In the realm of finance, the risk-neutral measure takes precedence in pricing financial derivatives. However, the real-world measure remains significantly valuable and indispensable across various domains. It plays a pivotal role in risk management and asset/liability applications, facilitating comprehensive evaluation and mitigation of risks. Real-world measures are useful for simulation-based …

The volatility smile is a phenomenon observed in the options market where implied volatility tends to be higher for out-of-the-money (OTM) options compared to at-the-money (ATM) options. It refers to the graphical shape of the volatility curve, resembling a smile when plotted against the strike prices of options. The volatility …

The Black-Scholes option pricing model is a widely used mathematical formula for calculating the theoretical value of European-style options. Developed by economists Fischer Black, Myron Scholes, and Robert Merton in 1973, the model takes into account various factors such as the current stock price, strike price, time to expiration, risk-free …

In option pricing theory, the risk-neutral measure is a measure that allows for the valuation of financial instruments such as options. The risk-neutral measure is obtained by assuming that investors are indifferent to the risk and that the expected rate of return on all assets is equal to the risk-free …

Implied volatility is an estimation of the future volatility of a security’s price. It is calculated using an option-pricing model, such as the Black-Scholes model, as it takes into account various factors including the current price of the underlying asset and its strike price. Implied volatility helps investors to gauge …

Option pricing is usually carried out in the risk-neutral world where the market participants are assumed to be indifferent between taking a certain payoff or investing in an asset with that same expected return. Mathematically, an option’s price is the expected value of its payoff in the risk-neutral measure discounted …

Implied volatility is a theoretical value that measures the expected volatility of a financial instrument over a specific period of time. It is derived from the price of a financial instrument and can be used to gauge market expectations. Implied volatility is often used by traders to make decisions about …

Volatility of an asset is a measure of how much the price of that asset varies over time. In other words, it is a measure of how “risky” an investment in that asset is. The higher the volatility, the greater the risk. There are two main types of volatility: historical …

The Black-Sholes-Merton model is a mathematical model for pricing financial options. The model is used to calculate the theoretical value of an option, which is the amount that the option holder would be paid if they exercised the option. The model takes into account factors such as the underlying asset’s …

Commodity derivatives are financial instruments whose value is based on an underlying commodity. These derivatives can be used for hedging purposes or for speculation. The most common types of commodity derivatives are futures, options, and swaps. Due to their seasonal nature, valuing commodity derivatives requires pricing models that are different …