Esscher transform
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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
- [math]\displaystyle{ f(x;h)=\frac{e^{hx}f(x)}{\int_{-\infty}^\infty e^{hx} f(x) dx}.\, }[/math]
More generally, if μ is a probability measure, the Esscher transform of μ is a new probability measure Eh(μ) which has density
- [math]\displaystyle{ \frac{e^{hx}}{\int_{-\infty}^\infty e^{hx} d\mu(x)} }[/math]
with respect to μ.
Basic properties
- Combination
- The Esscher transform of an Esscher transform is again an Esscher transform: Eh1 Eh2 = Eh1 + 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:
- [math]\displaystyle{ E_h(\mathcal{N}(\mu,\,\sigma^2)) =\mathcal{N}(\mu + h\sigma^2,\,\sigma^2).\, }[/math]
Examples
| Distribution | Esscher transform |
|---|---|
| Bernoulli Bernoulli(p) | [math]\displaystyle{ \,\frac{e^{hk}p^k(1-p)^{1-k}}{1-p+pe^h} }[/math] |
| Binomial B(n, p) | [math]\displaystyle{ \,\frac{{n\choose k}e^{hk}p^k(1-p)^{n-k}}{(1-p+pe^h)^n} }[/math] |
| Normal N(μ, σ2) | [math]\displaystyle{ \,\frac{1}{\sqrt{2 \pi \sigma^2}}e^{-\frac{(x-\mu-\sigma^2 h)^2}{2\sigma ^2}} }[/math] |
| Poisson Pois(λ) | [math]\displaystyle{ \,\frac{e^{hk-\lambda e^h}\lambda^k}{k!} }[/math] |
See also
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.
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