Now a trick is used to find the risk-free expectation \(\hat E[\ln\frac{S_T}{S_0}]\) using the prices of call options and put options with the same maturity \(T\): for any \(S_0>0\)
\begin{align}
\int_{0}^{S_0} \frac{1}{k^2} (k- S_T)^{+} dk + \int_{S_0}^{\infty} \frac{1}{k^2} (S_T - k)^{+} dk = \ln ...

Let \(S_t\) be the price of the stock at time \(T\), \(0\le t \le T\). Only \(S_0\) is known, and \(S_t\) is modeled as a Geometric Brownian Motion. That is, \(dS = r S dt + \sigma S dz\), where \(r\) is the risk-free interest rate. For example, the forward price with maturity \(T\) is \(F_0 = \hat ...

The Cboe Volatility Index (VIX), often dubbed the market's "fear gauge," provides a real-time measure of the market's expectation of 30-day forward-looking volatility for the S&P 500 Index.

Unlike a simple average, the VIX is calculated using a methodology known as the Bracket Method , which ...

In 1993, Cboe Global Markets, Incorporated (Cboe) introduced the Cboe Volatility Index (VIX Index). Originally designed to measure the market’s expectation of 30-day volatility implied by at-the-money S&P 100 Index (OEX Index) option prices, the VIX Index soon became the premier benchmark for U.S ...

Just testing
LaTeX: \(e^{i\pi} + 1 = 0\)

Monte Carlo methods are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. They operate on the principle that, given a sufficiently large number of samples, the distribution of those samples will approximate the true probability distribution ...

In the post https://mathmotives.com/viewtopic.php?t=8 we talked about the Laws for powers:

Laws of Powers: let \(a > 0, b > 0\) be two positive numbers, \(m, n\) be two integers. Then

- Products of Powers: \(a^{m} \cdot a^{n} = a^{m + n}\);
- Powers of Powers: \( (a^{m})^{n} = a^{m \cdot n ...

When learning the numbers, we start with counting, and then introduce additions, multiplications and powers.
- additions: continued counting. Example: \(3 + 4 = 7\) and \( 3 + 4 + 5 = 12 \);
- multiplications: repeated additions. Example: \(3 + 3 + 3 + 3 + 3 = 5 \cdot 3 \);
- powers: repeated ...

In mathematics, the Fundamental Theorem of Arithmetic, also called the Unique Factorization Theorem and Prime Factorization Theorem, states that every integer \(n\) greater than \(1\) is either prime or can be represented uniquely as a product of prime numbers, up to the order of the factors:
\[n=p ...

In statistics, linear regression is a model that estimates the relationship between a scalar response (dependent variable) and one or more explanatory variables (regressor or independent variable).