People have been trading options for a very long time, but there wasn’t a good way of quantify the value of an option.

There are two main types of uses for Black-Scholes Formula

1. you can use all variables and determine the value of options
2. you can get the price of options being traded, then compute the $σ$—the market’s estimation of volatility (how much they want the insurance policy that is the options)

constituents

• \(S_0\): stock price
• \(X\): exercise price
• \(r\): risk-free interest rate
• \(T\): maturity time
• \(\sigma\): standard-deviation of log returns—“volatility”

Black-Scholes Formula for an European “Call” Option

Here is the scary formula:

To evaluate the lower bound:

\begin{equation} \alpha_{a}^{k+1} (s) = R(s,a) + \gamma \sum_{s’}^{} T(s’|s,a) \alpha_{a}^{k}(s’) \end{equation}

we are essentially sticking with an action and do conditional plan evaluation of a policy that do one action into the future

The bloch sphere is a sphere encoding all possible probabilities of a qubit shared between two axis, \(|u\big>\) and \(|d\big>\).

You will notice that its a unit sphere, in which any magnitude has size \(1\). Hence, probabilities would result as projected onto each of the directions.