A general closed form option pricing formula

Abstract

A new method to retrieve the risk-neutral probability measure from observed option prices is developed and a closed form pricing formula for European options is obtained by employing a modified Gram–Charlier series expansion, known as the Gauss–Hermite expansion. This expansion converges for fat-tailed distributions commonly encountered in the study of financial returns. The expansion coefficients can be calibrated from observed option prices and can also be computed, for example, in models with the probability density function or the characteristic function known in closed form. We investigate the properties of the new option pricing model by calibrating it to both real-world and simulated option prices and find that the resulting implied volatility curves provide an accurate approximation for a wide range of strike prices. Based on an extensive empirical study, we conclude that the new approximation method outperforms other methods both in-sample and out-of-sample.

Appendix

Proof of Lemma 1

If one denotes by $\tilde{p}\left( {\tilde{x}} \right)$ the standardized density, one has that $p\left( x \right) = \frac{1}{\sigma }\tilde{p}\left( {\frac{x - \mu }{\sigma }} \right)$. Using a well-known property of the “physicists” Hermite polynomials, namely

$$ e^{{ - \frac{{x^{2} }}{2}}} \cdot H_{n} \left( x \right) = \left( {x - \frac{d}{dx}} \right)^{n} e^{{ - \frac{{x^{2} }}{2}}} , $$

the Gauss–Hermite expansion of $\tilde{p}\left( {\tilde{x}} \right)$ can be written as

$$ \tilde{p}\left( {\tilde{x}} \right) = z\left( {\tilde{x}} \right)\sum\limits_{n = 0}^{\infty } {a_{n} H_{n} \left( {\tilde{x}} \right)} = \sum\limits_{n = 0}^{\infty } {a_{n} \left( {\tilde{x} - \frac{d}{{d\tilde{x}}}} \right)^{n} } z\left( {\tilde{x}} \right). $$

Therefore, the Fourier transform of $\tilde{p}$ is

$$ \sum\limits_{n = 0}^{\infty } {a_{n} i^{n} \left( {\phi - \frac{d}{d\phi }} \right)^{n} \exp \left( { - \frac{1}{2}\phi^{2} } \right) = } \exp \left( { - \frac{1}{2}\phi^{2} } \right)\sum\limits_{n = 0}^{\infty } {a_{n} i^{n} H_{n} \left( \phi \right)} . $$

The Fourier transform of p follows immediately. □

Proof of Proposition 1

If one denotes by p_t+τ(S_t+τ) the terminal underlying asset price risk-neutral density and by p(x) the standardized log-return risk-neutral density then we have that:

$$ \begin{aligned} c\left( {S_{t} ,K,r,q,\mu ,\sigma ,\tau } \right) & = e^{ - r\tau } \int\limits_{ - \infty }^{\infty } {\hbox{max} \left( {S_{t + \tau } - K,0} \right)p_{t + \tau } \left( {S_{t + \tau } } \right)dS_{t + \tau } } \\ & = e^{ - r\tau } \int\limits_{ - \infty }^{\infty } {\hbox{max} \left( {S_{t} e^{\mu \tau + \sigma \sqrt \tau x} - K,0} \right)p\left( x \right)dx} \\ & = e^{ - r\tau } \int\limits_{{ - d_{2} }}^{\infty } {\left( {S_{t} e^{\mu \tau + \sigma \sqrt \tau x} - K} \right)p\left( x \right)dx} = S_{t} e^{ - q\tau }\Pi _{1} - Ke^{ - r\tau }\Pi _{2} \\ \end{aligned} $$

with

$$ \Pi _{1} = e^{(\mu - r + q)\tau } \int_{{ - d_{2} }}^{\infty } {e^{\sigma \sqrt \tau x} p\left( x \right)dx} $$

and $\Pi _{2} = \int_{{ - d_{2} }}^{\infty } {p\left( x \right)dx} .$ Taking into account the Gauss–Hermite expansion of the log-return risk-neutral density, one has that

$$ \Pi _{1} = \exp \left\{ {\left( {\mu - (r - q) + \frac{{\sigma^{2} }}{2}} \right)\tau } \right\}\sum\limits_{n = 0}^{\infty } {a_{n} I_{n} } $$

with

$$ \begin{aligned} I_{n} & = e^{{ - \frac{{\sigma^{2} \tau }}{2}}} \cdot \mathop \int\nolimits_{{ - d_{2} }}^{\infty } e^{\sigma \sqrt \tau x} z\left( x \right)H_{n} \left( x \right)dx \\ & = e^{{ - \frac{{\sigma^{2} \tau }}{2}}} \cdot \mathop \int\nolimits_{{ - d_{2} - \sigma \sqrt \tau }}^{\infty } e^{{\sigma \sqrt \tau \left( {x + \sigma \sqrt \tau } \right)}} z\left( {x + \sigma \sqrt \tau } \right)H_{n} \left( {x + \sigma \sqrt \tau } \right)dx \\ & = \mathop \int\nolimits \limits_{{ - d_{1} }}^{\infty } H_{n} \left( {x + \sigma \sqrt \tau } \right)z\left( x \right)dx \\ \end{aligned} $$

and Π₂ = ∑ ^∞ _n=0 a_nJ_n with $J_{n} = \int_{{ - d_{2} }}^{\infty } {H_{n} \left( x \right)z\left( x \right)dx}$. Using the properties of Gauss–Hermite polynomials namely $H_{n + 1} \left( x \right) = 2x H_{n} \left( x \right) - 2n H_{n - 1} \left( x \right)$, $H_{n}^{'} \left( x \right) = 2n H_{n - 1} (x)$ and integration by parts, one can obtain the recursion equations for I_n and J_n as follows:

$$ \begin{aligned} I_{n + 1} & = \int\limits_{{ - d_{1} }}^{\infty } {H_{n + 1} \left( {x + \sigma \sqrt \tau } \right)z\left( x \right)dx} \\ & = \int\limits_{{ - d_{1} }}^{\infty } {\left[ {2\left( {x + \sigma \sqrt \tau } \right)H_{n} \left( {x + \sigma \sqrt \tau } \right) - H^{\prime}_{n} \left( {x + \sigma \sqrt \tau } \right)} \right]z\left( x \right)dx} \\ & = - 2\int\limits_{{ - d_{1} }}^{\infty } {H_{n} \left( {x + \sigma \sqrt \tau } \right)z^{\prime}\left( x \right)dx} + 2\sigma \sqrt \tau \int\limits_{{ - d_{1} }}^{\infty } {H_{n} \left( {x + \sigma \sqrt \tau } \right)z\left( x \right)dx} \\ & \quad - \,\int\limits_{{ - d_{1} }}^{\infty } {H^{\prime}_{n} \left( {x + \sigma \sqrt \tau } \right)z\left( x \right)dx} \\ & = 2H_{n} \left( { - d_{1} + \sigma \sqrt \tau } \right)z\left( { - d_{1} } \right) + 2\sigma \sqrt \tau I_{n} + 2nI_{n - 1} \\ \end{aligned} $$

where we have used $z^{\prime}\left( x \right) = - x z\left( x \right)$.

$$ \begin{aligned} J_{n + 1} & = \int\limits_{{ - d_{2} }}^{\infty } {H_{n + 1} \left( x \right)z\left( x \right)dx} = \int\limits_{{ - d_{2} }}^{\infty } {\left[ {2xH_{n} \left( x \right) - H_{n}^{\prime } \left( x \right)} \right]z\left( x \right)dx} \\ & = 2H_{n} \left( { - d_{2} } \right)z\left( { - d_{2} } \right) + \int\limits_{{ - d_{2} }}^{\infty } {H_{n}^{\prime } \left( x \right)z\left( x \right)dx} \\ & = 2H_{n} \left( { - d_{2} } \right)z\left( { - d_{2} } \right) + 2n\int\limits_{{ - d_{2} }}^{\infty } {H_{n - 1}^{{}} \left( x \right)z\left( x \right)dx} \\ & = 2H_{n} \left( { - \,d_{2} } \right)z\left( { - \,d_{2} } \right)\, + \,2nJ_{n - 1} . \\ \end{aligned} $$

Finally, we have $I_{0} = \int_{{ - d_{1} }}^{\infty } z \left( x \right)dx = N\left( {d_{1} } \right)$ and similarly $J_{0} = \int_{{ - d_{2} }}^{\infty } z \left( x \right)dx = N\left( {d_{2} } \right)$. □

1.1 Discussion about the convergence of the series from the pricing formulas

We assume that the standardized density of log-returns for horizon τ, under the risk-neutral measure, is described by a continuous, smooth and positive function p(x) such that $\mathop \int\nolimits \nolimits_{ - \infty }^{\infty } p\left( x \right)dx\, = \,1$. Denote $\hat{p}_{N}^{{}} \left( x \right): = \sum\nolimits_{k = 0}^{N} {a_{k}^{{}} } H_{k}^{{}} \left( x \right)z\left( x \right)$ the Gauss–Hermite expansion truncated after N terms. It is known that $\hat{p}_{N}^{{}} \left( \cdot \right)$ converges pointwise (even uniformly on compact intervals) and in $L^{2} \left( {\mathbf{\mathbb{R}}} \right)$ to p( · ).

Consider a constant b ≥ 0 such that $\mathop \int\nolimits \nolimits_{ - \infty }^{\infty } e^{bx} p\left( x \right)dx\, < \,\infty$. For example, if $b = \sigma \sqrt \tau$ this condition is due to the martingale restriction (i.e. E [S_Te^−r(T−t)] = S_t).

For example consider the limit

$$ \mathop {\lim }\limits_{N \to \infty } \int\limits_{ - \infty }^{\infty } {e^{bx} \hat{p}_{N}^{{}} \left( x \right)dx} = \int\limits_{ - \infty }^{\infty } {e^{bx} p\left( x \right)dx} $$

This result cannot be proved using the Cauchy–Schwarz inequality because the function e^bx is not in $L^{2} \left( {\mathbf{\mathbb{R}}} \right)$. We present here an argument based on Abel-summability and Tauberian theorems (Hardy 1949).

Let

$$ \overline{{H_{k}^{{}} }} : = \frac{{H_{k}^{{}}}}{{\sqrt {\int\limits_{ - \infty }^{\infty } {H_{k}^{{}} \left( x \right)^{2} \omega \left( x \right)dx} } }} {\sqrt {2\pi }} $$

and

$$ \overline{{a_{k}^{{}} }} : = a_{k}{\frac{1}{\sqrt {2\pi }}}^{{}} \sqrt {\int_{ - \infty }^{\infty } {H_{k}^{{}} \left( x \right)^{2} \omega \left( x \right)dx} } $$

where $\omega \left( x \right) = e^{{ - x^{2} }} = 2\pi z\left( x \right)^{2}$. We have that

$$ \int\limits_{ - \infty }^{\infty } {\overline{{H_{k}^{{}} }} \left( x \right)\overline{{H_{l}^{{}} }} \left( x \right){\frac{1}{{2\pi }}}\omega \left( x \right)dx} = \delta_{kl} $$

and $\overline{{a_{k}^{{}} }} = \int_{ - \infty }^{\infty } {\overline{{H_{k}^{{}} }} z\left( x \right)p\left( x \right)dx} .$ Therefore, it follows that

$$ \begin{aligned} \int\limits_{ - \infty }^{\infty } {e^{bx} \hat{p}_{N}^{{}} \left( x \right)dx} & = \sum\limits_{k = 0}^{N} {a_{k}^{{}} } \int\limits_{ - \infty }^{\infty } {e^{bx} H_{k}^{{}} \left( x \right)z\left( x \right)dx} \\ & = \sum\limits_{k = 0}^{N} {\overline{{a_{k}^{{}} }} } \int\limits_{ - \infty }^{\infty } {e^{bx} \overline{{H_{k}^{{}} }} \left( x \right)z\left( x \right)dx} = \sum\limits_{k = 0}^{N} {\overline{{a_{k}^{{}} }} \overline{{q_{k}^{{}} }} } \\ \end{aligned} $$

here $\overline{{q_{k}^{{}} }} : = \int_{ - \infty }^{\infty } {e^{bx} \overline{{H_{k}^{{}} }} z\left( x \right)dx}$.

If we denote by $\overline{{\psi_{k}^{{}} }} : = \overline{{H_{k}^{{}} }} z$ the normalized Hermite functions (a complete orthonormal basis of $L^{2} \left( {\mathbf{\mathbb{R}}} \right)$) one has that:

$$ \begin{aligned} \overline{{a_{k}^{{}} }} & = \int\limits_{ - \infty }^{\infty } {p\left( x \right)\overline{{H_{k}^{{}} }} z\left( x \right)dx} = \left\langle {p,\overline{{\psi_{k}^{{}} }} } \right\rangle_{{L^{2} \left( {\mathbf{\mathbb{R}}} \right)}} < \infty \\ \overline{{q_{k}^{{}} }} & = \int\limits_{ - \infty }^{\infty } {h\left( x \right)\overline{{H_{k}^{{}} }} z\left( x \right)dx} = \left\langle {h,\overline{{\psi_{k}^{{}} }} } \right\rangle_{{L^{2} \left( {\mathbf{\mathbb{R}}} \right)}} < \infty \\ \end{aligned} $$

Denote by $h\left( x \right) = e^{bx} \notin L^{2} \left( {\mathbf{\mathbb{R}}} \right)$ and

$$ h_{n} \left( x \right) = \sqrt {1 + \frac{1}{n}} e^{{\frac{{b^{2} }}{2}\frac{1}{n}}} e^{{b\sqrt {1 - \frac{1}{{n^{2} }}} x - \frac{1}{2}\frac{1}{n}x^{2} }} \in L^{2} \left( {\mathbf{\mathbb{R}}} \right). $$

Let $\overline{{q_{k,n}^{{}} }} : = \left\langle {h_{n} ,\overline{{\psi_{k}^{{}} }} } \right\rangle_{{L^{2} \left( {\mathbf{\mathbb{R}}} \right)}}$. It follows that

$$ \mathop {\lim }\limits_{k \to \infty } \overline{{q_{k,n}^{{}} }} = 0,\sum\limits_{k = 0}^{\infty } {\overline{{a_{k}^{{}} }} \overline{{q_{k,n}^{{}} }} } = \left\langle {p,h_{n} } \right\rangle_{{L^{2} \left( {\mathbf{\mathbb{R}}} \right)}} < \infty $$

and

$$ \sum\limits_{k = 0}^{\infty } {\left| {\overline{{a_{k}^{{}} }} \overline{{q_{k,n}^{{}} }} } \right|} \le \left( {\sum\limits_{k = 0}^{\infty } {\left| {\overline{{a_{k}^{{}} }} } \right|^{2} } } \right)^{1/2} \left( {\sum\limits_{k = 0}^{\infty } {\left| {\overline{{q_{k,n}^{{}} }} } \right|^{2} } } \right)^{1/2} < \infty . $$

We have that (e.g. Gradshteyn et al. 2000, eq. 7.374.8, 7.376.1)

$$ \begin{aligned} \overline{{q_{k}^{{}} }} & : = \left\langle {h,\overline{{\psi_{k}^{{}} }} } \right\rangle_{{L^{2} \left( {\mathbf{\mathbb{R}}} \right)}} = e^{{\frac{{b^{2} }}{2}}} i^{k} \overline{{H_{k}^{{}} }} \left( { - bi} \right) \\ \overline{{q_{k,n}^{{}} }} & : = \left\langle {h_{n} ,\overline{{\psi_{k}^{{}} }} } \right\rangle_{{L^{2} \left( {\mathbf{\mathbb{R}}} \right)}} = \left( {\frac{{1 - \frac{1}{n}}}{{1 + \frac{1}{n}}}} \right)^{{\frac{k}{2}}} e^{{\frac{{b^{2} }}{2}}} i^{k} \overline{{H_{k}^{{}} }} \left( { - bi} \right) = \left( {\frac{{1 - \frac{1}{n}}}{{1 + \frac{1}{n}}}} \right)^{{\frac{k}{2}}} \overline{{q_{k}^{{}} }} \\ \end{aligned} $$

Since $0 < h_{n} < 2e^{{\frac{{b^{2} }}{2}}} \hbox{max} \left( {1,h} \right)$, $\left\langle {p,\hbox{max} \left( {1,h} \right)} \right\rangle_{{L^{2} \left( {\mathbf{\mathbb{R}}} \right)}} < \infty$ and p > 0 it follows that

$$ \mathop {\lim }\limits_{n \to \infty } \sum\limits_{k = 0}^{\infty } {\overline{{a_{k}^{{}} }} \overline{{q_{k,n}^{{}} }} } = \mathop {\lim }\limits_{n \to \infty } \left\langle {p,h_{n} } \right\rangle_{{L^{2} \left( {\mathbf{\mathbb{R}}} \right)}} = \left\langle {p,\mathop {\lim }\limits_{n \to \infty } h_{n} } \right\rangle_{{L^{2} \left( {\mathbf{\mathbb{R}}} \right)}} = \left\langle {p,h} \right\rangle_{{L^{2} \left( {\mathbf{\mathbb{R}}} \right)}} < \infty $$

Therefore, the series $\sum\nolimits_{k = 0}^{\infty } {\overline{{a_{k}^{{}} }} \overline{{q_{k}^{{}} }} }$ is Abel-summable to the limit $\left\langle {p,h} \right\rangle_{{L^{2} \left( {\mathbf{\mathbb{R}}} \right)}}$. A series ∑ ^∞ _k=0 c_k is called Abel-summable to the limit l if lim_z↗1 ∑ ^∞ _k=0 c_kz^k = l (e.g. Hardy 1949). If the Gauss–Hermite coefficients are such that one of the Tauberian conditions (Hardy 1949, p. 149) is satisfied, e.g. $\overline{{a_{k}^{{}} }} \overline{{q_{k}^{{}} }} = O\left( {k^{ - 1} } \right)$, then it follows from the corresponding Tauberian theorem (Hardy 1949, ch. VII) that the series $\sum\nolimits_{k = 0}^{\infty } {\overline{{a_{k}^{{}} }} \overline{{q_{k}^{{}} }} }$ converges (in the usual sense) and $\sum\nolimits_{k = 0}^{\infty } {\overline{{a_{k}^{{}} }} \overline{{q_{k}^{{}} }} } = \left\langle {p,h} \right\rangle_{{L^{2} \left( {\mathbb{R}} \right)}}$. It is worth pointing out that, even if the Tauberian conditions cannot be verified, one can Abel-sum the series $\sum\nolimits_{k = 0}^{\infty } {\overline{{a_{k}^{{}} }} \overline{{q_{k}^{{}} }} }$ to get the desired result $\left\langle {p,h} \right\rangle_{{L^{2} \left( {\mathbf{\mathbb{R}}} \right)}}$.

1.2 Additional tables

Tables 6, 7 and 8 provide a more in-depth picture of the in-sample pricing performance of the methods under analysis. Table 9 deals with the comparison regarding the out-of-sample implied volatility root mean squared errors. Tables 10 and 11 present the in-sample and the out-of-sample mean relative absolute hedging errors.

Table 6 The in-sample mean absolute pricing error for the subsample January 2014–December 2014

Table 7 The in-sample IVRMSE

Table 8 The in-sample IVRMSE for the subsample January 2014–December 2014

Table 9 The out-of-sample IVRMSE

Table 10 The in-sample mean relative absolute daily hedging error

Table 11 The out-of-sample mean relative absolute daily hedging error