Esscher transform

Last updated March 30, 2019

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 ).

Actuarial science is the discipline that applies mathematical and statistical methods to assess risk in insurance, finance and other industries and professions. Actuaries are professionals trained in this discipline. In many countries, actuaries must demonstrate their competence by passing a series of rigorous professional examinations.

In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample in the sample space can be interpreted as providing a relative likelihood that the value of the random variable would equal that sample. In other words, while the absolute likelihood for a continuous random variable to take on any particular value is 0, the value of the PDF at two different samples can be used to infer, in any particular draw of the random variable, how much more likely it is that the random variable would equal one sample compared to the other sample.

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

Probability measure Measure of total value one, generalizing probability distributions

In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity. The difference between a probability measure and the more general notion of measure is that a probability measure must assign value 1 to the entire probability space.

{\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_h_₁ E_h_₂ = E_h_₁ + 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:

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

Examples

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

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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.

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