Geometric Brownian Motion (GBM) is a widely used mathematical model for simulating the random behavior of asset prices in financial markets. It assumes that the price of an asset follows a continuous-time stochastic process, where the logarithmic returns are normally distributed. GBM is foundational in option pricing models like Black-Scholes-Merton. …

Bitcoin ETF options started trading last week. The debut of Bitcoin ETF options was met with significant bullish sentiment, as over 80% of trades were call options. Investors exhibited strong optimism about Bitcoin’s future price, with many purchasing options at a strike price of $100,000. This launch marks an important …

Catastrophe bonds, or CAT bonds, are a type of risk-linked security designed to transfer the financial risk of natural disasters from insurers to investors. These bonds are typically issued by insurance or reinsurance companies to cover significant losses caused by events such as hurricanes, earthquakes, or floods. Investors in CAT …

It’s well-known that there is a strong negative correlation between the S&P 500 index and the VIX. But what exactly is the causal relationship between them? In the financial media, we often see analysts using the VIX as an indicator to time and predict market direction. Is this approach valid? …

VIX Exchange-Traded Products (ETPs) are financial instruments designed to provide exposure to the CBOE Volatility Index (VIX), which measures the market’s expectation of 30-day forward-looking volatility derived from S&P 500 options. VIX ETPs, such as ETFs and ETNs, allow investors to gain exposure to volatility without directly trading options or …

The Implied Volatility Surface (IVS) represents the variation of implied volatility across different strike prices and maturities for options on the same underlying asset. It provides a three-dimensional view where implied volatility is plotted against strike price (moneyness) and time to expiration, capturing market sentiment about expected future volatility. Unlike …

VIX futures are financial derivatives that allow traders to speculate on or hedge against future volatility in the stock market, specifically the S&P 500 index. They are based on the CBOE Volatility Index (VIX), which measures market expectations of near-term volatility. Unlike spot VIX, which reflects current market volatility, VIX …

The Hull-White one-factor (1F HW) interest rate model is a widely used model in finance for simulating the evolution of interest rates over time. It is based on a mean-reverting stochastic process where short-term interest rates revert to a long-term mean at a certain speed. The model incorporates two key …

A convertible bond is a type of hybrid security that acts like a traditional bond but includes an option to convert the bond into a predetermined number of shares of the issuing company’s stock. This feature offers investors the potential to benefit from rising stock prices while providing downside protection …

Dividend futures are financial derivatives that allow investors to speculate on or hedge against the future dividend payments of a specific company or index. Unlike traditional equity futures, which are based on the price of the underlying asset, dividend futures are directly linked to the dividends that a company or …