title: General Option Pricing
author: Keith A. Lewis
institute: KALX, LLC
classoption: fleqn
fleqn: true
abstract: European option pricing and greeks
thanks: Thank you Peter Carr and Bill Goff for your valuable feedback.
...
\newcommand{\Var}{\operatorname{Var}}
\newcommand{\RR}{𝑹}
European option valuation involves calculating the expected value of
the option payoff using the underlying at expiration.
Greeks are derivatives of the option value with respect to model parameters.
This short note derives formulas for these that can be used for any underlying.
Let $F$ have mean $f$ and variance $f^2s^2$. We can write $F = f(1 + s X)$
where $X$ has mean 0 and variance 1. If $X$ is standard normal we have the
Bachelier model. Let $\Phi$ be the cdf of $X$.
Note $F\le k$ iff $X \le (k - f)/fs = x(k)$. $k(x) = f(1 + s x)$.
$dx(k)/df = 1/fs$.
$v = E[\nu(F)]$.
$dv/df = E[\nu'(F)(1 + sX)]$.
$d^nv/df^n = E[\nu''(F)(1 + sX)^n]$.
$dv/ds = E[\nu'(F)fX]$.
Let $\Phi(x, n) = E[X^n 1(X\le x)]$ be the partial moments.
By Carr-Madan $f(x) = f(a) + f'(a)x + \int_{-\infty}^a f''(k)(k - x)^+,dk + \int_a^{\infty} f''(k)(x - k)^+,dk$.
$E[\delta_a(g(X))] = E[\delta_a(Y)]$, $Y = g(X)$.
$\Psi(y) = P(Y\le y) = P(g(X)\le y) = P(X\le g^{-1}(y)) = \Phi(g^{-1}(y))$
$\psi(y) = \Phi'(y) = \phi(g^{-1}(y))dg^{-1}(y)/dy = \phi(g^{-1}(y))/g'(g^{-1}(y))$.
$F = f(1 + sX)$, $g(x) = f(1 + sx)$.
Let $f(x) = (x - a)^2/2 1(x \le a)$
$f'(x) = -(x - a)^2/2 \delta_a(x) + (x - a) 1(x \le a)$
$f''(x) = -(x - a)^2/2 \delta'_a(x) - 2(x - a) \delta_a(x) + 1(x \le a)$.
$E[(k - F)^+] = E[(k - F)1(F \le k)] = k\Phi(x(k)) - E[F1(X \le x(k))]$.
$E[F1(X \le x)] = E[f(1 + s X)1(X\le x)]
fP(X\le x) + fs E[X 1(X\le x)]$
$p = E[(k - F)^+] = (k - f)\Phi(x(k)) + fs\Phi(x(k), 1)$
$dp/df = E[-(F/f)1(F\le k)] = E[-(1 + s X)1(F\le k)]
= -\Phi(x(k)) - s \Phi(x(k), 1)$.
$d^2p/df^2 = $
$d/df E[1(F\le k)] =
$dp/ds = E[-X1(F\le k)] = - \Phi(x(k), 1)$.
$1 + s X = e^{sZ - s^2/2}$, $X = (e^{sZ - s^2/2} - 1)/s$
$s^2 = (e^{s^2} - 1)$, $s^2 = \log(s^2 + 1)$.