Option Pricing Model Biases: Bayesian and Markov Chain Monte Carlo Regression Analysis

Abstract

We investigate systematic and unsystematic option pricing biases in (a) pure jump Lévy, (b) jump-diffusion, (c) stochastic volatility, and (d) GARCH models applied to the Black–Scholes–Merton model. We use options data for trades on the S&P500 index from the CBOE. In addition to standard ordinary least square regression, we employ Bayesian regression and Markov Chain Monte Carlo regression to investigate the moneyness and maturity biases of the models. We demonstrate the usefulness of these advanced methodologies as compared to the benchmark techniques.

Appendix: Fundamental Models of Previous Studies and Their Dynamics

This Appendix provides a short description of the models corresponding to different approaches to incorporating skewness and kurtosis.

The Black–Scholes–Merton (BSM) framework uses geometric Brownian motion to model an underlying asset. Thereby, the BSM provides a closed form solution to European options. The price of a European option is determined (Shiryaev 1999):

$$\nu \left( {0,S_{0} } \right) = {\text{e}}^{ - rT} E\left[ {V\left( {S_{T} } \right)} \right]$$

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where V (S_T) represents the payoff of the option at expiration, T, r is the continuously-compounded rate of interest, and E denotes the expectation under the risk-neutral probability that is derived from the risk-neutral process:

$$\frac{{{\text{d}}S_{t} }}{{S_{t} }} = r{\text{d}}t + \sigma {\text{d}}B$$

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The BSM considers a normal distribution for log-returns and therefore fails to incorporate any effect resulting from skewness and kurtosis of returns. A more realistic pricing model is obtained by replacing simple Brownian motion with a richer Lévy process that incorporates the empirical evidence of jumps in returns.² In practice, of these Lévy model prices must be computed through (time-consuming) numerical inversion of characteristic functions.

Here, we briefly revisit the other option pricing models where the skewness is characterized quite differently. For pricing options, we use risk-neutral characterization of each model.

1.1 Heston (1993) Stochastic Volatility Model

The Heston (1993) stochastic volatility model assumes a diffusion process for the stock price given, using the usual symbols, by

$$\frac{{{\text{d}}S_{t} }}{{S_{t} }} = \mu {\text{d}}t + \sqrt \vartheta_{t} {\text{d}}B_{t}^{1}$$

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coupled with a CIR process for the volatility $\sqrt {\vartheta_{t} }$ given by

$${\text{d}}\vartheta_{t} = \kappa \left[ {\theta - \vartheta_{t} } \right]{\text{d}}t + \sigma \sqrt {\vartheta_{t} } {\text{d}}B_{t}$$

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with

$${\text{d}}B_{t} {\text{d}}B_{t}^{1} = \rho {\text{d}}t$$

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The model entails a flexible distributional structure, where the correlation (ρ) between volatility and asset returns controls the level of asymmetry, and the volatility variation coefficient (σ) controls the level of kurtosis. A non-zero correlation incorporates skewness into the model. The solution, expressed in standard symbols, is given by:

$$c_{{{\text{HS}}}} = S_{t} \left( {\frac{1}{2} + \frac{1}{\pi }\int_{0}^{\infty } {{\text{Re}} \left[ {\frac{{K^{ - iz} f_{1} }}{iz}} \right]{\text{d}}z} } \right) - K{\text{e}}^{rt} \left( {\frac{1}{2} + \frac{1}{\pi }\int_{0}^{\infty } {{\text{Re}} \left[ {\frac{{K^{ - iz} f_{2} }}{iz}} \right]{\text{d}}z} } \right)$$

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where $f_{j} = \exp \left\{ {C_{j} + D_{j} \vartheta + izx} \right\}$ with

$$\begin{aligned} & x = x_{t} = \log \left( {S_{t} } \right) \\ & \vartheta = \vartheta_{t} \\ & C_{j} = irz\left( {T - t} \right) + \frac{\kappa \theta }{{\sigma^{2} }}\left\{ {\left( {b_{j} - i\rho \sigma z + d_{j} } \right)\left( {T - t} \right) - 2\log \left[ {\frac{{1 - g_{j} {\text{e}}^{{d_{j} \left( {T - t} \right)}} }}{{1 - g_{j} }}} \right]} \right\} \\ & D_{j} = \frac{{b_{j} - iz\rho \sigma + d_{j} }}{{\sigma^{2} }}\left[ {\frac{{1 - {\text{e}}^{{d_{j} \left( {T - t} \right)}} }}{{1 - g_{j} {\text{e}}^{{d_{j} \left( {T - t} \right)}} }}} \right] \\ & g_{j} = \frac{{b_{j} - iz\rho \sigma + d_{j} }}{{b_{j} - iz\rho \sigma - d_{j} }} \\ & d_{j} = \sqrt {\left( {iz\rho \sigma - b_{j} } \right)^{2} - \left( {2iu_{j} z - z^{2} } \right)\sigma^{2} } \\ & u_{1} = \frac{1}{2},u_{2} = - \frac{1}{2} \\ & b_{1} = \kappa + \lambda - \rho \sigma ,b_{2} = \kappa + \lambda \\ \end{aligned}$$

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1.2 Heston-Nandi (2000) GARCH Model

For a particular type of GARCH process, Heston and Nandi (2000) provide a closed form pricing formula for European options, where returns are generated by the process:

$$\begin{aligned} & \log \left( {\frac{{S_{t + 1} }}{{S_{t} }}} \right) = r + \lambda \sigma_{t}^{2} + \sigma_{t + 1} z_{t + 1} ;\quad \, z_{t + 1} \sim N\left( {0,1} \right) \\ & \sigma_{t + 1}^{2} = \omega + \alpha \left( {z_{t} - \theta \sigma_{t} } \right)^{2} + \beta \sigma_{t}^{2} \\ \end{aligned}$$

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The variance persistence of return process can be derived as $\beta + \alpha \theta^{2}$ from the above GARCH characterization, and the process becomes mean-reverting if $\beta + \alpha \theta^{2} < 1$. Heston and Nandi (2000) show that a risk-neutral characterization can be obtained if $\lambda = - \tfrac{1}{2}$, and $\theta^{*} = \theta + \lambda + \tfrac{1}{2}$. It can further be shown that α determines kurtosis and θ determines skewness in the model. Given a moment generating function of the form:

$$f\left( z \right) = S_{t}^{z} \exp \left\{ {A\left( {t;t + T,z} \right) + B\left( {t;t + T,z} \right)\sigma_{t + 1}^{2} } \right\}$$

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the price can be obtained in a closed form GARCH (CFG) as

$$c_{{{\text{CFG}}}} = S_{t} \left( {\frac{1}{2} + \frac{1}{\pi }\int_{0}^{\infty } {{\text{Re}} \left[ {\frac{{K^{ - iz} f^{*} \left( {iz + 1} \right)}}{{izf^{*} \left( 1 \right)}}} \right]{\text{d}}z} } \right) - K{\text{e}}^{rt} \left( {\frac{1}{2} + \frac{1}{\pi }\int_{0}^{\infty } {{\text{Re}} \left[ {\frac{{K^{ - iz} f^{*} \left( i \right)}}{iz}} \right]{\text{d}}z} } \right)$$

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where f^* is the risk-neutral version of f, and A(t;t + T,z) and B(t;t + T,z) are given by the recursive relations:

$$A\left( {t;t + T,z} \right) = A\left( {t + 1;t + T,z} \right) + zr + B\left( {t + 1;t + T,z} \right)\omega - \frac{1}{2}\ln \left( {1 - 2\alpha B\left( {t + 1;t + T,z} \right)} \right)$$

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$$B\left( {t;t + T,z} \right) = z\left( {\lambda + \theta } \right) - \frac{1}{2}\theta^{2} + \beta B\left( {t + 1;t + T,z} \right) + \frac{{\tfrac{1}{2}\left( {z - \theta } \right)^{2} }}{{1 - 2\alpha B\left( {t + 1;t + T,z} \right)}}$$

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1.3 Pure Jump Lévy Models

The Lévy models we consider here assume that all possible movements in stock price are caused by jumps. Thus, the Lévy measure of these processes ensures that the frequent arrival of small jumps sufficiently captures the diffusion process. Hence, the process is effectively a pure jump processes (Geman 2002).

Consistent with the Lévy–Kintchine formula, the distribution of $X_{{\left( {t_{2} - t_{1} } \right)}} = \log \left( {\tfrac{{S_{{t_{2} }} }}{{S_{{t_{1} }} }}} \right)$ is characterized by the characteristic function of an infinitely divisible random variable:

$$E\left[ {{\text{e}}^{{isX_{{\left( {t_{2} - t_{1} } \right)}} }} } \right] = \exp \left\{ {\left( {t_{2} - t_{1} } \right)\left[ {ias - \frac{1}{2}s^{2} b^{2} } \right] + \int_{{\Re \backslash \left\{ 0 \right\}}} {\left[ {{\text{e}}^{isx} - 1 - isx{\rm I}_{{\left\{ { - 1,1} \right\}}} \left( x \right)} \right]\nu \left( {{\text{d}}x} \right)} } \right\}$$

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where t₁ can naturally be zero. Scalars $a,b \in \Re$ and the measure ν satisfies $\nu \left( {\left\{ 0 \right\}} \right) = 0$ and $\int_{{\Re \backslash \left\{ 0 \right\}}} {\left( {\left| x \right|^{2} \wedge 1} \right)} \nu \left( {{\text{d}}x} \right) < \infty$, which means that though numerous small jumps may not be integrable, squares of those jumps are always integrable—the condition which helps us extract a square integrable martingale process in the limit. In the case of pure jump processes, b is always zero.³ For example, the variance gamma process characterizes the random variable X₁ through the parameters (σ, θ, γ) and the Lévy measure:

$$\nu_{vg} \, = \frac{1}{\gamma \left| x \right|}\exp \left( {\frac{x\theta }{{\sigma^{2} }} - \frac{\left| x \right|}{\sigma }\sqrt {\frac{2}{\gamma } + \frac{{\theta^{2} }}{{\sigma^{2} }}} } \right){\text{d}}x$$

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When integrated for jumps of all possible sizes, Eq. (20) implies that the total rate is infinite, i.e. $\int_{0}^{\infty } {\nu_{vg} } \left( {{\text{d}}x} \right) = \infty$. However, for any $\varepsilon > 0$, we have $\int_{\varepsilon }^{\infty } {\nu_{vg} } \left( {{\text{d}}x} \right) < \infty$, implying that sufficiently small jumps are numerous while jumps exceeding any threshold $\varepsilon > 0$ are finite, arriving in compound Poisson fashion. The Lévy measure, when used in Eq. (21) with a = b = 0, yields the following closed form characteristic function of the process X_t:

$$\Phi_{{X_{t} }} \left( s \right) = \left( {\frac{1}{{1 - is\theta \gamma + \tfrac{1}{2}s^{2} \sigma^{2} \gamma }}} \right)^{{\frac{t}{\gamma }}}$$

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For this pure jump Lévy model the skewness of log returns over an interval of length one, is given by

$${\text{skew}}\left( {X_{1} } \right) = \frac{{\theta \gamma \left( {3\sigma^{2} + 2\gamma \sigma^{2} } \right)}}{{\left( {\sigma^{2} + \gamma \theta^{2} } \right)^{\frac{3}{2}} }}$$

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The risk-neutral version of the characteristic function (21) required in the formula of Carr and Madan (1999) for pricing the options is given by

$$\Phi_{{X_{t} }}^{rn} \left( s \right) \, = \exp \left\{ {i\left[ {r + \frac{1}{\gamma }\ln \left( {1 - \theta \gamma - \frac{1}{2}\sigma^{2} \gamma } \right)} \right]st - \frac{t}{\gamma }\ln \left( {1 - is\theta \gamma + \frac{1}{2}s^{2} \sigma^{2} \gamma } \right)} \right\}$$

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This risk-neutral form results from mean-correction of the drift part, thereby introducing a drift to a driftless process (Schouten 2003).

1.4 A Jump-Diffusion Model

We choose Kou’s double exponential model, motivated by the findings of Ramezani and Zeng (1999), who find that a double exponential jump-diffusion model fits stock market data better than the normal-diffusion model of Merton. Kou (2002) assumes that, in addition to drifted diffusion, the log-return has occasional jumps following a double exponential distribution DE (p, η₁, η₂). Here p is the probability of an upward jump and η₁ and η₂ govern the decay of the tails for the distribution of negative and positive jumps, respectively. The Lévy measure is given by

$$\nu \left( {{\text{d}}x} \right) = \left[ {p\lambda \eta_{1} {\text{e}}^{{ - \eta_{1} x}} I_{x < 0} + \left( {1 - p} \right)\lambda \eta_{2} {\text{e}}^{{ - \eta_{2} \left| x \right|}} I_{x > 0} } \right]{\text{d}}x$$

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where $\int_{ - \infty }^{\infty } \nu \left( {{\text{d}}x} \right) < \infty$, and unlike pure jump processes, Kou’s jump diffusion model is a finite activity model. The Lévy measure, through Eq. (13), this time with non-zero a and b, provides a closed form characteristic function:

$$\Phi_{{X_{t} }} \left( s \right) = \exp \left\{ {t\left( {ias - \frac{1}{2}b^{2} s^{2} + is\lambda \left[ {\frac{{p\eta_{1} }}{{\eta_{1} + is}} + \frac{{\left( {1 - p} \right)\eta_{2} }}{{\eta_{2} + is}} - 1} \right]} \right)} \right\}$$

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The skewness in this model is not explicitly characterized. However Kou suggests that the feature of heavier tails becomes more pronounced with an increase of either the jump size expectation (1/η_j) or jump rate (λ). The mean-corrected characteristic function is obtained as

$$\Phi_{{X_{t} }}^{rn} \left( s \right) = \exp \left\{ \begin{gathered} i\left( {r - \frac{1}{2}b^{2} - \lambda \left[ {\frac{{p\eta_{1} }}{{\eta_{1} + 1}} + \frac{{\left( {1 - p} \right)\eta_{2} }}{{\eta_{2} + 1}} - 1} \right]} \right)st \hfill \\ - \frac{1}{2}b^{2} s^{2} t + is\lambda t\left[ {\frac{{p\eta_{1} }}{{\eta_{1} + is}} + \frac{{\left( {1 - p} \right)\eta_{2} }}{{\eta_{2} + is}} - 1} \right] \hfill \\ \end{gathered} \right\}$$

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We consider the logarithm of the prices, $s_{t} = \log \left( {S_{t} } \right)$ and k = log(K) where K is the strike price of the option. As in Carr and Madan (1999) the value of a European call with maturity T can be expressed as a function of k:

$$C_{T} \left( k \right) = \int_{k}^{\infty } {e^{ - rT} } \left( {e^{s} - e^{k} } \right)q_{T} \left( s \right){\text{d}}s$$

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Here q_T(s) is the risk-neutral density of the log prices. In order to overcome the non-square integrability of C_T(k), Carr and Madan (1999) introduce modified call prices:

$$c_{T} \left( k \right) = {\text{e}}^{\alpha k} C_{T} \left( k \right),\alpha > 0$$

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where α is the dampening factor. Following Carr and Madan (1999), an analytic expression for the pricing formula (21) can be obtained as

$$C_{T} \left( k \right) = \frac{{{\text{e}}^{ - \alpha k} }}{\pi }\int_{0}^{\infty } {{\text{e}}^{ - iuk} } \psi_{T} \left( u \right){\text{d}}u$$

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where ψ_T has an analytic expression:

$$\psi_{T} \left( u \right) = \frac{{{\text{e}}^{ - rT} \Phi_{T} \left( {u - \left( {\alpha + 1} \right)i} \right)}}{{\alpha^{2} + \alpha - u^{2} - i\left( {2\alpha + 1} \right)u}}$$

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here Ф is the characteristic function of the model for which prices are computed. In our empirical study, we shall consider Ф under risk-neutral dynamics for all our Lévy models.