Hestonmodellen

The Hestonmodel is a mathematical model used in quantitative finance to describe the evolution of an asset's price and its volatility. It was developed by Steven Heston in 1993 and is widely employed for option pricing and risk management.

The core of the Heston model lies in its assumption that the asset price follows a stochastic process, meaning it's influenced by random fluctuations. Crucially, it also posits that the asset's variance, which measures the degree of its price dispersion, is itself a stochastic process. This means that volatility is not constant but changes over time in a random manner.

Specifically, the model typically involves two correlated stochastic differential equations. The first equation describes the dynamics of the asset price, influenced by its expected return and a random shock proportional to its current price and volatility. The second equation describes the dynamics of the variance, often modeled using a mean-reverting process, meaning it tends to return to an average level over time. The correlation between the asset price and its variance is a key parameter, as it captures the tendency for volatility to increase when prices fall (leverage effect).

The Heston model's ability to capture time-varying volatility and its correlation with price movements makes it a more realistic representation of financial markets compared to simpler models like the Black-Scholes model, which assumes constant volatility. This has led to its widespread adoption in pricing complex derivatives, particularly options, and in calculating risk measures like Value at Risk (VaR). However, the model's complexity can make calibration and computation more challenging.