Heston Model

The Heston model is a famous mathematical model used in quantitative finance to describe the dynamics of asset prices, particularly in the context of options pricing. It aims to capture the volatility observed in options markets, where the volatility of options with different strikes and maturities can deviate from the volatility implied by the Black-Scholes model.

It aims to provide more accurate pricing and risk management of options, especially when the Black-Scholes model fails to capture market behavior adequately. It was introduced by American economist Steven Heston in 1993 and has become a standard model in stochastic volatility modeling.

• The Heston model incorporates a stochastic volatility process, allowing the underlying asset's volatility to fluctuate. This feature captures the observed volatility clustering and means reversion in real-world financial markets.
• It is known for capturing the volatility, representing the market's expectation of volatility. By incorporating stochastic volatility and correlation, the model generates option prices reflecting the implied volatility of different strike prices.
• It introduces a correlation between the underlying asset's price and its volatility. This correlation captures the empirical observation that volatility changes often accompany asset price changes.

The Heston model is a widely used mathematical model in quantitative finance, specifically in options pricing and risk management. Its origin can be traced back to Steven Heston's seminal paper published in 1993, "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options." Heston developed the model to address the limitations of the Black-Scholes model, which assumes constant volatility and cannot capture the volatility smile observed in options markets.

One of the critical contributions of the model is its incorporation of stochastic volatility. By modeling volatility as a random process that follows the CIR (Cox-Ingersoll-Ross) process, the model acknowledges the empirical evidence that volatility in financial markets is not constant but exhibits time-varying behavior. In addition, this stochastic volatility feature allows the model to capture volatility clustering, mean reversion, and the observed volatility smile, which is critical for accurate pricing and hedging options.

Including a correlation between the asset price and volatility is another significant aspect of the model. This correlation captures the fact that changes in the asset price are often accompanied by changes in volatility, as observed in real-world markets. By accounting for this correlation, the model provides a more realistic depiction of the relationship between asset prices and their associated volatilities, improving option pricing accuracy.

Here are the main assumptions of the Heston Model:

1. Geometric Brownian Motion: The underlying asset price follows a geometric Brownian motion. This assumption implies that the asset price changes continuously and follows a log-normal distribution. It is a common assumption in many financial models, including the Black-Scholes model.
2. Stochastic Volatility: The underlying asset's volatility is not constant but follows a stochastic process. Specifically, the model employs the Cox-Ingersoll-Ross (CIR) process to describe the evolution of volatility. In addition, the CIR process incorporates mean reversion, ensuring the volatility reverts to a long-term average value over time.
3. Constant Interest Rates: The model assumes that interest rates are stable and do not fluctuate over the option's life. This assumption simplifies the analysis and allows for closed-form solutions in the model.
4. No Dividends: The model assumes that the underlying asset does not pay dividends during the option's life. This assumption is common in options pricing models and simplifies the calculations.
5. Correlation between Asset Price and Volatility: The model introduces a correlation between asset price and volatility. This correlation captures the empirical observation that changes in asset prices tend to be accompanied by changes in volatility.
6. Normal Distribution: The model assumes that the random components in the model, such as asset price returns and volatility changes, are typically distributed. This assumption allows for tractable mathematical analysis and closed-form solutions.

Let us understand it better with the help of examples:

Example #1

Suppose we have a stock called "XYZ" with a current price of $100. We want to price a European call option with a strike price of $110 and a maturity of 1 year using the Heston model.

Assuming the following parameters in our Heston model:

• Initial stock price: S₀ = $100
• Long-term average volatility: v̄ = 0.15 (15%)
• Mean reversion speed: κ = 1.5
• Volatility of volatility: ξ = 0.3 (30%)
• Correlation between asset price and volatility: ρ = -0.4
• Risk-free interest rate: r = 0.05 (5%)

Now, we can simulate the model to generate a price path for the stock and its associated volatility over time. We can obtain a range of future stock prices and volatilities using numerical methods like Monte Carlo simulation.

Once we have the simulated stock price paths and volatilities, we can calculate the payoff of the European call option at the maturity date. If the stock price at maturity (S_T) is above the strike price of $110, the payoff is the difference between the stock price and the strike price. Otherwise, the fix is zero.

Next, we discount the expected payoff to the present using the risk-free interest rate. Finally, taking the average of the discounted gains gives us the estimated price of the European call option according to the Heston model.

Example #2

One real-world example of the Heston model being used in recent news is its application in the pricing and risk management of options during the COVID-19 pandemic. The extreme market volatility experienced in early 2020 due to the pandemic highlighted the limitations of traditional option pricing models, such as the Black-Scholes model, which assumes constant volatility.

During the volatile market conditions, the model became particularly relevant with its ability to capture stochastic volatility and the correlation between asset price and volatility. As a result, financial institutions and market participants utilized the Heston and other stochastic volatility models to price accurately and hedge options in turbulent markets.

During that time, the news and financial reports mentioned the importance of incorporating stochastic volatility models, including the model, to adjust option prices and account for the increased market uncertainty and volatility associated with the COVID-19 crisis.

Here are some critical limitations of the Heston Model:

1. Simplifying Assumptions: It relies on several simplifying assumptions to make the mathematics tractable. For instance, it assumes constant interest rates, no dividends, and normally distributed random components.
2. Calibration Challenges: It requires calibration of its parameters to fit market data accurately. However, calibrating the model can be challenging due to the parameters' interdependence and the model's complexity.
3. Computational Complexity: It is computationally intensive, particularly regarding pricing options numerically. The model involves solving partial differential equations or utilizing numerical methods such as Monte Carlo simulation. These computational requirements can be time-consuming and resource-intensive, especially for large-scale applications or real-time risk management.
4. Model Misspecification: Like any specific financial model, it represents a simplified representation of reality. It assumes particular dynamics for asset prices and volatilities, which may only partially capture some market features and anomalies. As a result, using the model, or any single model, may result in model misspecification if the actual market dynamics deviate significantly from the model assumptions.
5. Limited Tail behavior assumes that asset returns and volatilities follow a normal distribution. However, empirical evidence suggests that asset returns often exhibit fat tails, meaning that extreme events occur more frequently than predicted by a normal distribution.
6. Market Liquidity and Frictions: It assumes frictionless markets with perfect liquidity, which may not reflect the reality of financial markets. Market frictions like transaction costs, bid-ask spreads, and liquidity constraints can impact option pricing and hedging strategies, introducing deviations from the model's idealized assumptions.

The Heston and Black-Scholes models are mathematical models used in quantitative finance, particularly in pricing options. While they share similarities, they also fundamentally differ in assumptions and capabilities. Here's a comparison between the Heston model and the Black-Scholes model:

#1 - Assumptions

The Black-Scholes model assumes constant volatility, no dividends, risk-free interest rates, and log-normal distribution of asset returns. It does not account for stochastic fluctuations or correlation between asset price and volatility.

The Heston model incorporates stochastic volatility, mean reversion in volatility, and correlation between asset price and volatility. In addition, it allows time-varying and random volatilities, capturing volatility clustering and mean regression observed in real-world markets.

#2 - Volatility Dynamics

The Black-Scholes model assumes a constant volatility parameter throughout the option's life, which does not capture the empirical evidence of changing market volatility levels.

The Heston model incorporates a stochastic volatility process that follows the Cox-Ingersoll-Ross (CIR) model. It allows the volatility to fluctuate over time, capturing the observed volatility clustering and mean reversion.

#3 - Volatility Smile

The Black-Scholes model assumes constant implied volatility, leading to a flat volatility smile. However, it does not account for the empirically observed phenomenon where implied volatility varies with the strike price, resulting in a non-flat volatility smile in options markets.

The Heston model is known for capturing the volatility smile, a key feature of options markets. By incorporating stochastic volatility and correlation, the model can generate volatility smiles that reflect the market's expectation of future volatility, providing a more accurate representation of option prices.

#4 - Complexity And Computations

The Black-Scholes model has a closed-form solution, enabling straightforward calculations of option prices and Greeks. Moreover, it is relatively computationally efficient.

The Heston model is more complex and needs a closed-form solution, requiring numerical methods such as Monte Carlo simulation or partial differential equation solving. As a result, it is computationally more demanding and time-consuming compared to the Black-Scholes model.

#5 - Flexibility And Realism

The Black-Scholes model is more straightforward and assumes constant volatility and other idealized conditions. It is suitable for situations where volatility is relatively stable, and market frictions are minimal.

The Heston model is more flexible and realistic, incorporating stochastic volatility and correlation. As a result, it is better suited for capturing the dynamics of volatile markets, accommodating changing volatility levels, and capturing the volatility smile.

This article has been a guide to what is Heston Model for Option Pricing. We explain the topic with examples, assumptions, limitations & vs Black Scholes. You may also find some useful articles here -

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