The Heston model, developed by Steven Heston in 1993, is a sophisticated financial model used for pricing欧式期权 and understanding the dynamics of underlying asset prices and their volatility. It is an extension of the classic Black-Scholes model, addressing some of its limitations, particularly the assumption of constant volatility. Key features of the Heston model include:
Unlike the Black-Scholes model, which assumes volatility is constant, the Heston model posits that volatility is random and follows its own stochastic process.
The model describes the evolution of two factors over time:
- The asset price (often a stock price), which is assumed to follow a geometric Brownian motion with a stochastic volatility component.
- The variance of the asset's returns, which is also modeled as a stochastic process.
The volatility process in the Heston model is mean-reverting, meaning that it tends to return to a long-term average level over time. This feature is characterized by the speed of reversion to the mean (kappa), the long-term variance (theta), and the volatility of the volatility (sigma).
The model includes a parameter (rho) that represents the correlation between the asset price returns and the variance changes. This allows for the modeling of scenarios where asset prices and volatility might move together or in opposite directions.
Despite its complexity, the Heston model has a closed-form solution for European option pricing, which makes it computationally efficient for practical use.
The model parameters can be calibrated to market data, such as option prices, to match observed market conditions. This calibration process involves finding the parameters that minimize the difference between model prices and market prices.
The Heston model is widely used in finance for pricing a variety of options, including equity options, FX options, and interest rate options. It is also used for risk management and for understanding the implied volatility surface.
The BSM model, also known as the Black-Scholes-Merton (BSM) model, is a foundational financial model used for pricing and understanding the behavior of financial derivatives, particularly European options. Developed independently by Fischer Black, Myron Scholes, and Robert Merton in the early 1970s, the model has become a cornerstone of financial economics. Here are the key features of the Black-Scholes-Merton model:
The BSM model provides a mathematical formula for pricing European call and put options on stocks that do not pay dividends.
It assumes that the stock price follows a lognormal distribution, meaning that the logarithm of the stock price is normally distributed.
The model assumes that the volatility of the stock's returns is constant over time. This is one of the model's main simplifications.
The BSM model is built on the principle of no-arbitrage, which states that it should not be possible to make a risk-free profit from price differences in the market.
The model uses the concept of risk-neutral probability, which simplifies the valuation of derivatives by removing the risk preference of investors.
The BSM model is known for its formula that calculates the theoretical price of an option based on these factors:
- The current stock price
- The option's strike price
- The time to expiration
- The risk-free interest rate
- The stock's volatility
The model is specifically designed for European options, which can only be exercised at expiration.
The BSM model makes several simplifying assumptions, such as no transaction costs, no taxes, and the ability to borrow and lend money at the risk-free rate.
Despite its simplifying assumptions, the BSM model has had a profound impact on financial theory and practice, providing a standard approach to option valuation and risk management.
The model has been extended in various ways to address its limitations, such as the inclusion of dividends (the Merton model), stochastic volatility (e.g., Heston model), and jumps in the asset price process (e.g., the Kou model).