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HESTON MODEL

By ProbabilityofProfit.com

The Heston Model was developed by American mathematician Steven Heston in 1993, and serves as an extension of the Black-Scholes model. Unlike its predecessor, the Heston Model introduces volatility as a random variable rather than a constant, reflecting the dynamic nature of financial markets.
This feature enables volatility to vary throughout the duration of the option, aligning more closely with real-world market behavior.

THEORY & METHODOLOGY

THEORY

The Heston Model, created by American mathematician Steven Heston in 1993, is a major advancement in financial modeling, building on the Black-Scholes model. Unlike the Black-Scholes model, which assumes volatility is constant, the Heston Model treats volatility as a variable that can change over time. This approach better mirrors the real-world behavior of financial markets, where volatility is anything but stable.
 
By allowing volatility to fluctuate, the Heston Model provides a more accurate and realistic view of market conditions. This makes it particularly useful for pricing options and other derivatives in volatile environments. Its ability to capture the unpredictable nature of market volatility offers traders and analysts a powerful tool for assessing risk and valuing derivatives under more realistic scenarios.

METHODOLOGY

The Heston Model follows a similar approach to the representation of both the underlying stock price and volatility by employing a stochastic process. 
  • Employs the Geometric Brownian motion for stock price dynamics similar to the Black-Scholes model.
  • Employs the Cox-Ingersoll-Ross (CIR) process for modeling stochastic volatility which is a mean-reverting square-root process that captures the tendency of volatility. 
This model more accurately represents market conditions than that of the Black-Scholes model through its use of various implied volatility conditions.

 

CALCULATIONS

HESTON ASSUMPTIONS

In the most fundamental form of the Heston model these assumptions are to be expected:
  • W1t is the Brownian motion of the asset price
  • W2t is the Brownian motion of the asset’s price variance
  • ρ is the correlation coefficient for W1t and W2t
  • St is the price of a specific asset at time t
  • √Vt is the volatility of the asset price
  • σ is the volatility of the volatility
  • r is the risk-free interest rate
  • θ is the long-term price variance
  • k is the rate of reversion to the long-term price variance
  • dt is the infinitely small positive time increment
     
Note that the Brownian motions are random processes that exhibit the following properties:
  • W0 = 0
  • Wt has independent movements
  • Wt is continuous in t
  • Increments of Wt – Ws have a normal distribution, mean zero, and variance|t – s|

FORMULAS

Heston Model Formulas: 
Dynamics of the asset price St
dSt = rStdt + √Vt StdW1t
  • It follows a geometric Brownian motion with a drift term rStdt and a volatility term, √Vt StdW1t where r is the risk-free interest rate, W1t is the Brownian motion of the asset price, and Vt is the volatility of the asset price.
Dynamics of the volatility Vt
dVt = k(θ-Vt)dt + σ √Vt dW2t
  • It follows a mean-reverting process towards a long-term mean θ with a rate k, along with a volatility term σ √Vt dW2t , where σ is the volatility of the volatility, k is the rate of reversion to the long-term variance, θ is the long-term variance, and W2t is the Brownian motion of the asset's price variance.

HESTON MODEL EXAMPLE

Let's consider a simplified example using the Heston model to describe the dynamics of the stock price and its volatility. Suppose we have a stock, let's call it XYZ, currently trading at $100 per share. We want to model how the price of XYZ evolves over time, along with its volatility, using the Heston model.

Here's how we can set up the parameters:
  • Initial stock price: S0 = $100
  • Risk-free interest rate: r = 0.05 (5% per annum)
  • Long-term variance: θ = 0.04
  • Rate of reversion to the long-term variance: k = 0.1
  • Volatility of volatility: σ = 0.3
  • Initial volatility: V0 = 0.04

Define The Time Frame:
  • Total time period: 1 year
  • Time steps: Divide the year into discrete intervals (e.g., daily steps, so 252 steps assuming 252 trading days in a year).

Generate Random Increments:
  • For each time step Δt , generate random increments dW1,t and dW2,t from a normal distribution with mean zero and variance Δt. Ensure the correlation between these increments if necessary. 

Update Stock Price and Volatility: 
  • Use the Heston model equations to update the stock price St and volatility Vt at each time step:
  • Iterate these equations for each time step to simulate the evolution of St and Vt over time. 

Simulate the Process:
  • Repeat the above steps for each time step across the entire year to generate a trajectory for the stock price and its volatility.

PROS & CONS

PROS

CONS

Incorporates stochastic volatility: Unlike the Black-Scholes model, which assumes constant volatility, the Heston Model incorporates stochastic volatility, allowing volatility to vary over time. This better reflects real-world market conditions where volatility is not constant.

Complexity: The Heston Model is more complex than the Black-Scholes model due to its incorporation of stochastic volatility.

Flexibility in modeling volatility: Through the use of the Cox-Ingersoll-Ross (CIR) process for modeling stochastic volatility, the Heston Model captures the mean-reverting nature of volatility thus reflecting the tendency of volatility to revert to a long-term average, which is observed in many financial markets.

 

Accurate representation of market conditions: The Heston Model's ability to incorporate various implied volatility conditions allows it to more accurately represent market dynamics compared to the Black-Scholes model

 

CONCLUSION

The Heston model stands out by allowing volatility to change over time, offering a more nuanced and accurate picture of financial markets compared to the Black-Scholes model. This makes it an invaluable tool for pricing options and other derivatives in situations where market volatility is anything but constant.
 
The Heston model’s sophisticated approach provides a deeper understanding of market dynamics, enhancing both pricing accuracy and risk management strategies. As you dive into options trading and derivative pricing, mastering the Heston model will empower you to better navigate and seize market opportunities. Thanks for exploring this guide, and best of luck with your trading!

REFERENCES

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