Explaining the volatility smile: non-parametric versus parametric option models

Abstract

We employ a “non-parametric” pricing approach of European options to explain the volatility smile. In contrast to “parametric” models that assume that the underlying state variable(s) follows a stochastic process that adheres to a strict functional form, “non-parametric” models directly fit the end distribution of the underlying state variable(s) with statistical distributions that are not represented by parametric functions. We derive an approximation formula which prices S&P 500 index options in closed form which corresponds to the lower bound recently proposed by Lin et al. (Rev Quant Financ Account 38(1):109–129, 2012). Our model yields option prices that are more consistent with the data than the option prices that are generated by several widely used models. Although a quantitative comparison with other non-parametric models is more difficult, there are indications that our model is also more consistent with the data than these models.

Appendix

1.1 Variances under jump diffusion and random volatility

In this appendix, we derive $V^{\text{Hes}}$ and $V^{\text{BCC}}$ used in regression. Note that $V^{\text{Hes}}$ and $V^{\text{BCC}}$ replace the notion of implied volatility in the Black–Scholes model. In both models, the volatility follows a mean-reverting square-root process as follows:

$$\left\{ {\begin{array}{*{20}l} {dS = \mu Sdt + \sqrt V SdW_{1} } \hfill \\ {dV = a(b - V)dt + g\sqrt V dW_{2} } \hfill \\ \end{array} } \right.$$

(14)

where $\mu$, $a$, $b$, and $g$ are constants and $W_{1}$ and $W_{2}$ are Brownian motions under the physical measure and $dW_{1} dW_{2} = \rho dt$ while in the empirical study $\rho = 0$. It is known that the expected volatility under 14 is:

$$E_{t} [V_{u} ] = b\left( {1 - e^{ - a(u - t)} } \right) + e^{ - a(u - t)} V_{t}$$

(15)

and hence the expected total volatility in the Heston model is:

$$\begin{aligned} V^{Hes} & = \int_{t}^{T} {E_{t} [V_{u} ]} \\ & = b(T - t) + (V_{t} - b)\frac{{1 - e^{ - a(T - t)} }}{a} \\ \end{aligned}$$

(16)

In a jump-diffusion case with a jump size $Y$ and an intensity parameter $\lambda$, we have the following distribution:

$${ \ln }\,S_{T} = \left\{ {\begin{array}{*{20}l} {e^{ - \lambda (T - t)} } \hfill & {{ \ln }\,S_{t} + \mu (T - t) - \frac{1}{2}\int_{t}^{T} {V_{u} du + \sqrt {V_{T} } dW_{u} } } \hfill & {\text{no jump}} \hfill \\ {1 - e^{ - \lambda (T - t)} } \hfill & {{ \ln }\,S_{t} + \ln Y + \mu (T - t) - \frac{1}{2}\int_{t}^{T} {V_{u} du + \sqrt {V_{T} } dW_{u} } } \hfill & {\text{jump}} \hfill \\ \end{array} } \right.$$

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where $V$ and $Y$ are random. The mean

$$E_{t} [{ \ln }\,S_{T} ] = { \ln }\,S_{t} + \mu (T - t) - \frac{1}{2}\int_{t}^{T} {E_{t} [V_{u} ]du + (1 - e^{ - \lambda (T - t)} )E[{ \ln }\,Y]}$$

(18)

and variance

$$V^{\text{BCC}} \approx \int_{t}^{T} {E_{t} [V_{u} ]du} + \left( {1 - e^{ - \lambda (T - t)} } \right)\left( {\mu_{Y}^{2} + \sigma_{Y}^{2} } \right)$$

(19)

1.2 Proof of Theorem

By Eq. 1, the option price must follow $C_{t} = E_{t} [M_{t,T} C_{T} ]$, and hence:

$$\begin{gathered} C_{t} = E_{t} [M_{t,T} C_{T} ] \\ = E_{t} [M_{t,T} ]E_{t} [C_{T} ] + \text{cov} [M_{t,T} ,C_{T} ] \\ = P_{t,T} E_{t} [C_{T} ] + \text{cov} [M_{t,T} ,C_{T} ] \\ \end{gathered}$$

(20)

Further expand the covariance term:

$$\begin{aligned} \text{cov} [M_{t,T} ,C_{T} ] & = \rho_{MC} \sigma_{M} \sigma_{C} = (\text{sgn} [\rho_{SC} ]\rho_{MS} + \varepsilon_{1} )\sigma_{M} \sigma_{C} \\ & = \text{sgn} [\rho_{SC} ]\rho_{MS} \sigma_{M} \sigma_{C} + \varepsilon_{1} \sigma_{M} \sigma_{C} \\ & = \text{sgn} [\rho_{SC} ]\rho_{MS} \sigma_{M} \sigma_{S} \frac{{\sigma_{C} }}{{\sigma_{S} }} + \varepsilon_{1} \sigma_{M} \sigma_{C} \\ & = \text{sgn} [\rho_{SC} ]\text{cov} [M_{t,T} ,S_{T} ]\frac{{\sigma_{C} }}{{\sigma_{S} }} + \varepsilon_{1} \sigma_{M} \sigma_{C} \\ & = \beta \text{cov} [M_{t,T} ,S_{T} ] + \varepsilon_{1} \sigma_{M} \sigma_{C} + \varepsilon_{2} \\ \end{aligned}$$

(21)

where $\sigma_{C} = \sqrt {\text{var} [C_{T} ]}$, $\sigma_{S}\,=\, \sqrt {\text{var} [S_{T} ]}$ and

$$\varepsilon_{1} = \rho_{MC} - \text{sgn} [\rho_{SC}^{{}} ]\rho_{MS}$$

$$\varepsilon_{2} = \left\{ {\text{sgn} [\rho_{SC} ]\frac{{\sigma_{C} }}{{\sigma_{S} }} - \beta } \right\}\text{cov} [M_{t,T} ,S_{T} ]$$

Given perfection correlation, $\beta = \tfrac{{\text{cov} [S_{T} ,C_{T} ]}}{{\text{var} [S_{T} ]}} = \text{sgn} [\rho_{SC}^{{}} ]\tfrac{{\sigma_{C} }}{{\sigma_{S} }}$. 21 becomes:

$$\text{cov} [M_{t,T} ,C_{T} ] = \beta \text{cov} [M_{t,T} ,S_{T} ] + \varepsilon_{1} \sigma_{M} \sigma_{C}$$

(22)

Also, under perfect correlation, $\rho_{MC} = \text{sgn} [\rho_{SC} ]\rho_{MS}$ and consequently $\varepsilon_{1} = 0$. We can then further simplify 22 to:

$$\begin{gathered} \text{cov} [M_{t,T} ,C_{T} ] = \beta \text{cov} [M_{t,T} ,S_{T} ] \\ = \beta \{ E_{t} [M_{t,T} S_{T} ] - E_{t} [M_{t,T} ]E_{t} [S_{T} ]\} \\ = \beta \{ S_{t} - P_{t,T} E_{t} [S_{T} ]\} \\ \end{gathered}$$

(23)

Substituting this result back into 20 and verifying that $\varepsilon_{1} \sigma_{M} \sigma_{C} + \varepsilon_{2} = \varepsilon$ complete the proof.

1.3 Results of approximation error

To gauge the magnitude of the errors, we create a standard binomial model where there are $n$ states and each state is $S_{j} = u^{j} d^{n - j}$ with probability $\hat{p}_{j} = \left( {_{j}^{n} } \right)\hat{p}^{j} (1 - \hat{p})^{n - j}$ in which $u = e^{{\sigma \sqrt {\varDelta t} }}$, $d = \tfrac{1}{u}$, and the risk neutral probability $\hat{p} = \tfrac{\exp (r\varDelta t) - d}{u - d}$. Then the call value can be computed as $C = e^{ - jr\varDelta t} \sum\nolimits_{j = 0}^{n} {} \hbox{max} \{ S_{j} - K,0\} \hat{p}_{j}$. Define the real probability as $p = \tfrac{\exp (\mu \varDelta t) - d}{u - d}$ where $\mu > r$. Following the state pricing theory and define state price as $\pi_{j} = \hat{p}_{j} e^{ - nr\varDelta t}$. Hence, kernel is: $M_{j} = \pi_{j} /p_{j}$. This way, $E[M] = \sum\nolimits_{j = 0}^{n} {} M_{j} = e^{ - nr\varDelta t}$. Now, we can simulate the option price and the errors with $n = 400$. The following is the table for the errors. We examine errors for the combinations of various moneyness levels (25 % in the money to 17 % out of the money), interest rates (3–9 %), and volatilities (0.2–0.6). Percentage errors are computed as $\tfrac{{\varepsilon_{1} + \varepsilon_{2} }}{C}$.

$r = 3\;\%$					$\sigma = 0.2$
	1.2500	1.1765	1.1111	1.0526	1.0000	0.9524	0.9091	0.8696	0.8333
Option price	18.7678	15.0116	11.7246	8.9380	6.6738	4.8632	3.4847	2.4474	1.6883
$\rho_{SC}$	0.9959	0.9909	0.9823	0.9689	0.9499	0.9242	0.8925	0.8543	0.8103
%error	0.0026	0.0053	0.0104	0.0194	0.0347	0.0610	0.1041	0.1753	0.2916
					$\sigma = 0.4$
Option price	24.0460	21.1811	18.6027	16.2725	14.2259	12.4022	10.7875	9.3660	8.1208
$\rho_{SC}$	0.9822	0.9753	0.9669	0.9569	0.9457	0.9330	0.9188	0.9034	0.8868
%error	0.0047	0.0066	0.0090	0.0122	0.0162	0.0212	0.0275	0.0352	0.0447
					$\sigma = 0.6$
Option price	30.1393	27.7423	25.5301	23.4889	21.6352	19.9316	18.3608	16.9060	15.5754
$\rho_{SC}$	0.9795	0.9745	0.9688	0.9626	0.9559	0.9488	0.9411	0.9329	0.9242
%error	0.0044	0.0055	0.0068	0.0083	0.0100	0.0119	0.0141	0.0165	0.0193

$r = 6\;\%$					$\sigma = 0.2$
	1.2500	1.1765	1.1111	1.0526	1.0000	0.9524	0.9091	0.8696	0.8333
Option price	20.6885	16.844	13.4109	10.4353	7.95713	5.92756	4.34122	3.11746	2.19909
$\rho_{SC}$	0.99585	0.9909	0.98232	0.96887	0.94986	0.92418	0.89247	0.85429	0.8103
%error	0.00193	0.00404	0.00794	0.01484	0.02658	0.04643	0.07867	0.13117	0.21581
					$\sigma = 0.4$
Option price	25.5	22.5839	19.9419	17.5397	15.4146	13.5094	11.8121	10.3088	8.98399
$\rho_{SC}$	0.98223	0.97529	0.96691	0.95691	0.9457	0.93299	0.91885	0.90339	0.88677
%error	0.00408	0.00575	0.00792	0.01072	0.01421	0.0186	0.02405	0.03077	0.03898
					$\sigma = 0.6$
Option price	31.3427	28.9226	26.6822	24.6088	22.7194	20.978	19.3682	17.8734	16.5025
$\rho_{SC}$	0.97951	0.97447	0.96883	0.96258	0.95592	0.94877	0.9411	0.93287	0.92424
%error	0.00411	0.00513	0.00632	0.00769	0.00924	0.011	0.013	0.01528	0.01782

$r = 9\;\%$					$\sigma = 0.2$
	1.2500	1.1765	1.1111	1.0526	1.0000	0.9524	0.9091	0.8696	0.8333
Option price	22.6283	18.7284	15.1812	12.0431	9.36818	7.12722	5.33085	3.91127	2.81952
$\rho_{SC}$	0.99585	0.9909	0.98232	0.96887	0.94986	0.92418	0.89247	0.85429	0.8103
%error	0.0014	0.00298	0.00591	0.01112	0.01994	0.03476	0.05861	0.09701	0.15815
					$\sigma = 0.4$
Option price	26.9729	24.0141	21.316	18.8485	16.6499	14.6672	12.8903	11.307	9.90323
$\rho_{SC}$	0.98223	0.97529	0.96691	0.95691	0.9457	0.93299	0.91885	0.90339	0.88677
%error	0.00355	0.00501	0.00691	0.00937	0.01241	0.01623	0.02097	0.0268	0.0339
					$\sigma = 0.6$
Option price	32.5547	30.1151	27.8499	25.7472	23.8247	22.0479	20.4009	18.868	17.4581
$\rho_{SC}$	0.97951	0.97447	0.96883	0.96258	0.95592	0.94877	0.9411	0.93287	0.92424
%error	0.0038	0.00474	0.00583	0.0071	0.00853	0.01016	0.012	0.01409	0.01643

Percentage wise, we see that the errors are large when the option is more out of the money (therefore low correlation), lower volatility, and lower interest rates. Hence in the above table, the largest error is observed on the most upper right (29 %) and the smallest error is on the most lower left (0.38 %). In dollar amount, the differences across various moneyness levels are not substantial. When the errors are large, the approximation formula gives substantially lower values than the correct price, resulting in higher implied volatility, causing the volatility smile to be more pronounced. In other words, the bias in the model will inflate the implied volatility generated by our model for out of the money calls. Hence, with the bias, our model is more conservative in resolving the smile puzzle.²²

1.4 Pricing formulas of the Black–Scholes, Heston, and Bakshi–Cao–Chen models

Following the notation of 14, the Black and Scholes (1973) call option formula on the SPX is:

$$C_{t,T,K}^{\text{BS}} = S_{t} N(d_{1} ) - e^{ - r(T - t)} KN(d_{2} ),$$

(24)

where

$$\begin{aligned} d_{1} & = \frac{{{ \ln }(S_{t} /K) + (r + \tfrac{{\sigma^{2} }}{2})(T - t)}}{{\sigma \sqrt {(T - t)} }} \\ d_{2} & = d_{1} - \sigma \sqrt {(T - t)} \\ \end{aligned}$$

We solve for the implied volatility of the Black–Scholes model, denoted as $\sigma^{*}$, by substituting the market price of the call option into the pricing equation.

The Heston (1993) model allows the volatility in the Black and Scholes (1973) model to be random over time. Heston shows that such a model has a closed form solution in the Fourier space:

$$C_{t,T,K}^{\text{Hes}} = S_{t} \varPi_{1} - e^{ - r(T - t)} K\varPi_{2}$$

(25)

where

$$\varPi_{j} = \frac{1}{2} + \frac{1}{\pi }\int\limits_{0}^{\infty } {\text{Re} \left[ {\frac{{e^{ - iu\;\ln K} \phi_{j} }}{iu}} \right]du} \quad j = 1,2$$

(26)

$i = \sqrt { - 1}$ and $\phi_{j}$, $j = 1,2$ is the characteristic function defined as:

$$\phi_{j} = e^{{B_{j} + D_{j} V + iux}}$$

(27)

where

$$\left\{ {\begin{array}{*{20}l} {B_{j} = riu(T - t) + \frac{ab}{{g^{2} }}\left\{ {(x_{j} - \rho giu + d_{j} )(T - t) - 2\;{ \ln }\left[ {\frac{{1 - y_{j} e^{{d_{j} (T - t)}} }}{{1 - y_{j} }}} \right]} \right\}} \hfill \\ {D_{j} = \frac{{x_{j} - \rho giu + d_{j} }}{{g^{2} }}\left[ {\frac{{1 - e^{{d_{j} (T - t)}} }}{{1 - y_{j} e^{{d_{j} (T - t)}} }}} \right]} \hfill \\ \end{array} } \right.$$

(28)

and

$$\left\{ {\begin{array}{*{20}l} {y_{j} = \frac{{x_{j} - \rho giu + d_{j} }}{{x_{j} - \rho giu - d_{j} }}} \hfill \\ {d_{j} = \sqrt {(\rho giu - x_{j} )^{2} - (ab)^{2} (2iu\xi_{j} - u^{2} )} } \hfill \\ \end{array} } \right.$$

(29)

$x_{1} = a - \rho g$, $x_{2} = a$, $\xi_{1} = {\raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} }$ and $\xi_{2} = - {\raise.5ex\hbox{$\scriptstyle 1$}\kern-.1em/ \kern-.15em\lower.25ex\hbox{$\scriptstyle 2$} }$. Here we assume that the volatility risk carries no risk premium. Bakshi et al. (1997) as well as other similar models add to the model jumps that are independent of volatility and stock price processes. Hence, for each characteristic function, $j = 1,2$ in A16, we multiply correspondingly the following characteristic function from the jumps:

$$\phi_{J1} (u) = \exp \left( {\begin{array}{*{20}l} { - iu\lambda e^{{ - \mu_{Y} + {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}\sigma_{Y}^{2} }} (T - t)\left( {e^{{\mu_{J} - {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}\sigma_{Y}^{2} }} - 1} \right)} \hfill \\ { + \lambda (T - t)\left( {e^{{iu\mu_{Y} - {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}\sigma_{Y}^{2} u^{2} }} - 1} \right)} \hfill \\ \end{array} } \right)$$

(30)

and

$$\phi_{J2} (u) = \exp \left( {\begin{array}{*{20}l} { - iu\lambda e^{{ - \mu_{Y} + {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}\sigma_{Y}^{2} }} (T - t)\left( {e^{{\mu_{Y} - {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}\sigma_{Y}^{2} }} - 1} \right)} \hfill \\ { + \lambda (T - t)e^{{ - \mu_{Y} + {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}\sigma_{Y}^{2} }} \left( {e^{{iu(\mu_{Y} - \sigma_{Y}^{2} ) - {\raise0.5ex\hbox{$\scriptstyle 1$} \kern-0.1em/\kern-0.15em \lower0.25ex\hbox{$\scriptstyle 2$}}\sigma_{Y}^{2} u^{2} }} - 1} \right)} \hfill \\ \end{array} } \right)$$

(31)

where $\lambda$, $\mu_{Y}$, and $\sigma_{Y}$ are the jump intensity, jump mean size and jump size standard deviation respectively. The full Bakshi et al. (1997) model also has random interest rates. However, since the impact of random interest rates on option prices is minimal for short maturity option contracts, we choose not to use it in our comparison. This reduces the complexity of the model and the number of parameters that should be estimated.