In actuarial science, the Esscher transform (Gerber & Shiu 1994) is a transform that takes a probability density f(x) and transforms it to a new probability density f(xh) with a parameter h. It was introduced by F. Esscher in 1932 (Esscher 1932).

Definition

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Let f(x) be a probability density. Its Esscher transform is defined as

 

More generally, if μ is a probability measure, the Esscher transform of μ is a new probability measure Eh(μ) which has density

 

with respect to μ.

Basic properties

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Combination
The Esscher transform of an Esscher transform is again an Esscher transform: Eh1 Eh2 = Eh1 + h2.
Inverse
The inverse of the Esscher transform is the Esscher transform with negative parameter: E−1
h
 = Eh
Mean move
The effect of the Esscher transform on the normal distribution is moving the mean:
 

Examples

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Distribution Esscher transform
Bernoulli Bernoulli(p)   
Binomial B(np)   
Normal N(μ, σ2)    
Poisson Pois(λ)    

See also

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References

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  • Gerber, Hans U.; Shiu, Elias S. W. (1994). "Option Pricing by Esscher Transforms" (PDF). Transactions of the Society of Actuaries. 46: 99–191.
  • Esscher, F. (1932). "On the Probability Function in the Collective Theory of Risk". Skandinavisk Aktuarietidskrift. 15 (3): 175–195. doi:10.1080/03461238.1932.10405883.