Johnson Distribution

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 > 0, xi, \lambda > 0 are real numbers and f is function.

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.

To compare with the lognormal model we want to match the first two moments of X and F = f\exp(sZ - s^2/2). We have E[F] = f and E[F^2] = f^2\exp(s^2). If the first two moments are m_1 and m_2 then f = m_1 and s^2 = \log(m_2/m_1^2).

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)}. Let {U = (Z - \gamma)/\delta} so {m = E[U] = - \gamma/\delta} and {s^2 = \operatorname{Var}(U) = 1/\delta^2}.

We have \begin{aligned} E[X] &= E[\xi + \lambda \sinh((Z - \gamma)/\delta)] \\ &= \xi + \lambda E[\sinh(U)] \\ &= \xi + \lambda E[\frac{e^U - e^{-U}}{2}] \\ &= \xi + \lambda \frac{e^{m + s^2/2} - e^{-m + s^2/2}}{2} \\ &= \xi + \lambda e^{1/2\delta^2} \sinh(- \gamma/\delta) \\ \end{aligned}

For small \gamma and \delta near 1 we have E[X] \approx \xi - \sqrt{e}\lambda\gamma/\delta.

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 cumulative distribution of the standard normal.

For share measure we calculate \begin{aligned} E[X 1(X\le x)] &= E[(\xi + \lambda \sinh((Z - \gamma)/\delta)) 1(Z\le z)]\\ &= \xi P(Z\le z) + \lambda E[\sinh(U) 1(Z\le z)]\\ &= \xi P(Z\le z) + \lambda E[\frac{e^U - e^{-U}}{2} 1(Z\le z)]\\ &= \xi P(Z\le z) + \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} Note {E[e^U] = e^{- \gamma/\delta + 1/2\delta^2}}, {E[e^{-U}] = e^{ \gamma)/\delta + 1/2\delta^2}}, and {\operatorname{Cov}(U, Z) = 1/\delta}. Letting z = - \gamma/\delta and s^2 = 1/\delta^2 we have E[X 1(X\le x)] = \xi P(Z\le z) + \frac{\lambda e^{s^2/2}}{2} \left(e^{z}P(Z \le z - s^2\delta) - e^{-z}P(Z \le z + s^2\delta)\right)

Note 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.