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)	$\frac{e^{hk}{p}^k(1-p{)}^{1-k}}{1-p+p{e}^h}$
Binomial B(n, p)	$\frac{(\genfrac{}{}{0ex}{}{n}{k})e^{hk}{p}^k(1-p{)}^{n-k}}{(1-p+p{e}^h{)}^n}$
Normal N(μ, σ2)	$\frac{1}{\sqrt{2\pi {\sigma }^2}}{e}^{-\frac{(x-\mu -{\sigma }^{2}h{)}^2}{2{\sigma }^2}}$
Poisson Pois(λ)	$\frac{e^{hk-\lambda e^h}{\lambda }^k}{k!}$

See also

• Esscher principle
• Exponential tilting

References

• Gerber, Hans U.; Shiu, Elias S. W. (1994). "Option Pricing by Esscher Transforms". 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.