A Gaussian semi-parametric implied volatility model

Abstract

Modeling the implied volatility has received extensive attention, as the implied volatility is an important parameter in option pricing. Usually the implied volatility can be approximated by fitting a polynomial about the strike and the maturity or by stochastic methods. In this article, a Gaussian semi-parametric model is proposed based on the quadratic polynomial semi-parametric model suggested by Borovkova. In the new model, the Gaussian function is used to construct a smooth term substituting the quadratic term in the polynomial model, and the arbitrage-free constraints are used to calibrate the model. The empirical tests show that the Gaussian semi-parametric model has a better performance in fitting and forecasting.

Introduction

In finance, an option is a contract which gives the owner of the option the right, but not the obligation, to buy or sell an underlying asset or instrument at a specified strike price on a specified date, depending on the form of the option. An option that conveys to the owner the right to buy at a specific price is referred to as a call; an option that conveys the right of the owner to sell at a specific price is referred to as a put.

Option pricing is one of the hot issues in the field of financial engineering in which the most famous and common used model is the Black–Scholes (B-S or Black–Scholes–Merton) model. ¹ The model assumes that the implied volatility is a constant across all strikes and maturities. But, in practice, the values of implied volatilities by inverting option prices with different strikes and maturities are varied. That is, in a three-dimensional coordinate system, the implied volatility is plotted as a function of the strike price and the time-to-maturity, which forms a non-flat surface called the implied volatility surface. This surface contains a lot of information of the option market, which is an important tool for option pricing and financial risk management. Thus, modeling the implied volatility becomes very important in the field of financial engineering.

There are two main construction methodologies to model implied volatility: the deterministic implied volatility model and the stochastic implied volatility model. The deterministic implied volatility model considers there is a deterministic functional relationship among the implied volatility, underlying asset price, the time-to-maturity and the strike price, and this relationship doesn’t change with time. There are three common used deterministic models: the sticky strike rule model and the sticky delta rule model proposed by Derman, ² and the stationary square root of time rule model proposed by Daglish et al. ³ Such model does not introduce the new risk source and is simple to model.

The stochastic implied volatility model is first proposed by Schonbucher, ⁴ which considers the change of implied volatility is driven by several risk sources, and assumes that these risk sources are different from the risk sources of underlying asset price. Ledoit and Santa-Clara, ⁵ Cont and Da Fonseca, ⁶ and some other researchers followed Schonbucher’s idea and put forward other new stochastic models. Because the implied volatility surface contains different strike price and maturity, whose change process is complex and multidimensional, building the model of implied volatility surface need to consider not only the random process of each of the implied volatility but also the relationship between every different random process, which is hard to practice in reality.

In recent years, with the option market data becoming abundant, researchers focus on the deterministic modeling methodologies.^7–10 The deterministic modeling include the parameter model, non-parameter model, and semi-parameter model.

Ncube ¹¹ analyzed the market data of FTSE100 index option and concluded that there is a correlation among the implied volatility, the strike price and the time-to-maturity. Dumas et al. ¹² used the market data of S&P500 index option to illustrate there is a linear relationship among them, and then put forward a set of parametric models in which the implied volatility is considered as some functions about the strike price and the time-to-maturity. Cassese and Guidolin ¹³ followed Dumas’s idea, replaced the strike price with the moneyness in above parametric models, and concluded that the new models have better fitting effects. However, the parameter model has bid error and low stability.

Fitting the implied volatility surface can also be tackled by a non-parametric approach. Bourke ¹⁴ built the implied volatility surface model in a non-parametric way by the nearest neighbor-weighted interpolation. The implied volatility surface obtained by a non-parametric method can have any complex shape. This is because non-parametric methods lack generalization power and attempt to fit observed implied volatilities exactly. However, this feature is a big disadvantage of non-parametric methods, and the reason why they are not widely used in practice.

To combine attractive features of the parametric method with the flexibility of the non-parametric one, in 2009, Borovkova and Permana ¹⁵ proposed a semi-parametric model which considered that, when fixing the time-to-maturity, the implied volatility is a quadratic polynomial of the strike price or the moneyness. Different from the parametric models in which the parameters are independent of the time-to-maturity, the parameters of the semi-parametric models vary with the time-to-maturities. In the other hand, the stochastic volatility inspired (SVI) model proposed by Gatheral, ¹⁶ fits the implied volatility by a function of the moneyness separately for each time-to-maturity. That means SVI is also a semi-parametric model.

The semi-parametric model proposed by Borovkova using a quadratic polynomial of the moneyness to fit the implied volatility curve can exhibit the feature of “volatility smile,” but in some cases, the curvature of the fitted implied volatility curve can’t display the extent of “volatility smile.”

In this article, we have done the following main contributions: first, the quadratic term in the Borovkova’s model is replaced by a new smooth term and a Gaussian semi-parametric model is proposed; second, arbitrage-free are used to calibrate the Gaussian semi-parametric model; third, implied volatility is interpolated based on the Gaussian semi-parametric model to obtain the implied volatility surface, in order to observe the tendency of the implied volatility; fourth, the Gaussian semi-parametric model is used to forecast implied volatility and the option price in the next trading day. In empirical analysis, the market dataset of AAPL stock options (from Yahoo finance website: http://finance.yahoo.com/quote/AAPL/options) is used to verify the effectiveness of the Gaussian semi-parametric model.

The article is organized as follows: The next section documents B-S model and implied volatility surface. In “Gaussian semi-parametric model” section, we provide a brief overview of parametric models and semi-parametric models, then propose a Gaussian semi-parametric implied volatility model. In “The semi-parametric model with no-arbitrage” section, we describe the arbitrage-free conditions of the implied volatility surface and propose a calibrated Gaussian semi-parametric model. “Empirical analysis” section outlines our empirical procedure. We show the new model’s fitting and forecasting performance. The final section concludes with a summary of the main results.

B-S model and implied volatility surface

The B-S model widely used in option pricing was first proposed by Fischer Black and Myron Scholes in 1973 ¹ under the following assumptions:

The market is frictionless and all participants at the markets are able to borrow and lend at the constant risk-free interest rate;

There is no transaction cost and tax;

There is no arbitrage in the market;

The underlying can be bought in any amount, including fractional units;

Finally and most importantly, the underlying follows a geometric Brownian motion with constant drift and constant volatility.

For a given underlying asset, the B-S model can be specified in a partial differential equation:

$$\frac{\partial V(S,t;K,T,\sigma )}{\partial t}+\mathit{rS}\frac{\partial V(S,t;K,T,\sigma )}{\partial S}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}+\frac{1}{2}{\sigma }^2{S}^2\frac{\partial {\hspace{0.17em}}^{2}V(S,t;K,T,\sigma )}{\partial S^2}=\mathit{rV}(S,t;K,T,\sigma )$$

(1) where r is the risk-free interest rate, σ is the implied volatility of underlying asset, $V(S,t;K,T,\sigma )$ is the theoretical price of an option at time t when the underlying asset price is S, and for which the strike price is K and the maturity is T. In this model, except the volatility, the other parameters and variables can be obtained directly from the financial markets or option contracts.

If the implied volatility σ is specified, then the option price V corresponding to the underlying asset price of S₀ at the present time $t=0$ can be uniquely determined by solving equation (1) subject to the appropriate initial and boundary conditions. The conditions for a standard European call are, for example,

$$\{V(S,T;K,T,\sigma )=max(S-K,0)\text{for}\hspace{1em}S\ge 0,V(0,t;K,T,\sigma )=0\text{for}\hspace{1em}0\le t\le T,\frac{\partial V(S,t;K,T,\sigma )}{\partial S}\to e^{-q(T-t)}\hspace{1em}\text{as}\hspace{1em}S\to \infty \text{for}\hspace{1em}0\le t\le T$$

(2)

With the equation (1) and conditions (2), the pricing formulae of European call option can be expressed as follows:

$$C=S\times N(d_1)-e^{-r(T-t)}K\times N(d_2)$$

(3)

$$d_1=\frac{\mathit{ln}(\frac{S}{K})+(r+\frac{{\sigma }^2}{2})(T-t)}{\sigma \sqrt{T-t}}$$

(4)

$$d_2=d_1-\sigma \sqrt{T-t}$$

(5) where $N(·)$ is the standard normal density function.

The B-S model assumes the volatility is a constant, that is, the implied volatilities for different options writing on an underlying asset have the same values. The implied volatilities, the strike price and the time-to-maturities may make up a 3D flat surface in a three-dimensional coordinate system. However, empirical studies show that the implied volatilities tend to differ across the strikes and the time-to-maturities. For example, Figure 1 shows the implied volatilities of the AAPL (Apple stock options) call options on 1 March 2016, which exhibits the systematic dependence of implied volatilities with the strike price and the time-to-maturities. Thus, how to reveal this dependence by modeling the implied volatility is important. This article mainly studies the semi-parametric models of the implied volatility surface.

Figure 1.

The implied volatilities of AAPL call options on 1 March 2016.

Gaussian semi-parametric model

Parametric models and semi-parametric models

Dumas et al. ¹² analyzed the data of S&P500 index options and developed a quadratic volatility function of the strike price and the time-to-maturity:

$$\text{Model}1:\hat{\sigma }(K,\tau )=x_1+x_{2}K+x_3{K}^2\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}+x_4\tau +x_{5}K\tau +\varepsilon$$

(6) where K is the strike price, $\tau =T-t$ is the time-to-maturity, $\hat{\sigma }(K,\tau )$ is the approximation of the implied volatility at the point $(K,\tau )$ , $x_1,\dots ,x_5$ are the parameters independent of K and τ, and ɛ is the model error.

As the implied volatility is sensitive to the strike price and underlying price movement, Cassese and Guidolin ¹³ suggested replacing the strike price with the moneyness and the implied volatility model is rewritten as following:

$$\text{Model}2:\hat{\sigma }(M,\tau )=x_1+x_{2}M+x_3{M}^2\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}+x_4\tau +x_{5}M\tau +\varepsilon$$

(7) where $M=\mathit{ln}(K/S)$ is the moneyness.

Models 1 and 2 use market data to construct a volatility surface fitting the implied volatility. Once the parameters are located, the approximated implied volatility can be calculated for each pair of K and τ, which is simple. However, the accuracy may be not good. Borovkova and Permana ¹⁵ considered that the shapes of the implied volatilities vary with different time-to-maturities and may not be captured by a surface accurately. They extended the Model 2 and proposed the idea of a semi-parametric model, when fixing the time-to-maturity, the implied volatility is a function of the moneyness:

$$\text{Model}3(\text{Borovkova}):{\hat{\sigma }}_{\tau }(M)=x_{1,\tau }+x_{2,\tau }M\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}+x_{3,\tau }{M}^2+{\varepsilon }_{\tau }$$

(8) where $\tau ,M$ are the same as that of Model 2; ${\hat{\sigma }}_{\tau }$ , $x_{i,\tau },(i=1,2,3)$ , ${\varepsilon }_{\tau }$ are the estimated implied volatility, model parametric, and model error at τ, respectively.

Another important semi-parametric model is the SVI model proposed by Gatheral, ¹⁶ which could fit the “volatility smile” quite well. When fixing the time-to-maturity, SVI model is a function of the moneyness M:

$$\text{Model}4:{\hat{\sigma }}_{\tau }^2(M)\sqrt{\tau }=x_{1,\tau }+x_{2,\tau }\{x_{3,\tau }(M-x_{4,\tau })\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}+\sqrt{(M-x_{4,\tau }){\hspace{0.17em}}^2+x_{4,\tau }^2}\}+{\varepsilon }_{\tau }$$

(9)

Semi-parametric model with the exponential parameter

In the semi-parametric models modeling the implied volatility with a polynomial or fractional order polynomial, the highest degree with the moneyness is an important factor to influence the curve curvature. Wu et al. ¹⁷ proposed a semi-parametric model with the exponential parameter, which obtain a better result when using $|M{|}^{\alpha }(0<\alpha <2)$ replacing M² of Model 3. Figure 2 shows the curve curvatures of $|M{|}^{\alpha }$ with different α.

Figure 2.

The bent effect of $|M{|}^{\alpha }$ with different α.

In Wu et al.,¹⁷M² in Model 3 is replaced with $|M{|}^{\alpha }$ and the following model is proposed:

$$\text{Model}5:{\hat{\sigma }}_{\tau }(M)=x_{1,\tau }+x_{2,\tau }M+x_{3,\tau }|M{|}^{{\alpha }_{\tau }}+{\varepsilon }_{\tau }$$

(10)

The parameters of this model include original parameters $x_{i,\tau }$ of Model 3 and an exponential parameter α, which will provide more flexibility for implied volatility curves.

Gaussian semi-parametric model with exponential parameter

The drawback of the Model 5 is that it is not smooth at the point $M=0$ , which will affect the use of arbitrage-free conditions.

As we know, the shape of the Gaussian function (11) has the form of Figure 3, where μ is mathematical expectation of x, and θ is the standard deviation of x. Its overturn is quite similar to the shapes of Figure 2.

$$f(x)=\frac{1}{\theta \sqrt{2\pi }}{e}^{-\frac{(x-\mu ){\hspace{0.17em}}^2}{2{\theta }^2}}$$

(11)

$$g(x)={max}_x(\frac{1}{\theta \sqrt{2\pi }}{e}^{-\frac{(x-\mu {)}^2}{2{\theta }^2}})-\frac{1}{\theta \sqrt{2\pi }}{e}^{-\frac{(x-\mu {)}^2}{2{\theta }^2}}$$

(12)

Figure 3.

Gaussian function with $\mu =0$ and $\theta =1$ .

The function image of $g(x)$ is shown as following, where $\mu =0$ , θ takes different values.

Figure 4 shows the characteristics of the implied volatility “smile” (options whose strike price differs substantially from the underlying asset’s price command implied volatilities), besides, the amplitude of “smile” can be determined by the parameters of Gaussian function. Based on this, the Gaussian function can be used to construct a smooth term $g(M)$ substituting $|M|$ :

Figure 4.

The function image of $g(x)$ with different $\theta$ .

After using $g(M)$ to replace $|M{|}^{\alpha }$ , the Model 5 is rewritten as the following model:

$$\text{Model}6:{\hat{\sigma }}_{\tau }(M)\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}=x_{1,\tau }+x_{2,\tau }(M-x_{3,\tau })+x_{4,\tau }[{max}_M(\frac{1}{\theta \sqrt{2\pi }}{e}^{-\frac{(M-x_{3,\tau }{)}^2}{2{\theta }^2}})\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}-\frac{1}{\theta \sqrt{2\pi }}{e}^{-\frac{(M-x_{3,\tau }{)}^2}{2{\theta }^2}}{]}^{{\alpha }_{\tau }}+{\varepsilon }_{\tau }$$

(13) where θ is the standard deviation of M, ${\alpha }_{\tau },x_{i,\tau }(i=1\dots ,4)$ are fitting parameters.

Solution to the model parameters

The implied volatility calculated by using a good implied volatility model would be close to the market data. Suppose there are n options with strikes $K_{\tau }^1,K_{\tau }^2,\dots ,K_{\tau }^n$ , writing on the underlying asset when fixing the time-to-maturity τ, the corresponding moneyness and market implied volatilities are $M_{\tau }^1,M_{\tau }^2,\dots ,M_{\tau }^n$ and ${\sigma }_{\tau }^1,{\sigma }_{\tau }^2,\dots ,{\sigma }_{\tau }^n$ , respectively. Finding the solution to the parameters of the Model 6 can be converted to the problem of minimizing the following function:

$$G(x_{1,\tau },x_{2,\tau }{x}_{3,\tau },x_{4,\tau ,}{\alpha }_{\tau })\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}=\sum _{i=1}^n({\hat{\sigma }}_{\tau }(M_{\tau }^i,x_{1,\tau },x_{2,\tau }{x}_{3,\tau },x_{4,\tau ,}{\alpha }_{\tau })-{\sigma }_{\tau }^i{)}^2$$

(14)

Using least squares technique, the minimization of the function (14) can be converted to solve the following system of nonlinear equations:

$$\{\frac{\partial G}{\partial x_{1,\tau }}=2\sum _{i=1}^n\frac{\partial {\hat{\sigma }}_{\tau }(M_{\tau }^i,x_{1,\tau },x_{2,\tau }{x}_{3,\tau },x_{4,\tau ,}{\alpha }_{\tau })}{\partial x_{1,\tau }}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}({\hat{\sigma }}_{\tau }(M_{\tau }^i,x_{1,\tau },x_{2,\tau }{x}_{3,\tau },x_{4,\tau ,}{\alpha }_{\tau })-{\sigma }_{\tau }^i)=0,\frac{\partial G}{\partial x_{2,\tau }}=2\sum _{i=1}^n\frac{\partial {\hat{\sigma }}_{\tau }(M_{\tau }^i,x_{1,\tau },x_{2,\tau }{x}_{3,\tau },x_{4,\tau ,}{\alpha }_{\tau })}{\partial x_{2,\tau }}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}({\hat{\sigma }}_{\tau }(M_{\tau }^i,x_{1,\tau },x_{2,\tau }{x}_{3,\tau },x_{4,\tau ,}{\alpha }_{\tau })-{\sigma }_{\tau }^i)=0,\frac{\partial G}{\partial x_{3,\tau }}=2\sum _{i=1}^n\frac{\partial {\hat{\sigma }}_{\tau }(M_{\tau }^i,x_{1,\tau },x_{2,\tau }{x}_{3,\tau },x_{4,\tau ,}{\alpha }_{\tau })}{\partial x_{3,\tau }}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}({\hat{\sigma }}_{\tau }(M_{\tau }^i,x_{1,\tau },x_{2,\tau }{x}_{3,\tau },x_{4,\tau ,}{\alpha }_{\tau })-{\sigma }_{\tau }^i)=0,\frac{\partial G}{\partial x_{4,\tau }}=2\sum _{i=1}^n\frac{\partial {\hat{\sigma }}_{\tau }(M_{\tau }^i,x_{1,\tau },x_{2,\tau }{x}_{3,\tau },x_{4,\tau ,}{\alpha }_{\tau })}{\partial x_{4,\tau }}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}({\hat{\sigma }}_{\tau }(M_{\tau }^i,x_{1,\tau },x_{2,\tau }{x}_{3,\tau },x_{4,\tau ,}{\alpha }_{\tau })-{\sigma }_{\tau }^i)=0,\frac{\partial G}{\partial {\alpha }_{\tau }}=2\sum _{i=1}^n\frac{\partial {\hat{\sigma }}_{\tau }(M_{\tau }^i,x_{1,\tau },x_{2,\tau }{x}_{3,\tau },x_{4,\tau ,}{\alpha }_{\tau })}{\partial \alpha }\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}({\hat{\sigma }}_{\tau }(M_{\tau }^i,x_{1,\tau },x_{2,\tau }{x}_{3,\tau },x_{4,\tau ,}{\alpha }_{\tau })-{\sigma }_{\tau }^i)=0$$

(15)

The most common used method to solve the system of nonlinear equations is Newton method, but it requires the equation system have reversible Jacobi matrix. The system (15) may not satisfy this request due to the varying market data. In this article, the modified Levenberg Marquardt (L-M) algorithm^17–19 is used to solve the system of the nonlinear equations (15), which does not require the Jacobi matrix reversible, and has global convergence.

For convenience, the system is rewritten as a vector form. Let $X_{\tau }=(x_{1,\tau },x_{2,\tau }{x}_{3,\tau },x_{4,\tau ,}{\alpha }_{\tau }){\hspace{0.17em}}^T$ be the unknown vector in the equation system (15), $F(X_{\tau })$ and $J(X_{\tau })$ be the corresponding equations and the Jacobi matrix separately. At the kth iteration, the corresponding of $\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}{X}_{\tau },\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}F(X_{\tau }),\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}J(X_{\tau })$ are $X_{\tau }^k,F_k,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}{J}_k$ separately, d_k be the iteration step length.

Here we give the algorithm for solution to the model parameters by the modified L-M algorithm. The details of the modified L-M algorithms can be found in Fan. ¹⁹

Algorithm 1: Solution to the model parameters by using the modified L-M algorithm

Step 1: For the given time-to-maturity τ

Prepare the tolerance $\varepsilon >0$ and the thresholds $0<p_1<p_2<1$ ;

Prepare the initial values: $X_{\tau }^0\in R^5,{\mu }_0>m>0,k:=0.$

Step 2: Do while ( $\parallel J_k^T{F}_k\parallel >\varepsilon$

{calculate

$$d_k=-(J_k^T{J}_k+{\mu }_k\parallel F_k\parallel ){-}^1{J}_k^T{F}_k$$

(16)

Step 3: Calculate r_k:

$$r_k=\frac{\parallel F_k{\parallel }^2-\parallel F(X_{\tau }^k+d_k){\parallel }^2}{\parallel F{\parallel }_k^2-\parallel F_k+J_k{d}_k{\parallel }^2}$$

(17)

Step 4:

$$X_{\tau }^{k+1}=\{X_{\tau }^k+d_k,r_k>p_1,X_{\tau }^k,r_k\le p_1$$

(18)

Step 5: Adjust parameter ${\mu }_k$

$${\mu }_{k+1}=\{4{\mu }_k,r_k<p_1,{\mu }_k,r_k\in [p_1<p_2],max\{\frac{{\mu }_k}{4},m\},r_k>p_2$$

(19)

Step 6: $k:=k+1$ }

Step 7: Output $X_{\tau }^k$ .

End-Algorithm

The model parameters solve by the modified L-M algorithm, then is taken into the implied volatility model, such as Model 6, to calculate the estimated implied volatility ${\hat{\sigma }}_{\tau }$ . Next the estimated implied volatility ${\hat{\sigma }}_{\tau }$ is put into the B-S model (formula (3)) to gain the estimated option price $\hat{C}$ .

The semi-parametric model with no-arbitrage

There should be no arbitrage in a complete options market. The arbitrage opportunities in the options market include the strike arbitrage and the calendar arbitrage. The strike arbitrage means obtaining profits from buying and selling at the same time two kinds of options writing on the same underlying asset with the same maturity date but different strike prices. The calendar arbitrage means obtaining profits from trading two kinds of option writing on the same underlying asset with the same strike price but different maturity date at the same time.

It is an important assumption of B-S option pricing model that there is no arbitrage opportunity in the trading market. However, the option implied volatilities and the option prices of the options market are subject to market supply and demand, so the market does exist the arbitrage. That indicates the implied volatility model obtained from fitting the market data cannot get an implied volatility predictor with no-arbitrage. In this section, we consider to calibrate the Model 6 with the arbitrage-free conditions and supply an implied volatility predictor with no-arbitrage for the options market.

Arbitrage-free conditions of the implied volatility surface

In 2009, Fengler ²⁰ developed the arbitrage-free conditions for option prices to judge whether there is a strike arbitrage opportunity when the time-to-maturity τ is fixed:

$$-e-{\hspace{0.17em}}_{t}T{r}_s\mathit{ds}\le \frac{V_i-V_{i-1}}{K_i-K_{i-1}}\le \frac{V_{i+1}-V_i}{K_{i+1}-K_i}\le 0,i=2,\dots ,n-1$$

(20) where n is the number of the options writing on the same underlying asset with different strike price $K_i,i=1,2,\dots ,n$ under the maturity τ, V_i is the corresponding option price. When the option contract $(K_i,\tau ,V_i),i=1,\dots ,n$ meets formula (20), the options will satisfy no-strike-arbitrage requirements.

Roper ²¹ extended above arbitrage-free conditions from option price to option implied volatilities. Denoting the total variance ${\nu }^2(M,\tau )={\hat{\sigma }}^2(M,\tau )\tau$ , where M is the moneyness, the arbitrage-free conditions for implied volatilities are as follows: ²¹

$$\{(\text{i})\text{Forevery}\tau >0,\nu (·,\tau )\text{istwicedifferentiable};(\text{ii})\text{Forevery}M\in R,\text{and}\tau >0,\nu (M,\tau )>0;(\text{iii})\text{Forevery}M\in R,\tau >0,\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}(1-\frac{M\partial {\hspace{0.17em}}_M\nu (M,\tau )}{\nu (M,\tau )}{)}^2-\frac{1}{4}{\nu }^2(M,\tau )(\partial {\hspace{0.17em}}_M\nu (M,\tau ){)}^2\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}\mathrm{\hspace{1em}\hspace{1em}\hspace{1em}}+\nu (M,\tau )\partial {\hspace{0.17em}}_{\mathit{MM}}2\nu (M,\tau )\ge 0;(\text{iv})\text{Forevery}M\in R,\nu (M,·)\text{isnon}-\text{decreasing};(\text{v})\text{Forevery}M\in R,\nu (M,0)=0$$

(21) where (i) of the equations (21) ensures the smoothness of the implied volatility function. (ii) of the equations (21) makes sure that the implied volatilities are positivity. (iii) of the equations (21) ensures that the implied volatility model is free of the strike arbitrage. (iv) of the equations (21) requires the implied volatility function is monotonicity in τ, which makes sure that the implied volatility model avoids the calendar arbitrage. (v) of the equations (21) is the boundary conditions of value at maturity.

As examples, Figures 5 and 6 show the arbitrage opportunities of market data. Figure 5 shows the option prices of AAPL call options with several different maturities. The red circle points in Figure 5 indicate that the data did not satisfy the conditions of the equations (20) and there have strike arbitrage opportunities. Figure 6 shows the total variance of AAPL call options on 1 March 2016 for different time-to-maturities. For some moneynesses, the total variance does not decreasing along the maturity, which breaks (iv) of the equations (21) and there have calendar arbitrage opportunities.

Figure 5.

Price plot for AAPL call option data with several maturities.

Figure 6.

Total variance plot for AAPL call option data on 1 March 2016 (matu. = time-to-maturity).

Calibrate the Gaussian semi-parametric model with arbitrage-free conditions

In this section, the Gaussian semi-parametric model proposed in “Gaussian semi-parametric model with exponential parameter” section is combined with equations (21) to predict no-arbitrage implied volatilities. The new model can be expressed as following:

Model 7:

Similar to Model 6, the Model 7 can be expressed as a vector form. Let $X_{\tau }=(x_{1,\tau },x_{2,\tau }{x}_{3,\tau },x_{4,\tau ,}{\alpha }_{\tau }){\hspace{0.17em}}^T$ . Suppose in a given trading date, the number of time-to-maturity is m, and ${\tau }_j<{\tau }_{j+1},(j=1,\dots ,m-1)$ ; the number of strike price is n_j with the time-to-maturity ${\tau }_j$ , the corresponding moneyness vector is $M_{{\tau }_j}=(M_{{\tau }_j}^1,\dots ,M_{{\tau }_j}^{n_j}){\hspace{0.17em}}^T$ . Then the problem of finding solution of the Model 7 can be transformed into the following inequality constrained nonlinear optimization problems

In this article, the sequential quadratic programming algorithm ²² is used to solve above nonlinear constrained optimization problems. For every ${\tau }_j,j=m,m-1,\dots ,1$ , equations (23) is a inequality constrained nonlinear optimization problem and the sequential quadratic programming algorithm has to be used once.

Empirical analysis

In this section, the options market data are used to verify the effectiveness of the Gaussian semi-parametric model. The effectiveness includes fitting accuracy and the forecasting ability. The market dataset for testing comes from AAPL stock options on 1 March 2016 and 2 March 2016.

The fitting accuracy and the forecasting ability are measured by the fitting error and the forecasting error, respectively. The forecasting error means when using the curves fitted by one day market data to calculate the implied volatilities on the next day, the forecast error will occur compared with the market data of the day. In this article, the root mean-squared error and the mean absolute estimation error are used:

The root mean-squared estimation error:

$$E_{\mathit{RMS}}=\sqrt{\frac{1}{n}\sum _{i=1}^n(\hat{y}-y){\hspace{0.17em}}^2}$$

(24)

1. The mean absolute estimation error:

   $$E_{\mathit{MA}}=\frac{1}{n}\sum _{i=1}^n|\hat{y}-y|$$
   (25)

where y is an market data and $\hat{y}$ is the corresponding estimated value. n is the number of the values for comparison.

The approximate implied volatilities calculated by the implied volatility models can be used to calculate the approximate option prices by the B-S model. In the following tests, implied volatilities and option prices are compared between the approximations calculated by the models and the market data, and their corresponding error indicators are as follows:

$$\{E_{{\mathit{RMS}}_{\_\sigma }}=\sqrt{\frac{1}{n}\sum _{i=1}^n(\hat{\sigma }-\sigma {)}^2};E_{{\mathit{RMS}}_{\_C}}=\sqrt{\frac{1}{n}\sum _{i=1}^n(\hat{C}-C{)}^2};E_{{\mathit{MA}}_{\_\sigma }}=\frac{1}{n}\sum _{i=1}^n|\hat{\sigma }-\sigma |;E_{\mathit{MA}\_{\hspace{0.17em}}_C}=\frac{1}{n}\sum _{i=1}^n|\hat{C}-C|$$

(26) where σ and C are the market data of implied volatility and option price, $\hat{\sigma }$ and $\hat{C}$ are the corresponding estimated value (see details in “Calibrate the Gaussian semi-parametric model with arbitrage-free conditions” section).

Fitting and forecasting results by the Gaussian semi-parametric model and comparison with the other models

In testing, the dataset of AAPL stock options on 1 March 2016 is used to fit the implied volatility curves by using the Models 3, 5, 6, and the SVI model (Model 4). The market implied volatilities with τ(time-to-maturity) = 45 days, 108 days, and 689 days are marked in Figures 7–10, which show the fitting curves of the Models 3, 5, 6, and the SVI model, respectively.

Figure 7.

The fitted implied volatility curves with Model 3 (matu. = time-to-maturity).

Figure 8.

The fitted implied volatility curves with Model 5 (matu. = time-to-maturity).

Figure 9.

The fitted implied volatility curves with Model 6 (matu. = time-to-maturity).

Figure 10.

The fitted implied volatility curves with SVI (matu. = time-to-maturity).

It can be found that the fitting results of the Model 6 and the SVI model are better than the Model 3. The Model 5 though has a similar effect but the curve for τ = 45 days is not smooth on $M=0$ .

Table 1 shows the fitting results of implied volatilities and option prices of different models on 1 March 2016 for different strikes with the same maturity 11 March 2016 (τ = 10 days). The AAPL spot price S = 100.53 ($), the risk-free interest rate r = 0.0028 on 1 March 2016. In the following tables, K is the strike price, $\sigma ,C$ are the market data of implied volatility and option price, $\hat{\sigma },\hat{C}$ , are the corresponding estimated value.

Table 1.

The fitting results of semi-parametric models.

K($)	σ	C($)	$\hat{\sigma }$				$\hat{C}$ ($)
			Model 3	Model 5	Model 6	SVI	Model 3	Model 5	Model 6	SVI
95	0.29	5.81	0.32	0.31	0.29	0.31	5.91	5.88	5.82	5.86
95.5	0.28	5.36	0.31	0.30	0.28	0.30	5.46	5.42	5.35	5.40
96	0.28	4.91	0.30	0.29	0.27	0.29	5.02	4.96	4.89	4.94
96.5	0.27	4.46	0.30	0.28	0.27	0.28	4.59	4.51	4.45	4.50
97	0.26	4.01	0.29	0.27	0.26	0.28	4.17	4.07	4.01	4.09
97.5	0.25	3.60	0.29	0.26	0.25	0.27	3.77	3.63	3.59	3.70
98	0.25	3.20	0.28	0.25	0.25	0.27	3.38	3.21	3.19	3.32
98.5	0.24	2.80	0.28	0.24	0.24	0.27	3.01	2.80	2.80	2.96
99	0.24	2.44	0.27	0.23	0.24	0.26	2.65	2.40	2.44	2.62
99.5	0.23	2.10	0.26	0.22	0.23	0.26	2.31	2.01	2.11	2.29
100	0.23	1.79	0.26	0.21	0.23	0.26	2.00	1.65	1.80	1.99
101	0.22	1.26	0.25	0.20	0.22	0.25	1.44	1.10	1.26	1.46
102	0.22	0.84	0.24	0.21	0.22	0.25	0.99	0.77	0.84	1.02
103	0.21	0.52	0.23	0.21	0.21	0.24	0.63	0.53	0.53	0.68
104	0.21	0.32	0.23	0.22	0.21	0.24	0.38	0.35	0.33	0.43
105	0.21	0.19	0.22	0.23	0.22	0.23	0.21	0.23	0.20	0.25
106	0.21	0.11	0.21	0.23	0.22	0.22	0.10	0.15	0.13	0.13
107	0.23	0.08	0.21	0.24	0.23	0.22	0.05	0.09	0.08	0.07
108	0.23	0.05	0.20	0.24	0.24	0.21	0.02	0.06	0.06	0.03

Then the fitted model functions on 1 March 2016 are used to forecast the implied volatilities and option prices on 2 March 2016. Table 2 shows the forecasting results of implied volatilities and option prices on 2 March 2016 for different strikes with the same maturity 11 March 2016 (τ = 9 days). The AAPL spot price S = 100.75 ($), the risk-free interest rate r = 0.0026 on 2 March 2016.

Table 2.

The forecasting results of semi-parametric models.

K($)	σ	C($)	$\hat{\sigma }$				$\hat{C}$ ($)
			Model 3	Model 5	Model 6	SVI	Model 3	Model 5	Model 6	SVI
95	0.31	6.01	0.32	0.32	0.30	0.31	6.05	6.03	5.98	6.02
95.5	0.30	5.56	0.31	0.31	0.29	0.30	5.60	5.57	5.51	5.55
96	0.28	5.06	0.31	0.30	0.28	0.29	5.15	5.11	5.04	5.09
96.5	0.27	4.61	0.30	0.29	0.27	0.28	4.71	4.65	4.59	4.63
97	0.26	4.15	0.29	0.27	0.26	0.28	4.29	4.20	4.14	4.21
97.5	0.25	3.70	0.29	0.26	0.25	0.27	3.87	3.76	3.71	3.80
98	0.25	3.30	0.28	0.25	0.25	0.27	3.47	3.33	3.30	3.41
98.5	0.24	2.87	0.28	0.24	0.24	0.27	3.09	2.91	2.90	3.04
99	0.23	2.49	0.27	0.23	0.24	0.27	2.72	2.50	2.53	2.68
99.5	0.23	2.14	0.27	0.22	0.23	0.26	2.38	2.11	2.18	2.35
100	0.22	1.81	0.26	0.21	0.23	0.26	2.05	1.74	1.86	2.04
101	0.22	1.25	0.25	0.20	0.22	0.25	1.47	1.12	1.29	1.49
102	0.21	0.82	0.24	0.20	0.22	0.25	1.00	0.77	0.85	1.03
103	0.21	0.51	0.23	0.21	0.21	0.24	0.64	0.52	0.53	0.68
104	0.21	0.30	0.23	0.22	0.21	0.24	0.37	0.34	0.32	0.42
105	0.21	0.18	0.22	0.22	0.22	0.23	0.20	0.21	0.19	0.24
106	0.21	0.10	0.21	0.23	0.22	0.23	0.10	0.13	0.11	0.13
107	0.23	0.08	0.21	0.23	0.23	0.22	0.04	0.08	0.07	0.06
108	0.25	0.06	0.20	0.24	0.24	0.22	0.02	0.05	0.04	0.03

Figures 11 and 12 show the fitting errors of different models on 1 March 2016 with different maturities and the forecasting errors on 2 March 2016 when using the curves fitted by the data of 1 March 2016 to calculate the implied volatilities on 2 March 2016. The experimental results show that the Gaussian semi-parametric model (the Model 6) has a similar good accuracy with the SVI model.

Figure 11.

The fitting errors of semi-parametric models.

Figure 12.

The forecasting errors of semi-parametric models.

Figure 13 shows the implied volatility surface by using the cubic spline interpolation method with the approximate implied volatilities for different maturities calculated by the Model 6, the market data comes from the AAPL stocks options on 1 March 2016. It can be found that the implied volatility functions have distinctly different shapes with different maturities, and the Gaussian semi-parametric method captures this feature remarkably well.

Figure 13.

The fitted implied volatility surface with Model 6 on 1 March 2016.

Fitting and forecasting results by the Gaussian semi-parametric model with arbitrage-free conditions

In this test, the data of AAPL stock options on 1 March 2016 is used to fit implied volatility curves by using arbitrage-free implied volatility Model 7. Figure 14 shows some arbitrage-free calibrated option prices with different maturities, and examined by the equation (20), there is absence of the strike arbitrage. Figure 15 shows the fitted total variance by the Model 7, in which there is no intersection among different maturities (i.e. non-decreasing in moneyness), that is, there is absence of the calendar arbitrage.

Figure 14.

The arbitrage-free price plot for AAPL call option data with several maturities on 1 March 2016.

Figure 15.

The arbitrage-free total variance plot for AAPL call option data on 1 March 2016 (matu. = time-to-maturity).

Table 3 shows the fitting errors and the forecasting errors of the Model 7 and the SVI model with no-arbitrage constraints. It can be found that the Model 7 and the SVI model have the similar fitting and forecasting ability under the no-arbitrage constraints.

Table 3.

The arbitrage-free fitting and forecasting results of Model 7 and SVI.

Maturity	Model	The arbitrage-free fitting results in 2016-3-1				The arbitrage-free forecasting results in 2016-3-2
		% $E_{\mathit{RMS}\_\sigma }$	% $E_{\mathit{RMS}\_C}$	% $E_{\mathit{MA}\_\sigma }$	% $E_{\mathit{MA}\_C}$	% $E_{\mathit{RMS}\_\sigma }$	% $E_{\mathit{RMS}\_C}$	% $E_{\mathit{MA}\_\sigma }$	% $E_{\mathit{MA}\_C}$
2016-3-11	Model7	4.03	10.19	3.12	7.75	4.16	6.47	3.06	5.48
	SVI	3.95	9.80	3.01	7.52	3.86	6.43	2.95	5.42
2016-3-18	Model7	2.72	5.70	1.85	4.17	16.79	4.69	3.17	2.22
	SVI	2.69	6.23	1.90	4.02	15.89	4.63	3.01	2.10
2016-3-24	Model7	2.17	2.82	1.29	1.90	1.52	2.95	0.74	2.08
	SVI	3.52	3.21	1.40	2.09	1.68	3.01	1.06	2.70
2016-4-15	Model7	6.44	21.70	4.99	14.04	1.52	4.12	1.15	2.44
	SVI	5.32	20.03	5.16	13.26	2.30	4.89	1.20	2.30
2016-5-20	Model7	3.15	7.86	2.56	5.49	1.68	6.63	1.15	5.46
	SVI	4.03	7.11	2.30	6.31	1.45	5.96	1.18	4.54
2016-6-17	Model7	3.27	4.08	1.99	2.91	1.06	7.59	0.88	4.68
	SVI	2.98	6.01	1.71	2.97	0.99	6.72	1.35	4.21
2016-7-15	Model7	2.12	3.30	1.34	2.30	1.18	8.14	0.89	6.04
	SVI	1.31	5.91	1.72	3.51	1.20	6.33	0.57	5.95
2016-10-21	Model7	1.23	4.66	0.87	3.99	0.75	10.94	0.65	9.40
	SVI	0.59	6.57	1.07	5.21	0.75	10.12	0.35	9.72
2017-1-20	Model7	0.76	3.40	0.61	2.88	0.96	7.01	0.50	5.30
	SVI	0.56	5.71	1.05	3.25	1.03	6.24	0.57	5.68
2017-6-16	Model7	0.76	3.01	0.53	2.23	1.17	35.28	0.56	15.22
	SVI	0.35	5.11	0.21	3.43	1.08	34.64	0.59	14.81

Conclusion

In this article, a Gaussian semi-parametric implied volatility model is proposed in which the Gaussian function is used to construct a smooth term substituting the quadratic term in the Borovkova’s model. And then the arbitrage-free constraints are used to calibrate the model. The empirical analysis shows that the Gaussian semi-parametric model can obtain a good performance in fitting and forecast, which is similar to the stochastic volatility inspired model that is based on the stochastic implied volatility model.