Heston Model

What Is the Heston Model?

The Heston Model, named after Steve Heston, is a type of stochastic volatility model used to price European options.

Understanding the Heston Model

The Heston Model, developed by associate finance teacher Steven Heston in 1993, is an option pricing model that can be utilized for pricing options on different securities. It is comparable to the more famous Black-Scholes option pricing model.

Overall, option pricing models are utilized by advanced investors to estimate and measure the price of a specific option, trading on an underlying security in the financial marketplace. Options, just like their underlying security, will have prices that change all through the trading day. Option pricing models try to examine and integrate the factors that cause vacillation of option prices to distinguish the best option price for investment.

As a stochastic volatility model, the Heston Model purposes statistical methods to compute and forecast option pricing with the assumption that volatility is inconsistent. The assumption that volatility is erratic, as opposed to consistent, is the key factor that makes stochastic volatility models unique. Different types of stochastic volatility models incorporate the SABR model, the Chen model, and the GARCH model.

Key Differences

The Heston Model has qualities that recognize it from other stochastic volatility models, in particular:

• It factors in a potential correlation between a stock's price and its volatility.
• It passes volatility as returning on to the mean.
• It gives a shut structure solution, meaning that the response is derived from an accepted set of mathematical operations.
• It doesn't need that stock prices follow a lognormal likelihood distribution.

The Heston Model is likewise a type of volatility smile model. "Smile" alludes to the volatility smile, a graphical representation of several options with indistinguishable expiration dates that show expanding volatility as the options become more in-the-money (ITM) or out-of-the-money (OTM). The smile model's name gets from the sunken state of the graph, which looks like a smile.

Heston Model Methodology

The Heston Model is a shut structure solution for pricing options that looks to beat a portion of the deficiencies introduced in the Black-Scholes option pricing model. The Heston Model is a device for advanced investors.

The calculation is as follows:

$\begin &dS_t = rS_tdt + \sqrt S_tdW_{1t} \ &dV_t = k ( \theta - V_t ) dt+ \sigma \sqrt dW_{2t} \ &\textbf \ &S_t = \text t \ &r = \text{risk-free interest rate -- theoretical rate on an} \ &\text \ &\sqrt = \text{volatility (standard deviation) of the asset price} \ &\sigma = \text \sqrt \ &\theta = \text \ &k = \text \theta \ &dt = \text \ &W_{1t} = \text \ &W_{2t} = \text{Brownian motion of the asset's price variance} \ \end$

Heston Model versus Black-Scholes

The Black-Scholes model for option pricing was presented during the 1970s and filled in as one of the primary models for assisting investors with determining a price associated with an option on a security. As a general rule, it assisted with advancing option investing as it made a model for investigating the price of options on different securities.

Both the Black-Scholes and Heston Model depend on underlying calculations that can be coded and modified through advanced Excel or other quantitative systems. The Black-Scholes call option formula is calculated by duplicating the stock price by the cumulative standard normal likelihood distribution function.

From that point, the net present value (NPV) of the strike price duplicated by the cumulative standard normal distribution is deducted from the subsequent value of the previous calculation.

In mathematical documentation,

Call = S * N(d₁) - K_e^{(- r * T) * N(d}₂⁾

On the other hand, the value of a put option could be calculated utilizing the formula:

Put = K_e^{(- r * T) * N(- d2)} - S * N(- d₁)

In the two formulas, S is the stock price, K is the strike price, r is the risk-free interest rate, and T is the opportunity to maturity.

The formula for d₁ is:

(ln(S/K) + (r + (Annualized Volatility)²/2) * T)/(Annualized Volatility * (T^0.5))

The formula for d₂ is:

d1 - (Annualized Volatility) * (T^0.5)

Special Considerations

The Heston Model is important in light of the fact that it tries to accommodate one of the primary limitations of the Black-Scholes model which holds volatility consistent. The utilization of stochastic factors in the Heston Model accommodates the thought that volatility isn't steady however erratic.

Both the essential Black-Scholes model and the Heston Model still just give option pricing estimates to an European option, which is an option that must be practiced on its expiration date. Different research and models have been read up for pricing American options through both Black-Scholes and the Heston Model. These varieties give estimates to options that can be practiced on any date leading up to the expiration date, similar to the case for American options.

Features

• This means that the model expects that volatility is erratic, as opposed to the Black-Scholes model that holds volatility steady.
• The Heston Model is an options pricing model that uses stochastic volatility.
• The Heston Model is a type of volatility smile model, which is a graphical representation of several options with indistinguishable expiration dates that show expanding volatility as the options become more ITM or OTM.