9.2 Estimation of the Option Implied SPD

Barle-Cakici's modification of Derman-Kani's Implied Binomial Tree (IBT) yields a proxy for the option implied SPD, $ f^*$, see Chapter 7. XploRe provides quantlets computing Derman-Kani's and Barle-Cakici's IBT's. Since the latter proved to be slightly more robust than the former, Jackwerth (1999), we decide to use Barle-Cakici's IBT to compute the option implied SPD. In the following subsection, we follow closely the notation used in Chapter 7. That is, $ N$ denotes the number of evenly spaced time steps of length $ \Delta t$ in which the tree is divided into (so we have $ N+1$ levels). $ F_{n,i}=e^{r\Delta
t}s_{n,i}$ is the forward price of the underlying, $ s_{n,i}$, at node $ i$ at, level $ n$. Each level $ n$ corresponds to time $ t_{n}=n\Delta t$.


9.2.1 Application to DAX Data

Using the DAX index data from M D *BASE , we estimate the $ 3$ month option implied IBT SPD $ f^*$ by means of the XploRe quantlets 18429 IBTbc and 18432 volsurf and a two week cross section of DAX index option prices for $ 30$ periods beginning in April $ 1997$ and ending in September $ 1999$. We measure time to maturity (TTM) in days and annualize it using the factor $ 360$, giving the annualized time to maturity $ \tau=$TTM$ /360$. For each period, we assume a flat yield curve. We extract from M D *BASE the maturity consistent interest rate.

We describe the procedure in more detail for the first period. First of all, we estimate the implied volatility surface given the two week cross section of DAX option data and utilizing the XploRe quantlet 18451 volsurf which computes the $ 3$ dimensional implied volatility surface (implied volatility over time to maturity and moneyness) using a kernel smoothing procedure. Friday, April $ 18$, $ 1997$ is the $ 3$rd Friday of April $ 1997$. On Monday, April $ 21$, $ 1997$, we estimate the volatility surface, using two weeks of option data from Monday, April $ 7$, $ 1997$, to Friday, April $ 18$, $ 1997$. Following, we start the IBT computation using the DAX price of this Monday, April $ 21$, $ 1997$. The volatility surface is estimated for the moneyness interval $ [0.8,1.2]$ and the time to maturity interval $ [0.0,1.0]$. Following, the XploRe quantlet 18458 IBTbc takes the volatility surface as input and computes the IBT using Barle and Cakici's method. Note that the observed smile enters the IBT via the analytical Black-Scholes pricing formula for a call $ C(F_{n,i}, t_{n+1})$ and for a put $ P(F_{n,i}, t_{n+1})$ which are functions of $ S_{t_1}=s_{1,1}$, $ K=F_{n,i}$, $ r$, $ t_{n+1}$ and $ \sigma_{impl}(F_{n,i},t_{n+1})$. We note, it may happen that at the edge of the tree option prices, with associated strike prices $ F_{n,i}$ and node prices $ s_{n+1,i+1}$, have to be computed for which the moneyness ratio $ s_{n+1,i+1}/F_{n,i}$ is outside the intverall $ [0.8,1.2]$ on which the volatility surface has been estimated. In these cases, we use the volatility at the edge of the surface. Note, as well, that the mean of the IBT SPD is equal to the futures price by construction of the IBT.

Finally, we transform the SPD over $ s_{N+1,i}$ into a SPD over log-returns $ u_{N+1,i}=\ln(s_{N+1,i}/s_{1,1})$ as follows:

$ \textrm{P}(s_{N+1,i} = x)$ $ =$ $ \textrm{P}\Big(\ln\big(\frac{s_{N+1,i}}{s_{1,1}}\big) =
\ln\big(\frac{x}{s_{1,1}}\big)\Big)$ $ =$ $ \textrm{P}\Big(u_{N+1,i} = u\Big)$
<>
where $ u=\ln(x/s_{1,1})$. That is, $ s_{N+1,i}$ has the same probability as $ u_{N+1,i}$. See Figure 9.1 for the SPD computed with parameters $ N=10$ time steps and interest rate $ r=3.23$.

Figure 9.1: Option implied SPD estimated on April $ 21$, $ 1997$, by an IBT with $ N=10$ time steps, $ S_0=3328.41$, $ r=3.23$ and $ \tau =88/360$.
\includegraphics[width=1.5\defpicwidth]{IBTSPDApril1997PS.ps}

A crucial aspect using binomial trees is the choice of the number of time steps $ N$ in which the time interval $ [t,T]$ is divided. In general one can state, the more time steps are used the better is the discrete approximation of the continuous diffusion process and of the SPD. Unfortunately, the bigger $ N$, the more node prices $ s_{n,i}$ possibly have to be overridden in the IBT framework. Thereby we are effectively losing the information about the smile at the corresponding nodes. Therefore, we computed IBT's for different numbers of time steps. We found no hint for convergence of the variables of interest, skewness and kurtosis. Since both variables seemed to fluctuate around a mean, we compute IBT's with time steps $ 10,20,\hdots,100$ and consider the average of these ten values for skewness and kurtosis as the option implied SPD skewness and kurtosis.

Applying this procedure for all $ 30$ periods, beginning in April $ 1997$ and ending in September $ 1999$, we calculate the time series of skewness and kurtosis of the $ 3$ month implied SPD $ f^*$ shown in Figures 9.3 and 9.4. We see that the implied SPD is clearly negatively skewed for all periods but one. In September $ 1999$ it is slightly positively skewed. The pattern is similar for the kurtosis of $ f^*$ which is leptokurtic in all but one period. In October $ 1998$ the density is platykurtic.