What Is the Heston Model?
Let me explain the Heston Model directly to you: it's a stochastic volatility model named after Steve Heston, designed specifically for pricing European options. Unlike the Black-Scholes model that assumes constant volatility, this one treats volatility as variable and arbitrary.
Key Takeaways
You should know that the Heston Model relies on stochastic volatility for options pricing, meaning it views volatility as random rather than fixed, setting it apart from Black-Scholes. It's also a volatility smile model, which graphs options with the same expiration showing higher volatility as they go more in-the-money or out-of-the-money.
Understanding the Heston Model
I developed an understanding of the Heston Model from its origins in 1993 by finance professor Steven Heston; it's an options pricing tool for various securities, similar to Black-Scholes but more advanced. As an advanced investor, you'll use such models to estimate option prices on underlying securities, factoring in daily price changes. What makes the Heston Model stand out is its stochastic approach, using statistics to forecast prices with arbitrary volatility—this is key, unlike constant volatility in other models. You might compare it to alternatives like SABR, Chen, or GARCH models.
Key Differences
Here's what sets the Heston Model apart: it considers potential correlations between a stock's price and its volatility, assumes volatility reverts to the mean, provides a closed-form solution from standard math operations, and doesn't demand that stock prices follow a lognormal distribution. Additionally, it's a volatility smile model, where the graph of options with matching expirations curves like a smile due to rising volatility for ITM or OTM options.
Heston Model Methodology
You need to grasp that the Heston Model offers a closed-form solution to fix Black-Scholes flaws, making it essential for advanced investors. The core equations are: dS_t = r S_t dt + √V_t S_t dW_{1t} and dV_t = k (θ - V_t) dt + σ √V_t dW_{2t}, where S_t is the asset price at time t, r is the risk-free rate, √V_t is the volatility of the asset price, σ is the volatility of √V_t, θ is the long-term variance, k is the reversion rate to θ, dt is a small time increment, W_{1t} is Brownian motion for the asset price, and W_{2t} for the variance.
Heston Model vs. Black-Scholes
Let me compare this to Black-Scholes, introduced in the 1970s to help derive option prices and boost options trading. Both models use programmable calculations, like in Excel. For Black-Scholes, the call option is S * N(d1) - K e^{-rT} * N(d2), and put is K e^{-rT} * N(-d2) - S * N(-d1), with d1 and d2 based on stock price S, strike K, rate r, time T, and volatility. You see, Heston improves on this by allowing arbitrary volatility.
Special Considerations
Remember, the Heston Model addresses Black-Scholes' constant volatility limit by introducing stochastic elements, making volatility arbitrary. Both primarily estimate European options, exercisable only at expiration, but variations exist for American options that can be exercised anytime.
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