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5

It comes from a direct application of the Fourier inversion theorem for a CDF: For a general one-dimensional CDF $F_X(x)$, the Fourier inversion theorem can be described as: \begin{align} F_X(x) &= \frac{1}{2} - \frac{1}{2\pi} \int_{-\infty}^\infty \frac{e^{-iux}\phi_X(u)}{iu} \: du\\ &=\frac{1}{2} - \frac{1}{\pi} \int_0^\infty \mathcal{R}\left[\frac{...

3

Call option gives the right to buy the stock. ATM call option struck at 100 would be worth zero in the lower state and would be worth 7.5 units of money in the upper state (you can buy at 100, when the stock is worth 107.5). As @emot points out, the risk-neutral probability is given at 80% for the upper state, so the option is worth 0.8 * 7.5 = 6. You can ...

0

Normally, the "time" variable unit is years. So the volatility is normalized to be a year volatility.

0

Really wish I knew why my previous version didn't work...but I found some modifications to GBM that got me the answer. import pandas as pd import numpy as np import matplotlib.pyplot as plt class MCPricing(): def __init__(self, S0, drift, sig, T, tIncrement, r) -> None: self.S0 = S0 self.mu = drift self.sig = sig self....

0

this implies that the ATMF implied from options prices, for European options, would be the strike where the price of call = price of put.

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