes pricing formula?

## Answer by StochasticNewby (score 0)

https://quant.stackexchange.com/a/74517

See the paper of Chan (1999, Ann Appl Prob): Pricing contingent claims on stocks driven by Lévy processes

The author explains that the Esscher measure is equivalent to the "minimum" relative entropy measure.

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.