Esscher transform

In actuarial science, the Esscher transform (Gerber Shiu) 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

$f(x;h)=\frac{e^{hx}f(x)}{\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{hx}f(x)dx}.$

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

$\frac{e^{hx}}{\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}{e}^{hx}d\mu (x)}$

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−1h = E_−h

Mean move

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

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

Examples

Distribution	Esscher transform
Bernoulli Bernoulli(p)	$k\mapsto \frac{e^{hk}{p}^k(1-p{)}^{1-k}}{1-p+p{e}^h}\text{ for }k=0,1.$
Binomial B(n, p)	$k\mapsto \frac{(\genfrac{}{}{0ex}{}{n}{k})e^{hk}{p}^k(1-p{)}^{n-k}}{(1-p+p{e}^h{)}^n}\text{ for }k=0,1,\dots ,n.$
Normal N(μ, σ2)	$x\mapsto \frac{1}{\sqrt{2\pi {\sigma }^2}}{e}^{-\frac{(x-\mu -{\sigma }^{2}h{)}^2}{2{\sigma }^2}}\text{ for }x\text{ real.}$
Poisson Pois(λ)	$k\mapsto \frac{e^{hk-\lambda e^h}{\lambda }^k}{k!}\text{ for }k=0,1,2,\dots .$

Esscher principle

The Esscher principle is an insurance premium principle used in actuarial sciences that derives from the Esscher transform. It is given by $\pi [X,h]=E[X{e}^{hX}]/E[e^{hX}]$, where $h$ is a strictly positive parameter. This premium is the net premium for a risk $Y=X{e}^{hX}/m_X(h)$, where $m_X(h)$ denotes the moment generating function. This risk measure does not respect the positive homogeneity property of coherent risk measure for $h>0$.

See also

• Exponential tilting

References

• Gerber, Hans U.; Shiu, Elias S. W. (1994). "Option Pricing by Esscher Transforms". Transactions of the Society of Actuaries 46: 99–191. http://pages.stern.nyu.edu/~dbackus/Disasters/Gerber_Shiu_94.pdf. 

• 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.