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(x; h) with a parameter h. It was introduced by F. Esscher in 1932 (Esscher 1932).

Definition

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

${\displaystyle f(x;h)={\frac {e^{hx}f(x)}{\int _{-\infty }^{\infty }e^{hx}f(x)dx}}.\,}$

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

${\displaystyle {\frac {e^{hx}}{\int _{-\infty }^{\infty }e^{hx}d\mu (x)}}}$

with respect to μ.

Basic properties

Combination

The Esscher transform of an Esscher transform is again an Esscher transform: E_h1 E_h2 = E_h1 + h2.

Inverse

The inverse of the Esscher transform is the Esscher transform with negative parameter: E⁻¹
_h = E_−h

Mean move

The effect of the Esscher transform on the normal distribution is moving the mean:

${\displaystyle E_{h}({\mathcal {N}}(\mu ,\,\sigma ^{2}))={\mathcal {N}}(\mu +h\sigma ^{2},\,\sigma ^{2}).\,}$

Examples

Distribution	Esscher transform
Bernoulli Bernoulli(p)	${\displaystyle \,{\frac {e^{hk}p^{k}(1-p)^{1-k}}{1-p+pe^{h}}}}$
Binomial B(n, p)	${\displaystyle \,{\frac {{n \choose k}e^{hk}p^{k}(1-p)^{n-k}}{(1-p+pe^{h})^{n}}}}$
Normal N(μ, σ2)	${\displaystyle \,{\frac {1}{\sqrt {2\pi \sigma ^{2}}}}e^{-{\frac {(x-\mu -\sigma ^{2}h)^{2}}{2\sigma ^{2}}}}}$
Poisson Pois(λ)	${\displaystyle \,{\frac {e^{hk-\lambda e^{h}}\lambda ^{k}}{k!}}}$

See also

• Esscher principle
• Exponential tilting

References

• 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.