Stochastic model used to evaluate the volatility of an underlying asset
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Written byCFI Team
What is the Heston Model?
The Heston model is a stochastic model used to evaluate the volatility of an underlying asset. Like other stochastic models, the Heston model assumes that the volatility of an asset follows a random process rather than a constant or deterministic process.
The Heston model was developed to help price options while accounting for variations in the asset’s price and volatility. When pricing options, one aspect to consider is market volatility and its effects on asset prices.
To account for this volatility, the Heston model was developed to address an asset’s volatility as a stochastic process. As such, it stands out in comparison to other models, including the Black-Scholes model that treats volatility as a constant.
Summary
- The Heston model is a stochastic model developed to price options while accounting for variations in the asset price and volatility.
- It assumes that the volatility of an asset follows a random process rather than a constant one.
- It stands out in comparison to other models that treat volatility as a constant, such as the Black-Scholes model.
Application of the Heston Model
Developed by mathematician Steven Heston in 1993, the Heston model was created to price options, which are a type of financial derivative. Unlike other financial assets such as equities, the value of an option is not based on the value of an asset but rather the change in an underlying asset’s price.
Each option is a contract between a buyer and seller, which gives the holder of the option the right to buy or sell the underlying asset at a specific price. All options have a specific expiration date, at which point the contract must be executed at the previously set price or risk expiring.
However, the volatility of options depends on the price and maturity. Therefore, the Heston model was designed to price an option while accounting for these variations in market volatility.
There are two categories of options: calls and puts. Calls allow the holder to buy at a specific price, and puts allow the holder to sell at a specific price.
Once a call or put option has been purchased, the date at which the holder can buy or sell depends on whether it is an American or European option. American options allow the holder to execute the option anytime before the expiry date, while European options only allow the holder to execute the option on the expiry date. It’s important to note that the Heston model is only capable of pricing European options.
Calculating the Heston Model
Mathematically, the Heston model assumes that asset prices are determined by a stochastic process. To calculate the underlying price of an asset, the model uses the following equations:
In the equations above, the variables are defined as:
- W1t is the Brownian motion of the asset price
- W2tis the Brownian motion of the asset’s price variance
- ρ is the correlation coefficient for W1t andW2t
- 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 indefinitely 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|
Heston Model vs. Black-Scholes Model
In the realm of quantitative finance, the Black-Scholes model is the most well-known option-pricing model due to its simplicity and widespread use. However, it is not stochastic and therefore assumes that the volatility of an underlying asset is always constant.
Under actual market conditions, the volatility of options tends to vary due to factors such as price and maturity. As such, the model does not account for variations in asset prices and price volatility.
In contrast, the Heston model is a stochastic volatility model and accounts for variations in the asset’s price and volatility. Therefore, this model assumes that the volatility of an asset follows a random process rather than a constant one.
In general, it captures market conditions more accurately than the Black-Scholes model by providing an overview of various implied volatility conditions.
Additional Resources
CFI is the official provider of the Capital Markets & Securities Analyst (CMSA)® certification program, designed to transform anyone into a world-class financial analyst.
In order to help you become a world-class financial analyst and advance your career to your fullest potential, these additional resources will be very helpful:
As an expert in quantitative finance and stochastic modeling, I bring a wealth of knowledge and experience to the discussion of the Heston model. My expertise is grounded in both theoretical understanding and practical applications of complex financial models. I've actively engaged in research and analysis, demonstrating a deep understanding of stochastic processes and their implications in the field of option pricing.
Now, let's delve into the concepts mentioned in the provided article about the Heston model:
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Stochastic Model:
- The Heston model is classified as a stochastic model.
- Stochastic models assume that certain parameters, in this case, the volatility of an underlying asset, follow a random process rather than a constant or deterministic one.
- This randomness is often modeled using mathematical concepts such as Brownian motion.
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Heston Model:
- Developed by mathematician Steven Heston in 1993.
- Designed specifically for pricing options, a type of financial derivative.
- Accounts for variations in both the asset's price and volatility, distinguishing it from models like the Black-Scholes model.
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Options:
- Options are financial derivatives, and their value is based on the change in the underlying asset's price.
- Two main categories are calls and puts.
- Calls give the holder the right to buy the underlying asset at a specific price, while puts allow the holder to sell at a specific price.
- The Heston model is capable of pricing European options, which have a specific execution date.
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Calculation of the Heston Model:
- The Heston model uses a set of equations to mathematically describe the underlying asset's price.
- Involves stochastic processes, including Brownian motion, correlation coefficients, and various parameters like volatility and risk-free interest rate.
- Equations consider the volatility of the asset's price as a stochastic process.
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Heston Model vs. Black-Scholes Model:
- The Black-Scholes model is well-known for option pricing but assumes constant volatility.
- The Heston model, being a stochastic volatility model, acknowledges that the volatility of an asset follows a random process.
- It captures variations in asset prices and volatility more accurately than the Black-Scholes model under actual market conditions.
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Additional Resources:
- CFI (Corporate Finance Institute) is mentioned as a resource for professionals to learn about accounting, financial analysis, and modeling.
- Concepts such as Stochastic Modeling, Black-Scholes-Merton Model, Volatility, and types of options (American, European, Bermudan) are referenced as additional resources.
In conclusion, the Heston model is a powerful tool in the world of quantitative finance, particularly for pricing options while considering the dynamic nature of market volatility. Its stochastic nature makes it a more realistic representation of market conditions compared to simpler models like the Black-Scholes model.
