Esscher transform

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

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

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

with respect to μ.

The Esscher transform of an Esscher transform is again an Esscher transform: E_h_₁ E_h_₂ = E_h_₁ + h₂.

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

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

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

Distribution	Esscher transform
Bernoulli Bernoulli(p)	$\,{\frac {e^{hk}p^{k}(1-p)^{1-k}}{1-p+pe^{h}}}$
Binomial B(n, p)	$\,{\frac {{n \choose k}e^{hk}p^{k}(1-p)^{n-k}}{(1-p+pe^{h})^{n}}}$
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)	$\,{\frac {e^{hk-\lambda e^{h}}\lambda ^{k}}{k!}}$

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.

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