8.3 Semiparametric estimation of the SPD


8.3.1 Estimating the call pricing function

The use of nonparametric regression to recover the SPD was first investigated by Aït-Sahalia and Lo (1998). They propose to use the Nadaraya-Watson estimator to estimate the historical call prices $ C_{t}(\cdot)$ as a function of the following state variables $ ( S_{t}, K, \tau, r_{t,
\tau},\delta_{t,\tau})^{\top }$. Kernel regressions are advocated because there is no need to specify a functional form and the only required assumption is that the function is smooth and differentiable, Härdle (1990). When the regressor dimension is $ 5$, the estimator is inaccurate in practice. Hence, there is a need to reduce the dimension or equivalently the number of regressors. One method is to appeal to no-arbitrage arguments and collapse $ S_{t}$, $ r_{t,\tau}$ and $ \delta_{t,\tau}$ into the forward price $ F_{t} = S_{t}e^{(r_{t,\tau}-\delta_{t,\tau})\tau}$ in order to express the call pricing function as:


$\displaystyle C(S_{t}, K, \tau, r_{t,\tau}, \delta_{t,\tau}) = C(F_{t,\tau}, K, \tau, r_{t,\tau}).$     (8.7)

An alternative specification assumes that the call option function is homogeneous of degree one in $ S_{t}$ and $ K$ (as in the Black-Scholes formula) so that:


$\displaystyle C(S_{t}, K, \tau, r_{t,\tau}, \delta_{t,\tau}) = K C(S_{t}/K,\tau, r_{t,\tau},
\delta_{t,\tau} ).$     (8.8)

Combining the assumptions of (8.7) and (8.8) the call pricing function can be further reduced to a function of three variables $ (\frac{K}{F_{t,\tau}},\tau,r_{t,\tau})$.

Another approach is to use a semiparametric specification based on the Black-Scholes implied volatility. Here, the implied volatility $ \sigma$ is modelled as a nonparametric function, $ \sigma(F_{t,\tau},K,\tau)$:

$\displaystyle C(S_{t}, K, \tau, r_{t,\tau}, \delta_{t,\tau}) = C_{BS} (F_{t,\tau},K,\tau, r_{t,\tau};\sigma(F_{t,\tau},K,\tau)).$ (8.9)

Empirically the implied volatility function mostly depends on two parameters: the time to maturity $ \tau$ and the moneyness $ M=K/F_{t,\tau}$. Almost equivalently, one can set $ M=\tilde{S_{t}}/K$ where $ \tilde{S_{t}}=S_{t}-D$ and $ D$ is the present value of the dividends to be paid before the expiration. Actually, in the case of a dividend yield $ \delta_{t}$, we have $ D=S_{t}(1-e^{-\delta_{t}})$. If the dividends are discrete, then $ D=\sum \limits_{t_{i}\leq t+\tau}D_{t_{i}}e^{-r_{t,\tau_{i}}}$ where $ t_{i}$ is the dividend payment date of the $ i^{th}$ dividend and $ \tau_{i}$ is its maturity.

Therefore, the dimension of the implied volatility function can be reduced to $ \sigma(K/F_{t,\tau}, \tau)$. In this case the call option function is:

$\displaystyle C(S_{t}, K, \tau, r_{t,\tau}, \delta_{t,\tau}) = C_{BS} (F_{t,\tau},K,\tau, r_{t,\tau};\sigma(K/F_{t,\tau},\tau)).$ (8.10)

Once a smooth estimate of $ \hat{\sigma}(\cdot)$ is obtained, estimates of $ \hat{C_{t}}(\cdot)$, $ \hat{\Delta}_{t} =
\frac{\partial \hat{C_{t}}(\cdot)} {\partial S_{t}}$, $ \hat{\Gamma}_{t} = \frac {\partial^{2}\hat{C_{t}}(\cdot)} {
\partial S_{t}^{2}}$, and $ \hat{f}_{t}^{*} = e^{r_{t,\tau} \tau}
\bigg[ \frac{\partial^{2} \hat {C_{t}}(\cdot)} {\partial K^{2}}
\bigg]$ can be calculated.


8.3.2 Further dimension reduction

The previous section proposed a semiparametric estimator of the call pricing function and the necessary steps to recover the SPD. In this section the dimension is reduced further using the suggestion of Rookley (1997). Rookley uses intraday data for one maturity and estimates an implied volatility surface where the dimension are the intraday time and the moneyness of the options.

Here, a slightly different method is used which relies on all settlement prices of options of one trading day for different maturities to estimate the implied volatility surface $ \sigma(K/F_{t,\tau}, \tau)$. In the second step, these estimates are used for a given time to maturity which may not necessarily correspond to the maturity of a series of options. This method allows one to compare the SPD at different dates because of the fixed maturity provided by the first step. This is interesting if one wants to study the dynamics and the stability of these densities.

Fixing the maturity also allows us to eliminate $ \tau$ from the specification of the implied volatility function. In the following part, for convenience, the definition of the moneyness is $ M=\tilde{S_{t}}/K$ and we denote by $ \sigma$ the implied volatility. The notation $ \frac {\partial
f(x_{1},\ldots,x_{n})}{\partial x_{i}}$ denotes the partial derivative of $ f$ with respect to $ x_{i}$ and $ \frac {\textrm{d}
f(x) }{\textrm{d} x}$ the total derivative of $ f$ with respect to $ x$.

Moreover, we use the following rescaled call option function:

$\displaystyle c_{it}$ $\displaystyle =$ $\displaystyle \frac{C_{it}} {\tilde{S_{t}}},$  
$\displaystyle M_{it}$ $\displaystyle =$ $\displaystyle \frac{\tilde{S_{t}}} {K_{i}}.$  

where $ C_{it}$ is the price of the $ i^{th}$ option at time $ t$ and $ K_{i}$ is its strike price.

The rescaled call option function can be expressed as:

$\displaystyle c_{it}$ $\displaystyle =$ $\displaystyle c(M_{it};\sigma(M_{it}))=\Phi(d_{1})- \frac{e^{-r \tau}
\Phi(d_{2})}{M_{it}},$  
$\displaystyle d_{1}$ $\displaystyle =$ $\displaystyle \frac{\log(M_{it}) + \left\{r_{t} +
\frac{1}{2}\sigma(M_{it})^{2}\right\}\tau} {\sigma(M_{it}) \sqrt{\tau}},$  
$\displaystyle d_{2}$ $\displaystyle =$ $\displaystyle d_{1} - \sigma(M_{it}) \sqrt{\tau}.$  

The standard risk measures are then the following partial derivatives (for notational convenience subscripts are dropped):

$\displaystyle \Delta=\frac{\partial C}{\partial S}=\frac{\partial C}{\partial \tilde{S}} = c(M,\sigma(M))+ \tilde{S}\frac{\partial c}{\partial \tilde{S}},$    

$\displaystyle \Gamma=\frac{\partial \Delta}{\partial S} = \frac{\partial^{2} C}...
...\partial \tilde{S}} + \tilde{S} \frac {\partial^{2} c}{\partial \tilde{S}^{2}}.$    

where
$\displaystyle \frac {\partial c} {\partial \tilde{S}}$ $\displaystyle =$ $\displaystyle \frac {\textrm{d} c} {\textrm{d}
M} \frac {\partial M} {\partial \tilde{S}} = \frac {\textrm{d} c} {\textrm{d} M} \frac
{1} {K},$  
$\displaystyle \frac {\partial^{2} c}{\partial \tilde{S}^{2}}$ $\displaystyle =$ $\displaystyle \frac {\textrm{d}^{2} c}
{\textrm{d} M^{2}} \bigg({\frac {1} {K}}\bigg)^{2}.$  

The SPD is then the second derivative of the call option function with respect to the strike price:

$\displaystyle f^{*}(\cdot)=e^{r\tau}\frac {\partial^{2} C}{\partial K^{2}}= e^{r\tau} \tilde{S} \frac {\partial^{2} c}{\partial K^{2}}.$ (8.11)

The conversion is needed because $ c(\cdot)$ is being estimated not $ C(\cdot)$. The analytical expression of (8.13) depends on:
$\displaystyle \frac {\partial^{2} c}{\partial K^{2}}$ $\displaystyle =$ $\displaystyle \frac
{\textrm{d}^{2} c} {\textrm{d} M^{2}} \bigg(\frac {M}
{K}\bigg)^{2} + 2 \frac {\textrm{d} c} {\textrm{d} M} \frac {M}
{K^{2}}$  

The functional form of $ \frac {\textrm{d} c}{\textrm{d} M}$ is:

$\displaystyle \frac{\textrm{d} c}{\textrm{d} M} = \Phi'(d_{1})\frac{\textrm{d} ...
...M} \frac{\textrm{d} d_{2}}{\textrm{d} M} +e^{-r\tau} \frac{\Phi(d_{2})}{M^{2}},$ (8.12)

while $ \frac {\textrm{d}^{2} c}{\textrm{d} M^{2}}$ is:
$\displaystyle \frac{\textrm{d}^{2} c}{\textrm{d} M^{2}}$ $\displaystyle =$ $\displaystyle \Phi'(d_{1})\bigg[
\frac{\textrm{d}^{2} d_{1}}{\textrm{d} M^{2}} - d_{1}\bigg(\frac{\textrm{d}
d_{1}}{\textrm{d} M}\bigg)^{2}\bigg]$  
  $\displaystyle -$ $\displaystyle \frac{e^{-r\tau} \Phi'(d_{2})}{M} \bigg[
\frac{\textrm{d}^{2} d_{...
...textrm{d} M} - d_{2}\bigg(\frac{\textrm{d} d_{2}}{\textrm{d} M}\bigg)^{2}\bigg]$  
  $\displaystyle -$ $\displaystyle \frac{2 e^{-r\tau} \Phi(d_{2})}{M^{3}} \bigg.$ (8.13)

The quantities in (8.14) and (8.15) are a function of the following first derivatives:

$\displaystyle \frac{\textrm{d} d_{1}}{\textrm{d}M}$ $\displaystyle =$ $\displaystyle \frac{\partial d_{1}}{\partial M} +
\frac{\partial d_{1}}{\partial \sigma} \frac{\partial \sigma}{\partial M},$  
$\displaystyle \frac{\textrm{d} d_{2}}{\textrm{d}M}$ $\displaystyle =$ $\displaystyle \frac{\partial d_{2}}{\partial M} +
\frac{\partial d_{2}}{\partial \sigma} \frac{\partial \sigma}{\partial M},$  
$\displaystyle \frac{\partial d_{1}}{\partial M}$ $\displaystyle =$ $\displaystyle \frac{\partial d_{2}}{\partial
M}=\frac{1} {M \sigma \sqrt{\tau}},$  
$\displaystyle \frac{\partial d_{1}}{\partial \sigma}$ $\displaystyle =$ $\displaystyle -\frac{\log(M)+r
\tau}{\sigma^{2}\sqrt{\tau}}+\frac{\sqrt{\tau}}{2},$  
$\displaystyle \frac{\partial d_{2}}{\partial \sigma}$ $\displaystyle =$ $\displaystyle -\frac{\log(M)+r \tau}{\sigma^{2}
\sqrt{\tau}}-\frac{\sqrt{\tau}}{2}.$  

For the remainder of this chapter, we define:

$\displaystyle V$ $\displaystyle =$ $\displaystyle \sigma(M),$  
$\displaystyle V'$ $\displaystyle =$ $\displaystyle \frac{\partial \sigma(M)} {\partial M},$  
$\displaystyle V''&=$   $\displaystyle \frac{\partial^{2} \sigma(M)} {\partial M^{2}}.$ (8.14)

The quantities in (8.14) and (8.15) also depend on the following second derivative functions:

$\displaystyle \frac{\textrm{d}^{2} d_{1}}{\textrm{d}M^{2}}$ $\displaystyle =$ $\displaystyle -\frac{1}{M\sigma\sqrt{\tau}}\bigg[\frac{1}{M}+\frac{V'}{\sigma}\...
...\bigg(\frac{\sqrt{\tau}}{2}-\frac{\log(M)+r \tau}{\sigma^{2} \sqrt{\tau}}\bigg)$  
  $\displaystyle +$ $\displaystyle V'\bigg[2V'\frac{\log(M)+r \tau}{\sigma^{3}
\sqrt{\tau}} -\frac{1}{M\sigma^{2}\sqrt{\tau}}\bigg],$ (8.15)


$\displaystyle \frac{\textrm{d}^{2} d_{2}}{\textrm{d}M^{2}}$ $\displaystyle =$ $\displaystyle -\frac{1}{M\sigma\sqrt{\tau}}\bigg[\frac{1}{M}+\frac{V'}{\sigma}\...
...\bigg(\frac{\sqrt{\tau}}{2}+\frac{\log(M)+r \tau}{\sigma^{2} \sqrt{\tau}}\bigg)$  
  $\displaystyle +$ $\displaystyle V'\bigg[2V'\frac{\log(M)+r \tau}{\sigma^{3} \sqrt{\tau}} -\frac{1}{M
\sigma^{2}\sqrt{\tau}}\bigg].$ (8.16)

Local polynomial estimation is used to estimate the implied volatility smile and its first two derivatives in (8.16). A brief explanation will be described now.


8.3.3 Local Polynomial Estimation

Consider the following data generating process for the implied volatilities:

$\displaystyle \sigma=g(M,\tau)+\sigma^{*}(M,\tau)\varepsilon,$      

where $ \textrm{E}(\varepsilon)=0$, $ \textrm{Var}(\varepsilon)=1$. $ M,\tau$ and $ \varepsilon$ are independent and $ \sigma^{*}(m_{0},\tau_{0})$ is the conditional variance of $ \sigma$ given $ M=m_{0},
\tau=\tau_{0}$. Assuming that all third derivatives of $ g$ exist, one may perform a Taylor expansion for the function $ g$ in a neighborhood of $ (m_{0},\tau_{0})$:


$\displaystyle g(m,\tau) \approx g(m_{0},\tau_{0})$ $\displaystyle +$ $\displaystyle \frac{\partial
g}{\partial
M}\bigg\vert _{m_{0},\tau_{0}}(m-m_{0}...
...}\frac{\partial^{2}
g}{\partial M^{2}}\bigg\vert _{m_{0},\tau_{0}}(m-m_{0})^{2}$  
  $\displaystyle +$ $\displaystyle \frac{\partial
g}{\partial\tau}\bigg\vert _{m_{0},\tau_{0}}(\tau-...
...artial^{2} g}{\partial
\tau^{2}}\bigg\vert _{m_{0},\tau_{0}}(\tau-\tau_{0})^{2}$  
  $\displaystyle +$ $\displaystyle \frac{1}{2} \frac{\partial^{2} g}{\partial M \partial
\tau}\bigg\vert _{m_{0},\tau_{0}}(m-m_{0})(\tau-\tau_{0}).$ (8.17)

This expansion suggests an approximation by local polynomial fitting, Fan and Gijbels (1996). Hence, to estimate the implied volatility at the target point $ (m_{0},\tau_{0})$ from observations $ \sigma_{j}$ $ (j=1,\ldots,n)$, we minimize the following expression:

\begin{displaymath}\begin{array}{l}
\sum_{j=1}^{n}\bigg\{\sigma_{j}-\Big[\beta_{...
...}
K_{h_{M},h_{\tau}}(M_{j}-m_{0},\tau_{j}-\tau_{0})
\end{array}\end{displaymath}     (8.18)

where $ n$ is the number of observations (options), $ h_{M}$ and $ h_{\tau}$ are the bandwidth controlling the neighborhood in each directions and $ K_{h_{M},h_{\tau}}$ is the resulting kernel function weighting all observation points. This kernel function may be a product of two univariate kernel functions.

For convenience use the following matrix definitions:

$\displaystyle X=\left(\begin{matrix}
1 & M_{1}-m_{0}&(M_{1}-m_{0})^{2}&\tau_{1}...
...\tau_{n}-\tau_{0})^{2}&(M_{n}-m_{0})(\tau_{n}-\tau_{0})\\
\end{matrix}\right),$      


\begin{displaymath}\begin{array}{l l l l}
\sigma=\left(\begin{matrix}\sigma_{1}\...
...}\beta_{0}\\ \vdots\\ \beta_{5}\end{matrix}\right).
\end{array}\end{displaymath}      

Hence, the weighted least squares problem (8.20) can be written as

$\displaystyle \min_{\beta}\left(\sigma-X\beta\right)^{\top }W\left(\sigma-X\beta\right).$     (8.19)

and the solution is given by
$\displaystyle \hat{\beta}=\left(X^{\top }WX\right)^{-1}X^{\top }W\sigma.$     (8.20)

A nice feature of the local polynomial method is that it provides the estimated implied volatility and its first two derivatives in one step. Indeed, one has from (8.19) and (8.20):

$\displaystyle \widehat{\frac{\partial g}{\partial
M}}\bigg\vert _{m_{0},\tau_{0}}$ $\displaystyle =$ $\displaystyle \hat{\beta_{1}},$  
$\displaystyle \widehat{\frac{\partial^{2} g}{\partial
M^{2}}}\bigg\vert _{m_{0},\tau_{0}}$ $\displaystyle =$ $\displaystyle 2\hat{\beta_{2}} .$  

One of the concerns regarding this estimation method is the dependence on the bandwidth which governs how much weight the kernel function should place on an observed point for the estimation at a target point. Moreover, as the call options are not always symmetrically and equally distributed around the ATM point, the choice of the bandwidth is a key issue, especially for estimation at the border of the implied volatility surface. The bandwidth can be chosen global or locally dependent on $ (M,\tau)$. There are methods providing "optimal" bandwidths which rely on plug-in rules or on data-based selectors.

In the case of the volatility surface, it is vital to determine one bandwidth for the maturity and one for the moneyness directions. An algorithm called Empirical-Bias Bandwidth Selector (EBBS) for finding local bandwidths is suggested by Ruppert (1997) and Ruppert et al. (1997). The basic idea of this method is to minimize the estimate of the local mean square error at each target point, without relying on asymptotic result. The variance and the bias term are in this algorithm estimated empirically.

Using the local polynomial estimations, the empirical SPD can be calculated with the following quantlet:


lpspd = 17162 spdbl (m,sigma,sigma1,sigma2,s,r,tau)
estimates the semi-parametric SPD.

The arguments for this quantlet are the moneyness $ ({\tt m})$, $ V$ $ ({\tt sigma})$, $ V'$ $ ({\tt sigma1})$, $ V''$ $ ({\tt sigma2})$, underlying price $ ({\tt s})$ corrected for future dividends, risk-free interest rate $ ({\tt r})$, and the time to maturity $ ({\tt tau})$. The output consist of the local polynomial SPD (lpspd.fstar), $ \Delta$ (lpspd.delta), and the $ \Gamma$ (lpspd.gamma) of the call-options.