July 23, 2025
N. L. Johnson considered how to apply classical statistical techniques involving normal distributions to almost normal distributions. Given an almost normal distribution X he looked for transformations of the form \gamma + \delta f((X - \xi)/\lambda) that would result in a normal distribution, where \gamma, \delta, \xi, \lambda are real numbers and f is function. We assume \delta and \lambda are positive
If N = \gamma + \delta f((X - \xi)/\lambda) is normal then we can adjust \gamma and \delta to make it standard normal.
Exercise. If \mu = E[N] and \sigma^2 = \operatorname{Var}(N) then \gamma' + \delta' f((X -\xi)/\lambda) is standard normal if \gamma' = (\gamma - \mu)/\sigma and \delta' = \delta/\sigma.
We will use Z = \gamma + \delta f((X - \xi)/\lambda) where Z is standard normal in what follows. If f is invertable then X = \xi + \lambda f^{-1}((Z - \gamma)/\delta).
Johnson’s S_U (unbounded) system is {f(y) = \sinh^{-1}(y) = \log(y + \sqrt{1 + y^2})} so {X = \xi + \lambda \sinh((Z - \gamma)/\delta) = \xi + \lambda \sinh(U)]} where {U = (Z - \gamma)/\delta}. Since E[e^N] = e^{E[N] + \operatorname{Var}(N)/2} for any normally distributed N we have E[\sinh U] = E[(e^U - e^{-U})/2] = (e^{E[U] + \operatorname{Var}(U)/2} - e^{-E[U] + \operatorname{Var}(U)/2})/2 = e^{\operatorname{Var}(U)/2}\sinh E[U]
In terms of the Johnson parameters E[\sinh U] = e^{1/2\delta^2}\sinh(-\gamma/\delta). Note this only depends on \gamma and \delta.
The expected value of X is E[X] = E[\xi + \lambda \sinh(U)]. For small \gamma and \delta near 1 we have E[X] \approx \xi - \sqrt{e}\lambda\gamma.
Clearly X\le x is equivalent to Z \le z where z = \gamma + \delta \sinh^{-1}((x - \xi)/\lambda). We have {P(X\le x) = \Phi(z)} where \Phi is the standard normal cumulative distribution.
Recall for N and M jointly normal we have E[e^N f(M)] = E[e^N] E[f(M + \operatorname{Cov}(M,N))]. For share measure we calculate
\begin{aligned} E[X 1(X\le x)] &= E[(\xi + \lambda \sinh(U) 1(Z\le z)] \\ &= \xi P(Z\le z) + \lambda E[\sinh(U) 1(Z\le z)] \\ &= \xi P(X\le x) + \lambda E[\frac{e^U - e^{-U}}{2} 1(Z\le z)] \\ &= \xi P(X\le x) + \frac{\lambda}{2} (E[e^U] P(Z + \operatorname{Cov}(U,Z)\le z) - E[e^{-U}] P(Z - \operatorname{Cov}(U,Z)\le z])) \\ \end{aligned}
Since {\operatorname{Cov}(U, Z) = 1/\delta}
E[X 1(X\le x)] = \xi P(X\le x) + \frac{\lambda e^{\operatorname{Var}(U)/2}}{2} \left(e^{E[U]}\Phi(z - 1/\delta) - e^{-E[U]}\Phi(z + 1/\delta)\right)
Note the put forward value is E[(k - X)^+] = kP(X\le k) - E[X 1(X\le k)] so we have a closed form solution for put values using the Johnson distribution.
To compute the second moment of X we will need \begin{aligned} E[\sinh^2(U)] &= E[((e^U - e^{-U})/2)^2] \\ &= E[(e^{2U} - 2 + e^{-2U})]/4 \\ &= (e^{-2\gamma/\delta + 2/\delta^2} - 2 + e^{2\gamma/\delta + 2/\delta^2})/4 \\ &= e^{2/\delta^2}\cosh(2\gamma/\delta)/2 - 1/2 \\ \end{aligned} Note E[\sinh^2(U)] does not depend on \xi and \lambda.
The second moment of X is \begin{aligned} E[X^2] &= E[(\xi + \lambda \sinh(U))^2] \\ &= \xi^2 + 2\xi\lambda E[\sinh(U)] + \lambda^2 E[\sinh^2(U)] \\ \end{aligned}
To compare with the lognormal model we want to match the first two moments of X and F = f\exp(\sigma\sqrt{t} Z - \sigma^2t/2). We have E[F] = f and E[F^2] = f^2\exp(\sigma^2 t).
Consider \gamma and \delta as given input parameters corresponding to skew and kertosis respectively. Given f (= se^{rt}) and \sigma we want \begin{aligned} f &= \xi + \lambda E[\sinh(U)] \\ f^2\exp(\sigma^2 t) &= \xi^2 + 2\xi\lambda E[\sinh(U)] + \lambda^2 E[\sinh^2(U)] \\ \end{aligned} Substituting {\xi = f - \lambda E[\sinh(U)]} and using u_n = E[\sinh^n(U)] gives a quadratic expression in \lambda \begin{aligned} f^2\exp(\sigma^2 t) &= (f - \lambda u_1)^2 + 2(f - \lambda u_1)\lambda u_1 + \lambda^2 u_2 \\ f^2\exp(\sigma^2 t) &= f^2 - 2f\lambda u_1 + \lambda^2 u_1^2 + 2f\lambda u_1 - 2\lambda^2 u_1^2 + \lambda^2 u_2 \\ f^2\exp(\sigma^2 t) &= (u_2 - u_1^2)\lambda^2 + f^2 \\ \end{aligned} This implies \lambda = \sqrt{(u_2 - u_1^2)/\operatorname{Var}(F)}.