April 25, 2024
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).
$d2p/df2 = $
$d/df E[1(Fk)] =
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).