& | ~ ^ << >>

&

Bitwise level AND

|

Bitwise level OR

~

Unary bitwise negation

^

Unary XOR

<<

Shift the number to the left. Fill unused slots with 0.

>>

Shift the number to the right

• for signed values, we perform an arithmetic right shift: fill the unused slots with the most significant bit from before (“fill with 1s”)
• for unsigned values, we perform a logical right shift

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