7.3 Example - Analysis of DAX data

We now use the IBT to forecast the future price distribution of the real stock market data. We use DAX index option prices data at January 4, 1999, which are included in M D *BASE , a database located at CASE (Center for Applied Statistics and Economics) at Humboldt-Universität zu Berlin, and provide some dataset for demonstration purposes. In the following program, we estimate the BS implied volatility surface first, while the quantlet 15404 volsurf , Fengler, Härdle and Villa (2001), is used to obtain this estimation from the market option prices, then construct the IBT using Derman and Kani method and calculate the interpolated option prices using CRR binomial tree method. Fitting the function of option prices directly from the market option prices is hardly ever attempted since the function approaches a value of zero for very high strike prices and option prices are bounded by non-arbitrage conditions.

15408 XFGIBT05.xpl

Figure 7.11 shows the price distribution estimation obtained by the Barle and Cakici IBT, for $ \tau =0.5$ year. Obviously, the estimated SPD by the Derman and Kani IBT can be obtained similarly.

Figure 7.11: The estimated stock price distribution, $ \tau =0.5$ year.
\includegraphics[width=1.1\defpicwidth]{realdk.ps}

In order to check the precision of the estimated price distribution obtained by the IBT method, we compare it to use DAX daily prices between January 1, 1997, and January 4. 1999. The historical time series density estimation method described in Aït-Sahalia, Wang and Yared (2000) is used here. Notice that Risk-neutrality implies two kinds of SPD should be equal, historical time series SPD is in fact the conditional density function of the diffusion process. We obtain the historical time series SPD estimation by the following procedure:
1.
Collect stock prices time series
2.
Assume this time series is a sample path of the diffusion process

$\displaystyle \frac{dS_{t}}{S_{t}}=\mu_{t} dt+\sigma(S_{t},t)dZ_{t},$

where $ dZ_t$ is a Wiener process with mean zero and variance equal to $ dt$.
3.
Estimate diffusion function $ \sigma(\cdot, \cdot)$ in the diffusion process model using nonparametric method from stock prices time series
4.
Make Monte-Carlo simulation for the diffusion process with drift function is interest rate and estimated diffusion function
5.
Estimate conditional density function $ g=p(S_T \vert S_t, \hat{\mu}, \hat{\sigma})$ from Monte-Carlo simulated process

Figure 7.12: SPD estimation by three methods, by historical estimation, and its $ 95\%$ confidence band (dashed), by B & C IBT, and by D & K IBT (thin), $ \tau =0.5$ year.
\includegraphics[width=1.1\defpicwidth]{realcom.ps}

From Figure 7.12 we conclude that the SPD estimated by the Derman and Kani IBT and the one obtained by Barle and Cakici IBT can be used to forecast future SPD. The SPD estimated by different methods sometimes have deviations on skewness and kurtosis. In fact the detection of the difference between the historical time series SPD estimation and the SPD recovered from daily option prices may be used as trading rules, see Table 7.1 and Chapter 9. In Table 7.1, SPD estimated from daily option prices data set is expressed by $ f$ and the time series SPD is $ g$. A far out of the money (OTM) call/put is defined as one whose exercise price is $ 10\%$ higher (lower) than the future price. While a near OTM call/put is defined as one whose exercise price is $ 5\%$ higher (lower) but $ 10\%$ lower(higher)than the future price. When skew($ f$) $ <$ skew($ g$), agents apparently assign a lower probability to high outcomes of the underlying than would be justified by the time series SPD (see Figure 7.13). Since for call options only the right `tail' of the support determines the theoretical price the latter is smaller than the price implied by diffusion process using the time series SPD. That is we buy calls. The same reason applies to put options.


Table 7.1: Trading Rules to exploit SPD differences.
Trading Rules to exploit SPD differences
Skewness ($ S1$) skew($ f$)$ <$ skew($ g$) sell OTM put,
      buy OTM call
Trade ($ S2$) skew($ f$) $ >$ skew($ g$) buy OTM put
      sell OTM call
Kurtosis ($ K1$) kurt($ f$)$ >$ kurt($ g$) sell far OTM and ATM
      buy near OTM options
Trade ($ K2$) kurt($ f$) $ <$ kurt($ g$) buy far OTM and ATM,
      sell near OTM options


Figure 7.13: Skewness Trade, skew($ f$)$ <$ skew($ g$).
\includegraphics[width=1.1\defpicwidth]{skewness.ps}

Figure 7.14: Kurtosis Trade, kurt($ f$)$ >$ kurt($ g$).
\includegraphics[width=1.1\defpicwidth]{kurtosis.ps}

From the simulations and real data example, we find that the implied binomial tree is an easy way to assess the future stock prices, capture the term structure of the underlying asset, and replicate the volatility smile. But the algorithms still have some deficiencies. When the time step is chosen too small, negative transition probabilities are encountered more and more often. The modification of these values loses the information about the smile at the corresponding nodes. The Barle and Cakici algorithm is a better choice when the interest rate is high.Figure 7.15 shows the deviation of the two methods under the situation that $ r=0.2$. When the interest rate is a little higher, Barle and Cakici algorithm still can be used to construct the IBT while Derman and Kani's cannot work any more. The times of the negative probabilities appear are fewer than Derman and Kani construction (see Jackwerth (1999)).

Figure 7.15: SPD estimation by Monte-Carlo simulation, and its $ 95\%$ confidence band (dashed), the B & C IBT, from the D & K IBT (thin), level =20, $ \tau =1$ year, $ r=0.20$.
\includegraphics[width=1.1\defpicwidth]{hirspd.ps}

Besides its basic purpose of pricing derivatives in consistency with the market prices, IBT is useful for other kinds of analysis, such as hedging and calculating of implied probability distributions and volatility surfaces. It estimate the future price distribution according to the historical data. On the practical application aspect, the reliability of the approach depends critically on the quality of the estimation of the dynamics of the underlying price process, such as BS implied volatility surface obtained from the market option prices.

The IBT can be used to produce recombining and arbitrage-free binomial trees to describe stochastic processes with variable volatility. However, some serious limitations such as negative probabilities, even though most of them appeared at the edge of the trees. Overriding them causes loss of the information about the smile at the corresponding nodes. These defects are a consequence of the requirement that a continuous diffusion is approximated by a binomial process. Relaxation of this requirement, using multinomial trees or varinomial trees is possible.