Stochastic Volatility Models: Heston and SABR in Practice

_TITLE: Stochastic Volatility Models: Heston and SABR in Practice _SLUG: stochastic-volatility-models-heston-sabr-exp14 _EXCERPT: Stochastic volatility models provide a more realistic framework for option pricing by treating volatility as a random process. This article provides an in-depth examination of two of the most widely used stochastic volatility models: the Heston model and the SABR model. _TAGS: stochastic volatility, Heston model, SABR model, option pricing, quantitative finance, volatility modeling, exp14

Introduction to Stochastic Volatility Models

In the landscape of quantitative finance, the accurate modeling of volatility is a subject of paramount importance. While the Black-Scholes-Merton model assumes constant volatility, empirical evidence overwhelmingly demonstrates that volatility is, in fact, time-varying and stochastic. Stochastic volatility models address this shortcoming by treating volatility as a random process, thereby providing a more realistic and flexible framework for pricing and hedging derivatives.

This article offers a detailed exploration of two of the most influential stochastic volatility models used in practice: the Heston model and the SABR model. We will dissect their mathematical formulations, compare their strengths and weaknesses, and discuss their practical applications in the pricing of complex derivatives.

The Heston Model

The Heston model, introduced by Steven Heston in 1993, is a continuous-time stochastic volatility model that has become a benchmark in the industry. It is particularly valued for its ability to capture the volatility smile and skew observed in equity markets and for the fact that it admits a semi-analytical solution for the price of a European option.

Mathematical Formulation

The Heston model is defined by a system of two correlated stochastic differential equations (SDEs):

dS_t = rS_t dt + √v_t S_t dW_t^1
dv_t = κ(θ - v_t)dt + ξ√v_t dW_t^2

dS_t = rS_t dt + √v_t S_t dW_t^1
dv_t = κ(θ - v_t)dt + ξ√v_t dW_t^2

where:

• S_t is the price of the underlying asset.
• v_t is the variance of the asset price.
• r is the risk-free interest rate.
• κ is the rate of mean reversion of the variance.
• θ is the long-term mean of the variance.
• ξ is the volatility of the variance (vol-of-vol).
• dW_t^1 and dW_t^2 are Wiener processes with correlation ρ.

The correlation ρ between the asset price and its volatility is a key feature of the Heston model. A negative correlation, as is typically observed in equity markets, gives rise to the volatility skew.

Pricing with the Heston Model

The price of a European call option in the Heston model can be calculated using the following formula, which involves the numerical integration of the characteristic function of the log-asset price:

C(S_t, v_t, T) = S_t * P_1 - Ke^(-r(T-t)) * P_2

C(S_t, v_t, T) = S_t * P_1 - Ke^(-r(T-t)) * P_2

where P_1 and P_2 are probabilities that are calculated using the characteristic function.

The SABR Model

The SABR model, which stands for Stochastic Alpha, Beta, Rho, was developed by Hagan, Kumar, Lesniewski, and Woodward in 2002. It has become the industry standard for pricing interest rate derivatives, but it is also widely used in other asset classes.

Mathematical Formulation

The SABR model describes the evolution of the forward price of an asset and its volatility:

dF_t = α_t * F_t^β * dW_t^1
dα_t = ν * α_t * dW_t^2

dF_t = α_t * F_t^β * dW_t^1
dα_t = ν * α_t * dW_t^2

where:

• F_t is the forward price of the asset.
• α_t is the stochastic volatility.
• β is the elasticity parameter, which determines the shape of the volatility smile.
• ν is the volatility of volatility.
• dW_t^1 and dW_t^2 are Wiener processes with correlation ρ.

The SABR Formula

One of the key advantages of the SABR model is the existence of an accurate and tractable approximation for the implied volatility, known as the SABR formula. This formula allows for the rapid calibration of the model to market data.

Heston vs. SABR: A Comparison

Conclusion

Stochastic volatility models, such as the Heston and SABR models, represent a significant advancement over the constant volatility assumption of the Black-Scholes model. By treating volatility as a random process, these models are able to capture the complex dynamics of the volatility smile and provide a more accurate framework for pricing and hedging derivatives. While the Heston model is a workhorse for equity derivatives, the SABR model has become the industry standard for interest rate products. The choice between these models, and their various extensions, depends on the specific application and the trade-off between analytical tractability and empirical fit.