Option Implied Risk-Neutral Density Estimation: A Robust and Flexible Method

Abstract

In practice, a reliable and flexible estimation of risk-neutral density from empirical data is a challenging task since it can not be observed directly from the market. In this study, we apply Bernstein polynomial basis to recover the risk-neutral density function from the observed price quotes of European-type option contingent on an underlying asset. More importantly, we perform an extensive simulation study to examine the flexibility and robustness of the proposed method in recovering different shapes of the true risk-neutral density function from noisy option price quotes. Also, we compare the proposed method with other two popular nonparametric methods namely the constrained local linear polynomial smoothing and the smoothed implied volatility smile reported in the literature. Accuracy and stability of the three nonparametric methods are assessed by the root mean integrated square error criterion. The simulation results show that the proposed method is flexible as it exhibits the various shapes of the true risk-neutral density function even when the volatility is high. Moreover, in comparison with the other two methods, the proposed approach is robust and yields more accurate densities even in the presence of noise. Finally, we demonstrate the applicability of the proposed method in recovering a smooth risk-neutral density function from the S&P 500 market index option data.

Technical Appendix: Alternative Techniques for RND Estimation

Here, we provide an overview of two existing methodologies for the RND estimation, namely the CLLPS and the SML methods.

1.1 Constrained Local Linear Polynomial Smoothing (CLLPS) Method

Aït-Sahalia and Duarte (2003) have proposed a fully nonparametric regression method based on constrained local polynomial kernel smoothing to recover the RND. To recover the RND, the authors have suggested the algorithm that contains the following three steps.

• Step 1 (constrained least squares regression) Constrained least squares regression is used to correct the option prices such that it satisfies a set of constraints in an arbitrary way. For a given sample of observations $\{K_{i},C_{i}\}_{1\le i \le n}$, where $C_{i}$ is the price of the call option with strike $K_{i}$, solve the constrained least squares regression problem

  $$\begin{aligned} \min _{\pmb m \in {\mathbb {R}}^{n}}\sum _{i=1}^{n}(C_{i}-m_{i})^{2}, \end{aligned}$$

  (32)

  subject to:

  $$\begin{aligned}&\frac{m_{i+2} - m_{i+1}}{K_{i+2}-K_{i+1}} \ge \frac{m_{i+1} - m_{i}}{K_{i+1}-K_{i}} ~~\forall ~~i = 1:n-2, \end{aligned}$$

  (33)
  $$\begin{aligned}&-e^{-r\tau } \le \frac{m_{2} - m_{1}}{K_{2}-K_{1}} ~~\text {and}~~ \frac{m_{n} - m_{n-1}}{K_{n-1}-K_{n}} \ge 0, \end{aligned}$$

  (34)

  Suppose, $\pmb m$ is the desired solution vector of (32).

• Step 2 (local linear polynomial Smoothing) To obtain local linear polynomial smoothing estimator ${\hat{m}}(K)$, solve the following local least squares problem

  $$\begin{aligned} \min _{\alpha _{0,1}, \alpha _{1,1}} \sum _{i=1}^{n}\bigg (\mathbf m _{i}-\alpha _{0,1}(K) - \alpha _{1,1}(K)\times (K_{i}-K)\bigg )^{2}K_{h}(K_{i}-K), \end{aligned}$$

  (35)

  where $\alpha _{0,1}$ and $\alpha _{1,1}$ are the unknown coefficients and $K_{h}(K_{i}-K)$ is the kernel function defined as $K_{h}(K_{i}-K) \equiv K(K_{i}-K)/h$. Here h is a smoothing parameter which can control the goodness of fit and smoothness of the estimator. Then ${\hat{m}}(K) ={\hat{\alpha }}_{0,1}(K)$ and its first derivatives ${\hat{m}}^{(1)}(K) \equiv {\hat{m}}_{11} \equiv 1!{\hat{\alpha }}_{1,1}(K)$.

• Step 3 (scaling and shifting process) Apply scaling and shifting process on the ${\hat{m}}^{\prime }_{11}$, where $\prime$ stands for differentiation. Finally, call price function is estimated as

  $$\begin{aligned} C_{{ ASD}}(K) = \int _{K}^{\infty }(S_{T}-K){\hat{m}}^{\prime }_{11}(S_{T})dS_{T}. \end{aligned}$$

  (36)

1.2 Smoothed Implied Volatility Smile (SML) Method

The SML method was first proposed by Shimko (1993) where the author has used a quadratic polynomial to approximate a continuous function for the implied volatility smile across strikes. To take the advantage of smoothing with a spline, Campa et al. (1998) have used a cubic spline. Instead of fitting the implied volatility smile across strike, Malz (1997) has suggested a method with lower order functional as the smoothing function in Black–Scholes option delta (i.e. change in Black–Scholes price with respect to underlying). Bliss and Panigirtzoglou (2002) have combined these two approaches. We write the full algorithm in the following steps followed by Lee (2014):

Step 1:

Convert all observed call prices $C_{i}$ into implied volatilities $\sigma _{K_{i}}$ by inverting the Black–Scholes option pricing formula, i.e.

$$\begin{aligned} \sigma _{K_{i}} = BS^{-1}(C_{i},K_{i},S_{t},t,r,q,T-t). \end{aligned}$$

(37)

Here, $S_{t}$ is the underlying asset with constant risk-free interest rate r and dividend rate q.

Step 2:

Next, calculate Black–Scholes delta $\delta _{i}$, associated with the strike price $K_{i}$, using the following equations

$$\begin{aligned} \sigma _{A}= & {} \frac{(K_{i+1}-S_{t})\sigma _{K_{i}}+(S_{t}-K_{i})\sigma _{K_{i+1}}}{K_{i+1}-K_{i}},~ \text {where}~ S_{t}\in [K_{i},K_{i+1}], \end{aligned}$$

(38)

$$\begin{aligned} \delta _{i}= & {} e^{-q(T-t)}N\bigg (\frac{\ln (\frac{S_{t}}{K_{i}}) + (r - q + \frac{\sigma ^{2}_{A}}{2})(T-t)}{\sigma _{A}\sqrt{T-t}}\bigg ), \end{aligned}$$

(39)

$$\begin{aligned} IV(\delta _{i})= & {} \sigma _{K_{i}}, \end{aligned}$$

(40)

where $\sigma _{A}$ is the at-the-money volatility, $N(\cdot )$ is the standard normal cumulative distribution function, and ${ IV}(\delta _{i})$ stands for the implied volatility associated with $\delta _{i}$.

Step 3:

Approximate the implied volatility smile using smoothing cubic spline functions

$$\begin{aligned} \min _{\varTheta }\bigg \{\lambda \sum _{i=1}^{n}w_{i}({ IV}(\delta _{i})- \tilde{{ IV}}(\delta _{i};\varTheta ))^{2} + (1-\lambda )\int _{0}^{e^{-q(T-t)}} f^{\prime \prime } (\delta ;\varTheta )^{2}d\delta \bigg \}, \end{aligned}$$

(41)

where $\varTheta$ is the matrix of polynomial parameters of the cubic spline and $f^{\prime \prime } (\delta ;\varTheta )$ is the second derivative of the spline function (with respect to delta). $\tilde{IV}(\delta _{i},\varTheta )$ is the fitted implied volatility at $\delta _{i}$, $w_{i}$ is the relative weights to each observation, and $\lambda$ (which can vary between 0 and 1) is the smoothing parameter which control goodness of fit and smoothness of the estimated function.

Step 4:

Convert the estimated implied volatility smile into the option pricing function in price-strike space by using the following steps

$$\begin{aligned} \delta (K)= & {} e^{-q(T-t)}N\left( \frac{\ln \left( \frac{S_{t}}{K}\right) + (r -q + \frac{\sigma ^{2}_{A}}{2})(T-t)}{\sigma _{A}\sqrt{T-t}}\right) , \end{aligned}$$

(42)

$$\begin{aligned} \sigma (K)= & {} \tilde{\textit{IV}}(\delta (K);\varTheta ), \end{aligned}$$

(43)

$$\begin{aligned} C(K)= & {} C_{BS}(K,\sigma (K),S_{t},t,r,q,T-t), \end{aligned}$$

(44)

where $\sigma (K)$ is the fitted implied volatility smile at $\delta (K)$ and C(K) is the fitted call pricing function obtained from the Black–Scholes option pricing formula.

Step 5:

Finally, compute the RND by using an analytical expression which is derived in the Section 2.2 of the work by Bu and Hadri (2007).