Unleashing the Power of the Heston Model for Option Pricing
The Heston Model, devised by Steven Heston in 1993, is a form of stochastic volatility model employed to price European options. Unlike the Black-Scholes model that assumes constant volatility, the Heston Model acknowledges the arbitrary nature of volatility.
Key Takeaways
- The Heston Model is an options pricing framework rooted in stochastic volatility.
- It deviates from the Black-Scholes model by assuming that volatility is variable, not constant.
- As a type of volatility smile model, the Heston Model presents a graph where options with identical expiration dates demonstrate rising volatility as they move more in-the-money (ITM) or out-of-the-money (OTM).
Understanding the Heston Model
The Heston Model is an advanced option pricing tool used for valuing options on various securities, paralleling but distinct from the more widespread Black-Scholes model. Designed for adept investors, option pricing models estimate the worth of options that fluctuate throughout the trading day. These models aim to understand and incorporate various factors affecting option prices to pinpoint the most lucrative prices.
As a stochastic volatility model, the Heston Model employs statistical techniques to predict option pricing with the presumption that volatility is dynamic. Stochastic models like the Heston Model are unique in their treatment of volatility. Other examples include the SABR model, the Chen model, and the GARCH model.
Key Differences of the Heston Model
The Heston Model exhibits several distinguishing characteristics:
- It considers the potential correlation between stock prices and their volatility.
- It models volatility as reverting to the mean over time.
- It offers a closed-form solution derived from specific mathematical operations.
- It rejects the necessity of stock prices following a lognormal probability distribution.
Moreover, the Heston Model is a type of volatility smile model, where the graph mirrors a concave shape resembling a smile. This graph plots options with identical expiration dates showcasing increasing volatility as they become more ITM or OTM.
Mastering the Heston Model Methodology
The Heston Model offers a closed-form solution for option pricing, addressing some limitations inherent in the Black-Scholes model. It’s predominantly a resource for well-versed investors.
Calculating Heston Model
\begin{aligned}
&dS_t = rS_tdt + \sqrt{ V_t } S_tdW_{1t} \\
&dV_t = k ( \theta - V_t ) dt + \sigma \sqrt{ V_t } dW_{2t} \\
&\textbf{where:} \\
&S_t = \text{Asset price at time } t \\
&r = \text{Risk-free interest rate} \\
&\sqrt{ V_t } = \text{Volatility of the asset price} \\
&\sigma = \text{Volatility of the } \sqrt{ V_t } \\
&\theta = \text{Long-term price variance} \\
&k = \text{Rate of reversion to } \theta \\
&dt = \text{Infinitesimal time increment} \\
&W_{1t} = \text{Brownian motion of the asset price} \\
&W_{2t} = \text{Brownian motion of the asset's price variance} \\
\end{aligned}
Heston Model vs Black-Scholes
The Black-Scholes model, introduced in the 1970s, revolutionized option pricing. However, it assumes constant volatility whereas the Heston Model adopts a more flexible approach. Both models are grounded in complex calculations capable of being programmed into advanced software systems.
For Black-Scholes, the call option formula:
Call = S \times N(d1) - K e^{-rT} \times N(d2)
The put option formula:
Put = K e^{-rT} \times N(-d2) - S \times N(-d1)
Where:
- S is the stock price.
- K is the strike price.
- r is the risk-free interest rate.
- T is the time to maturity.
d1 and d2 are calculated as:
\begin{aligned}
&d1 = \frac{\ln(S/K) + (r + \frac{Vol^2}{2})T}{Vol \sqrt{T}} \\
&d2 = d1 - Vol \sqrt{T}
\end{aligned}
Special Considerations
The Heston Model stands out by addressing a major limitation of Black-Scholes—the assumption of constant volatility. Stochastic variables in the Heston Model reflect the dynamic nature of volatility. However, both models traditionally giver option pricing estimates for European options—these are exercisable only upon expiration. Enhanced variations cater to American options, which are exercisable any time before expiration.
Related Terms: European Options, Stochastic Volatility, Black-Scholes, Volatility Smile, GARCH Model, Brownian Motion.
References
- Corporate Finance Institute. “Heston Model”.
