5.2 Implied Volatility Surface

Implied volatilities are derived from the BS pricing formula for European options. Recall that European call and put options are derivatives written on an underlying asset $ S$ driven by the price process $ S_t$, which yield the pay-off $ \textrm{max}(S_T-K,0)$ and $ \textrm{max}(K-S_T,0)$ respectively, at a given expiry time $ T$ and for a prespecified strike price $ K$. The difference $ \tau=T-t$ between the day of trade and day of expiration (maturity) is called time to maturity. The pricing formula for call options, Black and Scholes (1973), is:

    $\displaystyle C_t(S_t,K,\tau,r,\sigma)=S_t\Phi(d_1)-Ke^{-r\tau}\Phi(d_2)$  
      (5.1)
    $\displaystyle d_1=\frac{\ln (S_t/K)+(r+1/2\sigma^2)\tau}{\sigma \sqrt{\tau}},\ d_2=d_1 -\sigma \sqrt{\tau},$  

where $ \Phi(\cdot)$ is the cumulative distribution function of the standard normal distribution, $ r$ is the riskless interest rate, and $ \sigma$ is the (unknown and constant) volatility parameter. The put option price $ P_t$ can be obtained from the put-call parity $ P_t=C_t - S_t + e^{-\tau r}K$.

For a European option the implied volatility $ \hat{\sigma}$ is defined as the volatility - $ \sigma$, which yields the BS price $ C_t$ equal to the price $ \tilde{C}_t$ observed on the market. For a single asset, we obtain at each time point $ t$ a two-dimensional function - the IV surface $ \hat{\sigma}_t(K,\tau)$. In order to standardize the volatility functions in time, one may rescale the strike dimension by dividing $ K$ by the future price $ F_t(\tau)$ of the underlying asset with the same maturity. This yields the so-called moneyness $ \kappa=K/F_t(\tau)$. Note that some authors define moneyness simply as $ \kappa=K/S_t$. In contrast to the BS assumptions, empirical studies show that IV surfaces are significantly curved, especially across the strikes. This phenomenon is called a volatility smirk or smile. Smiles stand for U-shaped volatility functions and smirks for decreasing volatility functions.

Figure 5.1: Implied volatility surface of ODAX on May 24, 2001.

We focus on the European options on the German stock index (ODAX). Figure 5.1 displays the ODAX implied volatilities computed from the BS formula (red points) and the IV surface on May 24, 2001 estimated using a local polynomial estimator for $ \tau \in [0,0.6]$ and $ \kappa \in [0.8,1.2]$. We can clearly observe the ``strings" of the original data on maturity grid $ \tau \ \in \{ 0.06111, 0.23611, 0.33333, 0.58611 \}$, which corresponds to $ 22$, $ 85$, $ 120$, and $ 211$ days to maturity. This maturity grid is structured by market conventions and changes over time. The fact that the number of transactions with short maturity is much higher than those with longer maturity is also typical for the IVs observed on the market.

The IV surface is a high-dimensional object - for every time point $ t$ we have to analyze a two-dimensional function. Our goal is to reduce the dimension of this problem and to characterize the IV surface through a small number of factors. These factors can be used in practice for risk management, e.g. with vega-strategies.

The analyzed data, taken from MD*Base, contains EUREX intra-day transaction data for DAX options and DAX futures (FDAX) from January 2 to June 29, 2001. The IVs are calculated by the Newton-Raphson iterative method. The correction of Hafner and Wallmeier (2001) is applied to avoid influence of the tax-scheme in the DAX. For robustness, we exclude the contracts with time to maturity of less than 7 days and maturity strings with less than 100 observations. The approximation of the ``riskless" interest rate with a given maturity is obtained on a daily basis from the linear interpolation of the 1, 3, 6, and 12 month EURIBOR interest rates (obtained from Datastream).

The resulting data set is analyzed using the functional data analysis framework. One advantage of this approach, as we will see later in this chapter, is the possibility of introducing smoothness in the functional sense and using it for regularization. The notation of the functional data analysis is rather complex, therefor the theoretical motivation and the basic notation will be introduced in the next section.