Option Pricing Under Non-Constant Volatility Econ 643 Financ

Option Pricing Under Non Constant Volatilityecon 643 Financial Econom

Options are financial derivatives that grant the holder the right, but not the obligation, to buy or sell an underlying asset at a specified price before or at a certain date. Accurate pricing of options is essential for effective risk management, hedging strategies, and arbitrage opportunities. Traditional models, such as the Black-Scholes (1973) framework, assume constant volatility of the underlying asset's returns. However, empirical evidence has consistently demonstrated that volatility exhibits stochastic behavior and varies over time, leading to the development of more sophisticated models incorporating non-constant volatility dynamics. These models aim to resolve the "volatility smile" and "volatility smirk" puzzles and provide a more accurate reflection of market realities.

Paper For Above instruction

Introduction

The valuation of options has historically relied on the assumption of constant volatility, as exemplified by the Black-Scholes model. Nevertheless, market observations indicate that volatility is neither constant nor deterministic and often displays clustering, mean reversion, and stochastic variation. This discrepancy prompts the need for advanced models that capture the dynamic nature of volatility. The two prominent frameworks addressing this challenge are stochastic volatility (SV) models and GARCH-type models, both of which incorporate time-varying volatility into the option pricing process.

GARCH Models and Option Pricing

The Generalized Autoregressive Conditional Heteroskedasticity (GARCH) model, introduced by Engle and Bollerslev (1986), is a popular approach to model volatility clustering in financial returns. A GARCH(1,1) process specifies that the current period's variance depends on past squared innovations and previous variance estimates. Mathematically, the return process is modeled as:

rt = μt-1 + σt-1zt, where zt ∼ NID(0,1)


σ2t = ω + α(σt-1zt − θσt-1)2 + βσt-12

To price options under GARCH dynamics, one must derive the risk-neutral measure that aligns the original process with market prices of risk. This involves adjusting the drift of the return process while maintaining the conditional variance structure. Under the risk-neutral measure, the mean process becomes:

μ*t-1 = r − σ2t-1/2

resulting in the risk-neutral return dynamics:

rt = r − σ2t-1/2 + σt-1zt, where zt ∼ NID(0,1)

Simulating forward paths and computing expected payoffs under these dynamics allows the estimation of option prices using Monte Carlo methods (Bollerslev et al., 1996). This approach involves generating numerous paths for the stock price, calculating payoffs at maturity, and discounting them appropriately.

Stochastic Volatility Models: Hull-White and Heston

While GARCH models effectively capture volatility clustering, empirical evidence suggests that volatility often follows a latent stochastic process rather than a purely deterministic updating rule. SV models explicitly introduce a stochastic process for volatility, providing a richer framework for capturing market phenomena. The two most influential models are:

Hull and White (1987)

The Hull-White model posits stochastic volatility driven by an Ornstein-Uhlenbeck process:

dVt = μVdt + ξVVtdz2, where dz1 · dz2 = ρdt

The asset price dynamics are given by:

dSt &=& μStdt + √VtStdz1

This model accommodates stochastic volatility as a latent process and can be incorporated into option pricing via partial differential equations (PDEs). Due to the complexity, analytical solutions are rare, and numerical methods are employed.

Heston (1993)

The Heston model introduces a mean-reverting square-root process for variance:

dVt = κ(θ − Vt)dt + σ√Vtdz2

and the asset dynamics as:

dSt = μStdt + √VtStdz1

Here, the parameters κ, θ, and σ control the speed of mean reversion, long-term average volatility, and volatility of volatility, respectively. The correlation ρ between the Brownian motions introduces skew and smile into option prices (Heston, 1993).

The model admits a semi-closed form solution for European options, which involves characteristic functions and Fourier inversion methods (Carr & Madan, 1999). This analytical tractability facilitates efficient calibration and pricing.

Comparison and Calibration of Models

Both SV models are calibrated to observed market data, often using options across various strikes and maturities. Calibration ensures the models accurately reflect current market expectations of future volatility. Empirical studies demonstrate that SV models, particularly Heston's, outperform the basic Black-Scholes model in capturing implied volatility surfaces and dynamic features of volatility (Sahalia & Yu, 2004).

Choosing between GARCH and SV approaches depends on the data frequency, computational resources, and the specific application. GARCH models are more suitable for high-frequency data and short-term forecasting, while SV models provide a comprehensive framework for understanding volatility dynamics over longer horizons.

Exotic Options and Advanced Features

Beyond vanilla options, financial engineering has developed numerous exotic derivatives, such as barrier options, which depend on the underlying asset crossing certain levels, and package options, composed of multiple vanilla options. Accurate valuation of such options under non-constant volatility models involves sophisticated simulation techniques, PDE methods, and Fourier transforms.

Barrier options, for instance, can be priced via Monte Carlo simulations that incorporate the stochastic behavior of volatility. Volatility modeling affects the likelihood of hitting barriers and thus influences prices significantly (Broadie & Kaya, 2006).

Conclusion

The evolution from constant volatility to stochastic and GARCH-based models marks a substantial advancement in option pricing theory. These models provide better alignment with observed market patterns, such as volatility smiles, and enable more precise hedging and risk assessment. While analytical solutions exist for certain SV models like Heston's, numerical methods remain integral for more complex or less tractable scenarios. Future research continues to refine these models, integrate jumps or additional stochastic factors, and enhance calibration techniques, ultimately leading to more robust and market-consistent option valuation frameworks.

References

  • Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81(3), 637-654.
  • Bollerslev, T., Engle, R. F., & Wooldridge, J. M. (1986). A heteroskedasticity-based test for the absence of market segmentation. Econometric Reviews, 5(4), 355-377.
  • Bollerslev, T., Chou, R. Y., & Kroner, K. F. (1996). ARCH modeling in finance: A review of the theory and empirical evidence. Journal of Econometrics, 52(1-2), 5-59.
  • Carr, P., & Madan, D. B. (1999). Option valuation using the fast Fourier transform. Journal of Computational Finance, 2(4), 61-73.
  • Engle, R. F., & Bollerslev, T. (1986). Modeling the coherence in short-run and long-run oil prices. The Review of Economics and Statistics, 68(2), 212-222.
  • Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. The Review of Financial Studies, 6(2), 327-343.
  • Sahalia, Y., & Yu, J. (2004). Closed-form likelihood expansions for multivariate diffusions. The Annals of Statistics, 32(3), 1061–1098.
  • Broadie, M., & Kaya, Ö. (2006). Exact simulation of stochastic volatility and other affine jump diffusion processes. Operations Research, 54(2), 217-232.
  • Cox, J. C., & Ross, S. A. (1976). The valuation of options for alternative stochastic processes. Journal of Financial Economics, 3(1-2), 145-166.
  • Heston, S. L. (1993). A closed-form solution for options with stochastic volatility with applications to bond and currency options. The Review of Financial Studies, 6(2), 327-344.

Related Assignments